# Prism Physics

> Free interactive physics lessons for JEE, NEET and first-year university students. Every concept is explained in plain language, animated, and paired with draggable graphs, worked examples, quizzes and hand-picked videos.

Each lesson explains one idea in plain language, derives its formulas, and includes interactive graphs, step-by-step worked examples, a short quiz and links to further videos, simulations and readings. Chapters follow the standard sequence of a first course in physics: mechanics, waves and sound, light and optics, heat and thermodynamics, electricity, magnetism and electromagnetism, and modern physics (atoms, X-rays, semiconductors, the nucleus and special relativity).

- [All chapters](https://physics.sillybits.io/chapters): course overview
- [About](https://physics.sillybits.io/about): who writes the lessons and how
- [Full text for LLMs](https://physics.sillybits.io/llms-full.txt): every lesson's description and key points in one file

## Chapter 1: Introduction to Physics

What physics is and how it uses mathematics, units and the SI system with its prefixes and modern definitions, dimensions and dimensional formulas, using dimensions to check equations, convert units and guess formulas, orders of magnitude and estimation, and the structure of matter.

- [What Is Physics? Nature's Rules, Mathematics and the Structure of Matter](https://physics.sillybits.io/chapters/introduction-to-physics/what-is-physics): Physics as the search for the basic rules behind natural events, Feynman's chess analogy, why mathematics is the language of physics but not physics itself, and how matter is built from atoms, nuclei and elementary particles.
- [Units, SI Base Units, Derived Units and SI Prefixes](https://physics.sillybits.io/chapters/introduction-to-physics/units-and-si): Why measurements need standard units, who decides them (CGPM), fundamental and derived quantities, the seven SI base units, the radian and steradian, SI prefixes from atto to exa, and the 2019 definitions based on fixed constants of nature.
- [Dimensions and Dimensional Formulas of Physical Quantities](https://physics.sillybits.io/chapters/introduction-to-physics/dimensions): What the dimensions of a physical quantity are, how to write a dimensional formula such as [F] = MLT⁻², and the dimensions of common mechanical quantities.
- [Uses of Dimensional Analysis: Checking Equations, Converting Units, Deriving Formulas](https://physics.sillybits.io/chapters/introduction-to-physics/uses-of-dimensions): The principle of homogeneity for checking equations, converting units between systems (SI ↔ CGS), deducing relations such as the pendulum period, and the limitations of the method.
- [Order of Magnitude, Powers of Ten and Fermi Estimates](https://physics.sillybits.io/chapters/introduction-to-physics/order-of-magnitude): Writing numbers as a × 10ᵇ, the order of magnitude of a quantity, comparing sizes across the universe, and quick order-of-magnitude estimates.

## Chapter 2: The Mathematical Toolkit

Before we can talk about motion, forces or fields we need a vocabulary. This chapter builds it: arrows that add in a special way, the calculus of change and accumulation, and the discipline of reporting a measurement truthfully.

- [Scalars and Vectors Explained with Interactive Diagrams](https://physics.sillybits.io/chapters/math-toolkit/scalars-and-vectors): What makes a quantity a vector rather than a scalar, why direction changes how quantities add, the triangle rule, and when two vectors are equal.
- [Vector Addition: Triangle Rule, Parallelogram Rule and Resultant Formula](https://physics.sillybits.io/chapters/math-toolkit/adding-vectors): Add vectors with the triangle and parallelogram rules, derive the resultant formula √(a² + b² + 2ab cos θ) and its direction, and see why equal vectors bisect the angle.
- [Multiplying a Vector by a Number, Unit Vectors and Vector Subtraction](https://physics.sillybits.io/chapters/math-toolkit/scaling-and-subtracting): How multiplying by a number stretches or reverses a vector, what a unit vector is, and how to subtract vectors by adding the reverse, with the magnitude of a − b.
- [Resolving Vectors into Components with i, j, k Unit Vectors](https://physics.sillybits.io/chapters/math-toolkit/components): Resolve a vector into x, y and z components using projections and direction cosines, write it with î, ĵ, k̂, and add vectors component by component.
- [Dot Product (Scalar Product) of Vectors with Interactive Projection](https://physics.sillybits.io/chapters/math-toolkit/dot-product): The scalar (dot) product a·b = ab cos θ as length times projection, its algebra, the component formula aₓbₓ + a_yb_y + a_zb_z, and its use for work and angles.
- [Cross Product (Vector Product) and the Right-Hand Rule in 3D](https://physics.sillybits.io/chapters/math-toolkit/cross-product): The vector (cross) product: magnitude ab sin θ as parallelogram area, direction by the right-hand rule, anti-commutativity, right-handed axes and the component formula, in 3D.
- [Derivatives as Rate of Change: Secant to Tangent, Visualised](https://physics.sillybits.io/chapters/math-toolkit/derivatives): The derivative dy/dx as the limit of Δy/Δx and the slope of a tangent, the derivatives of common functions, and the sum and product rules, with animated secant lines.
- [Maxima and Minima Using Derivatives](https://physics.sillybits.io/chapters/math-toolkit/maxima-and-minima): Find maxima and minima by setting dy/dx = 0 and use the second derivative to tell a peak from a valley, with a ball thrown upward and an optimisation problem.
- [Integration as Area Under a Curve: Riemann Sums and Antiderivatives](https://physics.sillybits.io/chapters/math-toolkit/integration): The definite integral as the limit of a Riemann sum, the area under y = x, antiderivatives and the fundamental theorem of calculus, with a table of standard integrals.
- [Significant Figures and Rounding Rules in Physics Measurements](https://physics.sillybits.io/chapters/math-toolkit/significant-figures): Significant digits from a scale's least count, which zeros count, the rules for multiplying, dividing, adding and subtracting measured values, and rounding an exact 5.
- [Errors in Measurement, Mean and Standard Deviation](https://physics.sillybits.io/chapters/math-toolkit/errors): Systematic versus random errors, why averaging helps, the standard deviation σ, and how to report a result as mean ± σ with 68%, 95% and 99.7% intervals.

## Chapter 3: Describing Motion

Motion is a relationship between an object and an observer. Using the toolkit from the previous chapter we describe position, velocity and acceleration in a line, in a plane, through the air, and from moving frames.

- [Rest, Motion and Frames of Reference](https://physics.sillybits.io/chapters/describing-motion/rest-and-motion): Why rest and motion are always relative to an observer, what a frame of reference is, and how to choose a convenient one, shown with an animated train and platform.
- [Distance vs Displacement: Scalar and Vector](https://physics.sillybits.io/chapters/describing-motion/distance-and-displacement): The difference between distance (path length, a scalar) and displacement (the straight arrow from start to finish, a vector), with a path you can draw and a circular track.
- [Average Speed and Instantaneous Speed from Graphs](https://physics.sillybits.io/chapters/describing-motion/speed): Average speed over an interval, instantaneous speed ds/dt at a moment, reading both from a distance–time graph, and finding distance as the area under a speed–time graph.
- [Velocity, Position Vectors and Average Velocity](https://physics.sillybits.io/chapters/describing-motion/velocity): Position vectors, average velocity (r₂ − r₁)/(t₂ − t₁), instantaneous velocity dr/dt tangent to the path, and how average velocity can differ sharply from average speed.
- [Acceleration, Deceleration and Signs in 1D Motion](https://physics.sillybits.io/chapters/describing-motion/acceleration): Average and instantaneous acceleration a = dv/dt, acceleration from turning at constant speed, and how the signs of v and a decide whether a body speeds up or slows down.
- [Equations of Motion for Constant Acceleration (v = u + at, s = ut + ½at²)](https://physics.sillybits.io/chapters/describing-motion/constant-acceleration): Derive v = u + at, x = ut + ½at² and v² = u² + 2ax by integration, use them with signs, and avoid confusing position with distance, with a synchronised x–t, v–t, a–t lab.
- [Free Fall and Motion Under Gravity](https://physics.sillybits.io/chapters/describing-motion/free-fall): Free fall as constant acceleration g ≈ 9.8 m/s² downward, sign conventions, maximum height u²/2g and time of rise u/g, with a strobe-photo simulator.
- [Motion in a Plane with Constant Acceleration](https://physics.sillybits.io/chapters/describing-motion/motion-in-a-plane): Treat two-dimensional motion as two independent straight-line motions along x and y that share one clock, resolve u and a into components, then recombine.
- [Projectile Motion: Time of Flight, Range and Maximum Height](https://physics.sillybits.io/chapters/describing-motion/projectile-motion): Projectile motion as constant horizontal velocity plus free fall: time of flight 2u sin θ/g, range u² sin 2θ/g, maximum height u² sin²θ/2g, the 45° rule and the parabolic path.
- [Relative Velocity: River Crossing, Rain and Changing Frames](https://physics.sillybits.io/chapters/describing-motion/relative-motion): Transform position, velocity and acceleration between frames, find relative velocity v₁ − v₂, and solve river-crossing and rain-umbrella problems interactively.

## Chapter 4: The Forces of Nature

What a force is, who exerts it on whom, and where forces come from. We meet gravity, the electromagnetic force hiding inside every contact, tension and spring, the nuclear and weak forces, and the boundaries where classical physics stops working.

- [What Is a Force? Interactions, Newtons and the Four Fundamental Forces](https://physics.sillybits.io/chapters/the-forces/what-is-a-force): Force as an interaction between two bodies, the newton as a unit, forces as vectors, the action–reaction pairs of Newton's third law, and the four fundamental forces of nature.
- [Newton's Law of Gravitation and the Acceleration Due to Gravity](https://physics.sillybits.io/chapters/the-forces/gravitational-force): Newton's law of gravitation F = Gm₁m₂/r², treating spheres as points, how g = GM/R² arises at Earth's surface, and why gravity between everyday objects is so tiny.
- [Electromagnetic Force: Coulomb's Law, Normal Force, Tension and Spring Force](https://physics.sillybits.io/chapters/the-forces/electromagnetic-forces): Coulomb's law between charges, and how electromagnetic forces between atoms appear as normal force, friction, tension in strings and the spring force F = k|Δx|.
- [Nuclear and Weak Forces, and Where Classical Physics Applies](https://physics.sillybits.io/chapters/the-forces/nuclear-weak-and-classical-limits): The strong nuclear force that holds nuclei together, the weak force behind beta decay, and the size and speed limits within which classical (Newtonian) physics works.

## Chapter 5: Newton's Laws of Motion

Newton's laws connect the forces on a body to how it moves. We learn when they hold (inertial frames), how to apply them reliably with free-body diagrams, how tension behaves, how to rescue them in accelerating frames with pseudo forces, and how to untangle the famous horse-and-cart puzzle.

- [Newton's First Law of Motion and Inertial Frames of Reference](https://physics.sillybits.io/chapters/newtons-laws/first-law-and-inertial-frames): Newton's first law (a = 0 if and only if the net force is zero), why it fails in a falling lift, what makes a frame inertial, and why uniformly moving frames are inertial too.
- [Newton's Second Law of Motion: F = ma](https://physics.sillybits.io/chapters/newtons-laws/second-law): Newton's second law a = F/m measured from an inertial frame, its instantaneous and vector nature, and how to use it with the kinematic equations to find forces and motion.
- [How to Draw Free-Body Diagrams and Apply Newton's Laws](https://physics.sillybits.io/chapters/newtons-laws/free-body-diagrams): A reliable four-step method for Newton's-law problems: choose the system, list the forces on it by other bodies, draw the free-body diagram, then pick axes and write ΣF = ma.
- [Newton's Third Law, Action–Reaction Pairs and Tension in Strings](https://physics.sillybits.io/chapters/newtons-laws/third-law-and-tension): Newton's third law and why action–reaction forces never cancel, what tension means at a point of a string, why a light string has the same tension everywhere, and the Atwood machine.
- [Pseudo Forces in Non-Inertial (Accelerating) Frames](https://physics.sillybits.io/chapters/newtons-laws/pseudo-forces): How to use Newton's second law from an accelerating frame by adding the pseudo force −ma₀, with a pendulum in an accelerating car and the apparent weight in a lift.
- [The Horse and Cart Paradox and the Meaning of Inertia](https://physics.sillybits.io/chapters/newtons-laws/horse-and-cart): Resolve the horse-and-cart paradox by choosing a system and listing every force on it, including friction from the ground, and understand inertia as resistance to acceleration measured by mass.

## Chapter 6: Friction

Friction is the sideways part of the contact force between surfaces. We separate kinetic from static friction, learn the laws they follow, see why friction can drive motion as well as resist it, and measure friction coefficients with a pulley or a tilting plank.

- [Friction as a Component of the Contact Force](https://physics.sillybits.io/chapters/friction/contact-force-and-friction): Split the contact force between two surfaces into the normal force and friction, find its size and angle, and distinguish kinetic friction (slipping) from static friction (no slipping).
- [Kinetic Friction: Direction, Magnitude and Coefficient](https://physics.sillybits.io/chapters/friction/kinetic-friction): The direction of kinetic friction (opposite to velocity relative to the other surface), its magnitude fₖ = μₖN, and why it hardly depends on speed or contact area.
- [Static Friction, Limiting Friction and Its Direction](https://physics.sillybits.io/chapters/friction/static-friction): Static friction that adjusts itself up to the limiting value μₛN, how to find its direction, and why friction often causes motion rather than opposing it.
- [Laws of Friction, Friction Coefficients and the Atomic Origin of Friction](https://physics.sillybits.io/chapters/friction/laws-of-friction): The four laws of friction, typical friction coefficients for common material pairs, and how microscopic contact points and molecular bonds produce friction and heat.
- [Measuring the Coefficient of Friction: Inclined Plane and Angle of Repose](https://physics.sillybits.io/chapters/friction/measuring-friction): Measure static and kinetic friction coefficients with a pulley-and-hanger table and with an inclined plane (μ = tan θ), and see why rolling friction is so much smaller than sliding.

## Chapter 7: Circular Motion

Angular position, velocity and acceleration; why every circling body accelerates toward the centre; the force that motion demands; how roads are banked; and what changes when we watch from a spinning frame, including the Earth itself.

- [Angular Velocity and Angular Acceleration in Circular Motion](https://physics.sillybits.io/chapters/circular-motion/angular-variables): Angular position, angular velocity ω = dθ/dt and angular acceleration α = dω/dt, the constant-α equations, and the links v = rω and aₜ = rα between angular and linear quantities.
- [Centripetal Acceleration: Why Circular Motion Needs a = v²/r](https://physics.sillybits.io/chapters/circular-motion/centripetal-acceleration): Derive the acceleration of a particle on a circle using radial and tangential unit vectors: v²/r toward the centre for uniform motion, plus dv/dt along the tangent when the speed changes.
- [Centripetal Force: F = mv²/r and Where It Comes From](https://physics.sillybits.io/chapters/circular-motion/centripetal-force): Newton's second law for circular motion, F = mv²/r toward the centre, and why 'centripetal force' is always supplied by a real force such as tension, friction, gravity or a normal force.
- [Car on a Circular Turn: Friction and Banking of Roads (tan θ = v²/rg)](https://physics.sillybits.io/chapters/circular-motion/turns-and-banking): How static friction provides the centripetal force on a level turn (μₛ ≥ v²/rg), and why roads are banked at tan θ = v²/rg so a car can turn without relying on friction.
- [Centrifugal Force, Rotating Frames and Earth's Rotation (Apparent g)](https://physics.sillybits.io/chapters/circular-motion/centrifugal-force): The centrifugal pseudo force mω²r needed in uniformly rotating frames, the Coriolis force, and how the Earth's rotation reduces apparent weight and tilts plumb lines.

## Chapter 8: Work & Energy

Kinetic energy and the work that changes it; how to calculate work for constant, spring and perpendicular forces; power; conservative forces and potential energy; and the conservation of mechanical energy, up to E = mc².

- [Work–Energy Theorem: Work Done and Kinetic Energy](https://physics.sillybits.io/chapters/work-and-energy/work-and-kinetic-energy): Kinetic energy ½mv², the work done by a force W = ∫F·dr, and the work–energy theorem: the total work done on a particle equals its change in kinetic energy.
- [Work Done by Constant Force, Gravity (mgh) and a Spring (½kx²)](https://physics.sillybits.io/chapters/work-and-energy/calculating-work): Work done by a constant force (Fd cos θ), by gravity (mgh, independent of path), by a spring (−½kx² from natural length) and by forces perpendicular to motion (zero), with work as area under F–x graphs.
- [Power in Physics: P = F·v, Watts and Horsepower](https://physics.sillybits.io/chapters/work-and-energy/power): Power as the rate of doing work, P = dW/dt = F·v, its SI unit the watt, the horsepower, and average versus instantaneous power, with lifts, inclines and engines.
- [Conservative and Non-Conservative Forces and Potential Energy](https://physics.sillybits.io/chapters/work-and-energy/potential-energy): Work done by internal forces in a system, the difference between conservative forces (gravity, springs, electric) and non-conservative ones (friction), and the definition ΔU = −W of potential energy.
- [Conservation of Mechanical Energy: Gravitational and Spring Potential Energy](https://physics.sillybits.io/chapters/work-and-energy/conservation-of-energy): Conservation of mechanical energy K + U, gravitational potential energy mgh, spring energy ½kx², the work of external and non-conservative forces, other forms of energy and E = mc².

## Chapter 9: Centre of Mass, Momentum & Collisions

The centre of mass of particles and solid bodies, how it moves under external forces alone, linear momentum and its conservation, rocket propulsion, elastic and inelastic collisions, the coefficient of restitution, and impulse.

- [Centre of Mass of a System of Particles](https://physics.sillybits.io/chapters/momentum-and-collisions/centre-of-mass): The centre of mass R = Σmᵢrᵢ / M of a set of particles, the two-particle inverse-mass rule, and combining parts by treating each as a point at its own centre of mass.
- [Centre of Mass of a Rod, Semicircular Ring and Semicircular Disc by Integration](https://physics.sillybits.io/chapters/momentum-and-collisions/centre-of-mass-of-bodies): Find the centre of mass of continuous bodies by integration: a uniform rod (midpoint), a semicircular wire (2R/π) and a semicircular plate (4R/3π), using symmetry and smart slicing.
- [Motion of the Centre of Mass: Ma(cm) = F(external)](https://physics.sillybits.io/chapters/momentum-and-collisions/motion-of-centre-of-mass): Show that M a(cm) = F(external): internal forces cancel in pairs, so the centre of mass moves like a single particle under the external forces alone, even when the body explodes.
- [Conservation of Linear Momentum and Recoil](https://physics.sillybits.io/chapters/momentum-and-collisions/momentum-conservation): Linear momentum p = mv, the system momentum P = Mv(cm), the principle of conservation of momentum when external forces cancel, and recoil in guns and radioactive decay.
- [Rocket Propulsion and the Rocket Equation v = u ln(M₀/M)](https://physics.sillybits.io/chapters/momentum-and-collisions/rocket-propulsion): Derive the acceleration and speed of a rocket from momentum conservation: dv/dt = ru/M and v = u ln(M₀/M), and why rockets speed up faster as fuel burns.
- [Elastic and Inelastic Collisions and the Coefficient of Restitution](https://physics.sillybits.io/chapters/momentum-and-collisions/collisions): What happens during a collision (modelled with a spring), why compression peaks when velocities match, the types of collision, perfectly inelastic collisions and their energy loss, and the coefficient of restitution e.
- [Elastic Collision Formulas in One and Two Dimensions](https://physics.sillybits.io/chapters/momentum-and-collisions/elastic-collisions): Final velocities in a one-dimensional elastic collision, the rule 'separation speed = approach speed', the special cases of equal, very heavy and very light masses, and glancing collisions in two dimensions.
- [Impulse and Impulsive Force: J = ∫F dt = Δp](https://physics.sillybits.io/chapters/momentum-and-collisions/impulse): Impulsive forces in collisions, impulse J = ∫F dt as the area under a force–time graph, and why it equals the change in momentum, with airbags, catching and follow-through.

## Chapter 10: Rotational Mechanics

Rotation of rigid bodies about a fixed axis: angular kinematics, torque, τ = Iα, equilibrium, angular momentum and its conservation, rotational kinetic energy and power, moments of inertia, the parallel and perpendicular axis theorems, and rolling.

- [Rigid Body Rotation About a Fixed Axis: θ, ω, α and v = rω](https://physics.sillybits.io/chapters/rotational-mechanics/rigid-body-rotation): What it means for a rigid body to rotate about a fixed axis, angular position, velocity and acceleration of the whole body, the constant-α equations, and v = rω, aₜ = rα for each particle.
- [Torque: τ = r × F, Lever Arm and Moment of a Force](https://physics.sillybits.io/chapters/rotational-mechanics/torque): Torque τ = r × F of a force about a point and about an axis, the lever arm (moment arm), when a force has zero torque, and why net force and net torque are different things.
- [Newton's Second Law for Rotation: τ = Iα and Moment of Inertia](https://physics.sillybits.io/chapters/rotational-mechanics/torque-and-angular-acceleration): Derive τ(ext) = Iα for a rigid body about a fixed axis, meet the moment of inertia I = Σmr², and see why it depends on how mass is spread around the axis.
- [Equilibrium of Rigid Bodies, Toppling and the Leaning Cyclist (tan θ = v²/rg)](https://physics.sillybits.io/chapters/rotational-mechanics/equilibrium-of-rigid-bodies): Conditions for equilibrium of a rigid body (zero net force and zero net torque), solving plank and support problems, toppling and stable/unstable/neutral equilibrium, and why a cyclist leans on a turn.
- [Angular Momentum L = Iω, dL/dt = τ and Conservation of Angular Momentum](https://physics.sillybits.io/chapters/rotational-mechanics/angular-momentum): Angular momentum of a particle (l = r × p) and of a rigid body (L = Iω), the law dL/dt = τ(ext), conservation of angular momentum when external torque vanishes, and angular impulse.
- [Rotational Kinetic Energy ½Iω², Work by a Torque and Power P = τω](https://physics.sillybits.io/chapters/rotational-mechanics/rotational-kinetic-energy): Rotational kinetic energy K = ½Iω² as the sum of ½mv² over a spinning body, energy conservation with wheels and pulleys, the work ∫τ dθ done by a torque and the power P = τω.
- [Moment of Inertia of Rod, Ring, Disc, Cylinder and Sphere (with Derivations)](https://physics.sillybits.io/chapters/rotational-mechanics/moment-of-inertia): Moments of inertia of point-mass systems and, by integration, of a uniform rod, rectangular plate, ring, disc, hollow and solid cylinder, and hollow and solid sphere.
- [Parallel Axis Theorem, Perpendicular Axis Theorem and Radius of Gyration](https://physics.sillybits.io/chapters/rotational-mechanics/parallel-and-perpendicular-axes): The parallel axis theorem I = I(cm) + Md², the perpendicular axis theorem Iz = Ix + Iy for flat bodies, and the radius of gyration k defined by I = Mk².
- [Rolling Without Slipping: v = Rω, Kinetic Energy of Rolling and the Rolling Race](https://physics.sillybits.io/chapters/rotational-mechanics/rolling): Combined rotation and translation: a(cm) = F/M with τ(cm) = I(cm)α, pure rolling v = Rω, slipping, kinetic energy ½I(cm)ω² + ½Mv², angular momentum of a rolling body, and rolling resistance.

## Chapter 11: Gravitation

Newton's law of universal gravitation, the Cavendish experiment, gravitational potential energy and potential, the gravitational field of rings, shells and spheres, how g varies, satellites and Kepler's laws, weightlessness, escape velocity and black holes.

- [Newton's Law of Universal Gravitation and the Moon–Apple Argument](https://physics.sillybits.io/chapters/gravitation/universal-law-of-gravitation): Kepler's three laws, Newton's comparison of the Moon's acceleration with an apple's (a 3600 : 1 ratio at 60 times the distance), and the law F = Gm₁m₂/r² with G = 6.67 × 10⁻¹¹ N·m²/kg².
- [The Cavendish Experiment: How G Was Measured (Torsion Balance)](https://physics.sillybits.io/chapters/gravitation/measuring-big-g): How the Cavendish torsion balance measures G: gravitational torque balanced by the twist of a fibre, the optical lever that doubles the angle, and how G gives the mass of the Earth.
- [Gravitational Potential Energy U = −Gm₁m₂/r of Particles and Systems](https://physics.sillybits.io/chapters/gravitation/gravitational-potential-energy): Derive U = −Gm₁m₂/r with zero at infinite separation, add the energies of all pairs in a system, and connect it with mgh for small heights.
- [Gravitational Potential of a Point Mass, Ring, Spherical Shell and Solid Sphere](https://physics.sillybits.io/chapters/gravitation/gravitational-potential): Gravitational potential V = U/m, with V = −GM/r for a point, −GM/√(a² + r²) on the axis of a ring, constant inside a shell, and −GM(3a² − r²)/2a³ inside a solid sphere.
- [Gravitational Field Intensity: Point Mass, Ring, Disc, Shell and Sphere; E = −dV/dr](https://physics.sillybits.io/chapters/gravitation/gravitational-field): The gravitational field E = F/m, its link to potential (E = −dV/dr), and the field of a point mass, a ring, a disc, a thin shell (zero inside) and a solid sphere (proportional to r inside).
- [Variation of g with Height, Depth, Latitude and Earth's Shape](https://physics.sillybits.io/chapters/gravitation/variation-of-g): How the acceleration due to gravity changes with height (≈ g(1 − 2h/R)), with depth (g(1 − d/R)), with latitude because the Earth spins, and with the Earth's shape and uneven density; and how g weighs the Earth.
- [Orbital Speed, Period, Energy, Geostationary Satellites, Kepler's Laws and Weightlessness](https://physics.sillybits.io/chapters/gravitation/satellites-and-keplers-laws): Orbital speed √(GM/a), period 2π√(a³/GM), total energy −GMm/2a, geostationary orbits, Kepler's three laws, and why astronauts feel weightless though gravity still acts.
- [Escape Velocity √(2GM/R), Binding Energy, Black Holes and Inertial vs Gravitational Mass](https://physics.sillybits.io/chapters/gravitation/escape-velocity-and-black-holes): Escape velocity √(2GM/R) from energy conservation, gravitational binding energy, black holes (escape speed ≥ c), the equivalence of inertial and gravitational mass, and tests of the inverse-square law.

## Chapter 12: Simple Harmonic Motion

Restoring forces and simple harmonic motion: the equation x = A sin(ωt + δ), period, frequency and phase, the reference circle, energy, angular SHM, simple, physical and torsional pendulums, combining oscillations, and damped and forced oscillations with resonance.

- [Simple Harmonic Motion: Restoring Force F = −kx and a = −ω²x](https://physics.sillybits.io/chapters/simple-harmonic-motion/what-is-shm): Periodic and oscillatory motion, the definition of SHM (a = −ω²x, or F = −kx), the restoring force, the spring–block system and why it swings equally far on both sides.
- [x = A sin(ωt + δ): Amplitude, Period T = 2π√(m/k), Frequency and Phase](https://physics.sillybits.io/chapters/simple-harmonic-motion/equation-of-shm): Solve a = −ω²x to get v = ω√(A² − x²) and x = A sin(ωt + δ); meet amplitude, period T = 2π√(m/k), frequency, angular frequency, phase and phase constant.
- [SHM as the Projection of Uniform Circular Motion (Reference Circle)](https://physics.sillybits.io/chapters/simple-harmonic-motion/shm-and-circular-motion): The projection of uniform circular motion on any diameter is simple harmonic motion: the reference circle explains x = A cos ωt, the meaning of ω and the phase.
- [Energy in Simple Harmonic Motion: U = ½kx², K and Total E = ½mω²A²](https://physics.sillybits.io/chapters/simple-harmonic-motion/energy-in-shm): Potential energy ½kx² for the linear restoring force, kinetic energy ½mv², and the constant total ½mω²A², with energy swapping between the two forms during each oscillation.
- [Simple Pendulum: T = 2π√(l/g), Small-Angle Approximation and Measuring g](https://physics.sillybits.io/chapters/simple-harmonic-motion/simple-pendulum): Angular SHM and the simple pendulum: the restoring torque mgl sin θ, the small-angle approximation, T = 2π√(l/g), how large amplitudes lengthen the period, and measuring g in the lab.
- [Physical Pendulum T = 2π√(I/mgl) and Torsional Pendulum T = 2π√(I/k)](https://physics.sillybits.io/chapters/simple-harmonic-motion/physical-and-torsional-pendulums): The physical (compound) pendulum, T = 2π√(I/mgl) with I about the pivot, and the torsional pendulum, T = 2π√(I/k) for a body twisting on a wire.
- [Superposition of SHMs: Phasor Addition and Lissajous Figures](https://physics.sillybits.io/chapters/simple-harmonic-motion/combining-oscillations): Two SHMs of the same frequency along one line combine into one SHM with amplitude √(A₁² + A₂² + 2A₁A₂cos δ) (phasor addition); at right angles they trace lines, ellipses and circles.
- [Damped Oscillations, Forced Oscillations and Resonance (Tacoma Narrows)](https://physics.sillybits.io/chapters/simple-harmonic-motion/damped-and-forced-oscillations): Damping force −bv and exponentially decaying amplitude A₀e^(−bt/2m), critical damping, forced oscillations under F₀ sin ωt, the resonance curve, and the Tacoma Narrows Bridge.

## Chapter 13: Fluid Mechanics

Pressure in fluids and how it grows with depth, Pascal's law and hydraulic machines, atmospheric pressure, barometers and manometers, Archimedes' principle and floating, fluids in accelerating containers, steady flow and the equation of continuity, Bernoulli's equation and its applications.

- [Pressure in a Fluid: P = P₀ + ρgh and Why It Grows with Depth](https://physics.sillybits.io/chapters/fluid-mechanics/pressure-in-fluids): What a fluid is, the ideal (incompressible, nonviscous) liquid, pressure as force per area at a point, dP = −ρg dz and P = P₀ + ρgh, and why points at the same level have the same pressure.
- [Pascal's Law and the Hydraulic Lift (F₂/F₁ = A₂/A₁)](https://physics.sillybits.io/chapters/fluid-mechanics/pascals-law): Pascal's law (a pressure change is transmitted undiminished throughout a liquid), the hydraulic lift and press, and why they multiply force but not work.
- [Atmospheric Pressure, the Mercury Barometer and the Manometer (Gauge Pressure)](https://physics.sillybits.io/chapters/fluid-mechanics/atmospheric-pressure): Atmospheric pressure, Torricelli's mercury barometer (P₀ = ρgH, 76 cm of mercury ≈ 1.01 × 10⁵ Pa = 1 atm), the U-tube manometer and gauge pressure.
- [Archimedes' Principle, Buoyant Force and Floatation](https://physics.sillybits.io/chapters/fluid-mechanics/archimedes-principle): The buoyant force as the resultant of pressure forces, a proof of Archimedes' principle by replacing the body with fluid, apparent weight, and the condition for floating.
- [Pressure and Buoyancy in Accelerating Fluids: Lifts and Tilting Surfaces (tan θ = a/g)](https://physics.sillybits.io/chapters/fluid-mechanics/accelerating-fluids): Pressure difference and buoyant force in a liquid accelerating vertically (g replaced by g + a₀), and the tilted free surface of a liquid accelerating horizontally, tan θ = a₀/g.
- [Steady and Turbulent Flow, Streamlines and the Equation of Continuity (A₁v₁ = A₂v₂)](https://physics.sillybits.io/chapters/fluid-mechanics/fluid-flow): Steady (streamline) and turbulent flow, streamlines and tubes of flow, the ideal-fluid assumptions, and the equation of continuity Av = constant as conservation of mass.
- [Bernoulli's Equation: P + ρgh + ½ρv² = Constant (Derivation and Meaning)](https://physics.sillybits.io/chapters/fluid-mechanics/bernoulli-equation): Derive Bernoulli's equation from the work–energy theorem for steady flow of an ideal liquid, and use it with the equation of continuity to find pressure differences.
- [Applications of Bernoulli's Equation: Torricelli's Theorem, Venturi Meter, Sprayers and Swing Bowling](https://physics.sillybits.io/chapters/fluid-mechanics/applications-of-bernoulli): Hydrostatics as a special case of Bernoulli's equation, Torricelli's speed of efflux √(2gh), the venturi meter, the aspirator (sprayer), and the swerve of a spinning ball.

## Chapter 14: Some Mechanical Properties of Matter

Interatomic forces, bonds and the states of matter; elasticity, stress, strain, Hooke's law and the elastic moduli; stress–strain curves; elastic energy and measuring Young's modulus; surface tension, surface energy, excess pressure and capillarity; viscosity, Poiseuille's equation, Stokes' law, terminal velocity and the Reynolds number.

- [Interatomic Forces, Ionic and Covalent Bonds, Crystalline and Amorphous Solids](https://physics.sillybits.io/chapters/properties-of-matter/molecular-structure): The interatomic potential-energy curve with its equilibrium separation, ionic and covalent bonds, why solids, liquids and gases differ, the four kinds of crystalline solid, and amorphous (glassy) solids.
- [Elasticity, Stress (Tensile, Compressive, Shear, Volume) and Strain](https://physics.sillybits.io/chapters/properties-of-matter/stress-and-strain): Elastic, plastic and partially elastic bodies, the microscopic origin of elasticity, longitudinal (tensile and compressive), shearing and volume stress, and the matching strains.
- [Hooke's Law, Young's Modulus, Shear Modulus, Bulk Modulus and Poisson's Ratio](https://physics.sillybits.io/chapters/properties-of-matter/hookes-law-and-moduli): Hooke's law for small deformations, Young's modulus Y = FL/AΔL, the shear modulus, the bulk modulus and compressibility, and Poisson's ratio, with typical values.
- [Stress–Strain Curve: Elastic Limit, Yield, Plastic Flow, Fracture, Ductile vs Brittle, Rubber Hysteresis](https://physics.sillybits.io/chapters/properties-of-matter/stress-strain-curve): The stress–strain curve of a metal wire (proportional limit, elastic limit, plastic region, permanent set, fracture), ductile and brittle materials, and the rubber curve with elastic hysteresis.
- [Elastic Potential Energy of a Stretched Wire and Searle's Method for Young's Modulus](https://physics.sillybits.io/chapters/properties-of-matter/elastic-energy-and-youngs-modulus): Elastic potential energy of a stretched wire, U = ½ × stress × strain × volume, and the laboratory determination of Young's modulus with a reference wire and vernier.
- [Surface Tension: Definition S = F/l, Molecular Origin and Demonstrations](https://physics.sillybits.io/chapters/properties-of-matter/surface-tension): Why surface molecules feel an unbalanced pull, surface tension S = F/l as a force per unit length across any line in the surface, classic demonstrations, and the tendency of surfaces to shrink.
- [Surface Energy, Excess Pressure Inside a Drop (2S/R) and a Soap Bubble (4S/R)](https://physics.sillybits.io/chapters/properties-of-matter/surface-energy-and-pressure): Surface energy per unit area equals surface tension, energy needed to break a drop into droplets, and the excess pressure 2S/R inside a drop or air bubble and 4S/R inside a soap bubble.
- [Contact Angle and Capillary Rise h = 2S cos θ / rρg](https://physics.sillybits.io/chapters/properties-of-matter/capillarity): The contact angle and the competition between cohesive and adhesive forces, and the rise or fall of liquid in a capillary tube, h = 2S cos θ/(rρg), with the meniscus correction.
- [Viscosity, Coefficient of Viscosity, Poiseuille's Equation and the Reynolds Number](https://physics.sillybits.io/chapters/properties-of-matter/viscosity): Viscosity as internal friction between fluid layers, F = −ηA dv/dz, units and dimensions of η, temperature dependence, Poiseuille's equation from dimensional analysis, critical velocity and the Reynolds number.
- [Stokes' Law F = 6πηrv, Terminal Velocity and Measuring Viscosity](https://physics.sillybits.io/chapters/properties-of-matter/stokes-law): The viscous drag on a slowly moving sphere, F = 6πηrv, the terminal velocity of a sphere falling through a fluid, and Stokes' method of measuring viscosity.

## Chapter 15: Wave Motion & Waves on a String

Wave motion and pulses, the travelling-wave equation, sine waves and their wavelength, frequency and speed, the speed of waves on a string and the power they carry, superposition and interference, reflection and transmission, standing waves, the normal modes of a string, the sonometer laws, and transverse, longitudinal and polarized waves.

- [What Is a Wave? Wave Pulses and the Travelling-Wave Equation y = f(t − x/v)](https://physics.sillybits.io/chapters/waves-on-a-string/wave-motion): Waves as transport of energy without bulk motion of matter, mechanical and nonmechanical waves, a pulse on a string, wave trains, and the equation of a travelling wave y = f(t − x/v) or g(x − vt).
- [Sinusoidal Waves: y = A sin(ωt − kx), Wavelength, Frequency, Wave Number and v = νλ](https://physics.sillybits.io/chapters/waves-on-a-string/sine-waves): Sine waves on a string, the difference between particle velocity and wave velocity, wavelength λ = vT, v = νλ, wave number k, crests and troughs, and the equivalent forms of the wave equation.
- [Speed of a Wave on a String v = √(F/μ) and Power Transmitted P = 2π²μvA²ν²](https://physics.sillybits.io/chapters/waves-on-a-string/wave-speed-and-power): Derive the speed of a transverse wave on a stretched string, v = √(F/μ), and the average power a sine wave transmits, proportional to the squares of amplitude and frequency.
- [Principle of Superposition and Interference of Waves (Constructive and Destructive)](https://physics.sillybits.io/chapters/waves-on-a-string/superposition-and-interference): Pulses crossing on a string, the principle of superposition for linear waves, and interference of two sine waves travelling the same way: resultant amplitude √(A₁² + A₂² + 2A₁A₂cos δ).
- [Reflection of Waves at Fixed and Free Ends, and at a Junction of Two Strings](https://physics.sillybits.io/chapters/waves-on-a-string/reflection-and-transmission): Reflection of a pulse at a fixed end (inverted) and a free end (not inverted), and partial reflection and transmission at the junction of a light and a heavy string.
- [Standing (Stationary) Waves: Nodes, Antinodes and How They Differ from Travelling Waves](https://physics.sillybits.io/chapters/waves-on-a-string/standing-waves): Two equal sine waves travelling in opposite directions form y = 2A cos kx sin ωt: nodes and antinodes λ/2 apart, and six differences between standing and travelling waves.
- [Vibrations of a String Fixed at Both Ends (ν = n/2L √(F/μ)) and at One End: Harmonics and Overtones](https://physics.sillybits.io/chapters/waves-on-a-string/modes-of-a-string): Resonance of a string driven by a tuning fork, the natural frequencies ν = (n/2L)√(F/μ) of a string fixed at both ends, harmonics, overtones and normal modes, plucked strings, and a string fixed at one end (odd harmonics only).
- [Laws of Transverse Vibration of a String (Length, Tension, Mass) and the Sonometer](https://physics.sillybits.io/chapters/waves-on-a-string/sonometer): The three laws of transverse vibration (ν ∝ 1/L, ν ∝ √F, ν ∝ 1/√μ) and how a sonometer with a paper rider, tuning forks and an auxiliary wire tests them.
- [Transverse and Longitudinal Waves, Water Waves and Polarization](https://physics.sillybits.io/chapters/waves-on-a-string/transverse-longitudinal-polarization): Transverse waves (string, light) and longitudinal waves (sound), water waves that are neither, and polarization of a wave on a string by a slit: linear, circular, elliptical and unpolarized waves.

## Chapter 16: Sound Waves

How sound is produced and travels, displacement and pressure waves, the speed of sound in solids, liquids and gases (Newton, Laplace, temperature), intensity and decibels, pitch, loudness and quality, interference, organ pipes, measuring the speed of sound, beats, diffraction, the Doppler effect, sonic booms, musical scales and the acoustics of buildings.

- [Nature of Sound: Compressions, Rarefactions, Audible Range and Wavefronts](https://physics.sillybits.io/chapters/sound-waves/nature-of-sound): How a vibrating source produces compressions and rarefactions, the pressure-wave equation, the audible range 20 Hz–20 kHz (infrasonic and ultrasonic), why sound is longitudinal and can't be polarized, and spherical and plane wavefronts.
- [Sound as a Displacement Wave and a Pressure Wave: p₀ = Bks₀](https://physics.sillybits.io/chapters/sound-waves/displacement-and-pressure): Describing sound by particle displacement s or by excess pressure p, the relation p₀ = Bks₀ through the bulk modulus, the π/2 phase difference, and why detectors respond to pressure.
- [Speed of Sound: v = √(B/ρ), Newton's Formula, Laplace's Correction and Temperature](https://physics.sillybits.io/chapters/sound-waves/speed-of-sound): Derive v = √(B/ρ) for fluids, v = √(Y/ρ) for rods, Newton's isothermal formula √(P/ρ) and Laplace's adiabatic correction √(γP/ρ), and how pressure, temperature and humidity affect the speed of sound in air.
- [Sound Intensity I = p₀²/2ρv, the Decibel Scale, Pitch, Loudness and Quality (Timbre)](https://physics.sillybits.io/chapters/sound-waves/intensity-and-loudness): Intensity of a sound wave, I = p₀²v/2B = p₀²/2ρv, the decibel scale β = 10 log(I/I₀), and how pitch, loudness and quality relate to frequency, intensity and waveform.
- [Interference of Sound Waves, Coherent Sources, Path Difference and Quincke's Tube](https://physics.sillybits.io/chapters/sound-waves/interference-of-sound): Superposing sound as pressure waves, coherent and incoherent sources, the phase difference 2πΔx/λ from a path difference, Quincke's interference tube, and how sound reflects from rigid boundaries and open ends.
- [Standing Sound Waves in Closed and Open Organ Pipes (Harmonics and Overtones)](https://physics.sillybits.io/chapters/sound-waves/organ-pipes): Standing longitudinal waves, pressure and displacement nodes and antinodes, resonance in a closed organ pipe (ν = (2n + 1)v/4l) and an open pipe (ν = nv/2l), and the end correction.
- [Resonance Column Method and Kundt's Tube: Measuring the Speed of Sound](https://physics.sillybits.io/chapters/sound-waves/measuring-speed-of-sound): The resonance-column method with end correction, v = 2(l₂ − l₁)ν, and Kundt's tube for the speed of sound in a gas or a solid rod.
- [Beats: Beat Frequency |ν₁ − ν₂| from Two Nearly Equal Frequencies](https://physics.sillybits.io/chapters/sound-waves/beats): Superposing two waves of slightly different frequency gives a tone at the average frequency whose amplitude rises and falls |ν₁ − ν₂| times per second.
- [Diffraction of Sound and the Doppler Effect: ν′ = ν(v + u₀)/(v − uₛ)](https://physics.sillybits.io/chapters/sound-waves/diffraction-and-doppler): Diffraction of sound around obstacles and through openings, and the Doppler effect for a moving source, a moving observer and both, with the change in wavelength.
- [Sonic Booms and Mach Cones, the Diatonic Scale, and Acoustics of Buildings (Echo, Reverberation)](https://physics.sillybits.io/chapters/sound-waves/sonic-boom-music-and-acoustics): Shock waves from supersonic sources (Mach cone, sin θ = v/u), the diatonic musical scale, and the acoustics of auditoriums: curved surfaces, echo, reverberation time and amplification problems.

## Chapter 17: Light Waves

Light as a transverse electromagnetic wave: Huygens' principle and the laws of reflection and refraction, Young's double-slit experiment and fringe width, optical path, thin-film colours, the Fresnel biprism, coherence, diffraction by a slit, a circular hole and an edge, the limit of resolution, scattering and the blue sky, and polarization with polaroids, Malus' law and Brewster's angle.

- [Nature of Light: Wave Theory, Refractive Index, Visible Spectrum and Ray Optics](https://physics.sillybits.io/chapters/light-waves/nature-of-light): The corpuscle and wave theories of light, why light needs no medium, the electric field as the waving quantity, refractive index μ = c/v, the visible spectrum and colours, monochromatic light, and when light can be treated as rays.
- [Huygens' Principle: Secondary Wavelets, Reflection and Snell's Law from Wave Theory](https://physics.sillybits.io/chapters/light-waves/huygens-principle): Secondary wavelets and the Huygens construction, and how it proves the laws of reflection and refraction (Snell's law, μ₂₁ = v₁/v₂), with the prediction that light is slower in water.
- [Young's Double-Slit Experiment: Bright and Dark Fringes, Intensity and Fringe Width w = Dλ/d](https://physics.sillybits.io/chapters/light-waves/youngs-double-slit): Young's pinhole and double-slit experiments, conditions for bright and dark fringes, the intensity I = 4I′cos²(δ/2), and the fringe width w = Dλ/d used to measure the wavelength of light.
- [Optical Path Length μΔx and Wavelength of Light in a Medium λ = λ₀/μ](https://physics.sillybits.io/chapters/light-waves/optical-path): Why a distance Δx in a medium of refractive index μ causes the same phase change as μΔx in vacuum, the wavelength in a medium λ₀/μ, and why every ray through a lens has the same optical path.
- [Thin Film Interference: Soap Bubbles, Oil Films, 2μd = nλ and the Reflection Phase Change](https://physics.sillybits.io/chapters/light-waves/thin-films): Interference of light reflected from and transmitted through a thin film, the optical path difference 2μd at normal incidence, the phase change of π on reflection from a denser medium, and why reflected and transmitted colours are complementary.
- [Fresnel Biprism, Coherent and Incoherent Sources, Wavetrains and Lasers](https://physics.sillybits.io/chapters/light-waves/biprism-and-coherence): The Fresnel biprism as a double-slit substitute, coherent versus incoherent sources, wavetrains from atoms, why the path difference must be small, the spread in wavelength of real sources, and lasers.
- [Single Slit Diffraction: Fraunhofer vs Fresnel, Minima b sin θ = nλ and Central Maximum](https://physics.sillybits.io/chapters/light-waves/single-slit-diffraction): Diffraction explained with Huygens' wavelets, Fraunhofer and Fresnel diffraction, the minima b sin θ = nλ for a single slit, the intensity I₀(sin β/β)², and why the spread grows as the slit narrows.
- [Circular Aperture Diffraction, Airy Disc, Straight-Edge Diffraction and the Rayleigh Criterion](https://physics.sillybits.io/chapters/light-waves/circular-aperture-and-resolution): Diffraction by a circular hole (sin θ = 1.22λ/b), the disc image formed by a lens, Fresnel diffraction at a straight edge, the limit of resolution and Rayleigh's criterion.
- [Scattering of Light: Rayleigh's 1/λ⁴ Law, Blue Sky, Red Sunset and Red Danger Signals](https://physics.sillybits.io/chapters/light-waves/scattering-of-light): Scattering as absorption and re-radiation by molecules, Rayleigh's law (∝ 1/λ⁴), the blue sky, red sunsets, red danger signals, the black sky at high altitude, and haze from larger particles.
- [Polarization of Light: Linear, Circular, Elliptical, Polaroids, Malus' Law and Brewster's Angle](https://physics.sillybits.io/chapters/light-waves/polarization-of-light): Linearly, circularly and elliptically polarized and unpolarized light, polaroids and the transmission axis, Malus' law I = I₀cos²θ, polarization by reflection (Brewster's law tan i = μ) and by scattering.

## Chapter 18: Geometrical Optics

Reflection by plane and spherical mirrors, the mirror formula and magnification, refraction at plane surfaces and apparent depth, critical angle, total internal reflection and optical fibres, prisms and minimum deviation, refraction at spherical surfaces, thin lenses, the lens maker's formula, power and lens combinations, and the defects (aberrations) of real images.

- [Laws of Reflection and Image Formation by a Plane Mirror](https://physics.sillybits.io/chapters/geometrical-optics/plane-mirrors): The two laws of reflection, how a plane mirror forms a virtual image as far behind it as the object is in front, and how normals and reflection work on curved surfaces.
- [Spherical Mirrors: Pole, Centre of Curvature, Focus, Paraxial Rays and the Sign Convention](https://physics.sillybits.io/chapters/geometrical-optics/spherical-mirrors): Concave and convex mirrors, pole, centre and radius of curvature, principal axis, focus and focal plane, paraxial rays, tracing images, real and virtual objects and images, and the coordinate sign convention.
- [Mirror Formula 1/u + 1/v = 1/f, f = R/2 and Magnification m = −v/u](https://physics.sillybits.io/chapters/geometrical-optics/mirror-formula): Deriving 1/u + 1/v = 2/R for a spherical mirror, f = R/2, ray diagrams for extended objects, and lateral magnification m = −v/u.
- [Refraction at a Plane Surface: Snell's Law, Real and Apparent Depth and Image Shift](https://physics.sillybits.io/chapters/geometrical-optics/refraction-and-apparent-depth): Snell's law with refractive indices, the image formed by refraction at a flat surface, μ = real depth / apparent depth, and the shift (1 − 1/μ)t of an object seen through a slab.
- [Critical Angle, Total Internal Reflection and Optical Fibres](https://physics.sillybits.io/chapters/geometrical-optics/total-internal-reflection): Light going from a denser to a rarer medium, the critical angle sin θc = 1/μ, total internal reflection, and optical fibres and light pipes for medicine and communication.
- [Refraction Through a Prism: Angle of Deviation, Minimum Deviation and μ = sin((A+δm)/2)/sin(A/2)](https://physics.sillybits.io/chapters/geometrical-optics/prism): Refraction through a prism, r + r′ = A, deviation δ = i + i′ − A, the condition i = i′ for minimum deviation, finding μ from A and δm, and δ = (μ − 1)A for a thin prism.
- [Refraction at a Spherical Surface: μ₂/v − μ₁/u = (μ₂ − μ₁)/R and Magnification](https://physics.sillybits.io/chapters/geometrical-optics/spherical-surfaces): Image formation by refraction at a single spherical surface, μ₂/v − μ₁/u = (μ₂ − μ₁)/R, its sign convention, and the lateral magnification m = μ₁v/(μ₂u).
- [Thin Lenses: Focus, Sign Convention, Lens Maker's Formula and the Lens Formula 1/v − 1/u = 1/f](https://physics.sillybits.io/chapters/geometrical-optics/thin-lenses): Thin lenses and their principal axis, optical centre, first and second foci, the sign convention, the lens maker's formula from two successive refractions, and the lens formula.
- [Ray Diagrams for Convex and Concave Lenses and Magnification m = v/u](https://physics.sillybits.io/chapters/geometrical-optics/lens-images): Locating images by ray diagrams (through the optical centre, parallel to the axis, through the first focus), real and virtual images, and lateral magnification m = v/u.
- [Power of a Lens in Dioptres, Lenses in Contact and Two Separated Thin Lenses](https://physics.sillybits.io/chapters/geometrical-optics/lens-power-and-combinations): Power of a lens in dioptres, two thin lenses in contact (P = P₁ + P₂), the deviation of a ray by a thin lens δ = h/f, and the equivalent focal length 1/F = 1/f₁ + 1/f₂ − d/f₁f₂ for separated lenses.
- [Aberrations: Spherical Aberration, Coma, Astigmatism, Curvature, Distortion and Chromatic Aberration](https://physics.sillybits.io/chapters/geometrical-optics/aberrations): Monochromatic aberrations (spherical aberration, coma, astigmatism, curvature of field, distortion) and chromatic aberration (axial and lateral), with ways to reduce them: stops, parabolic mirrors, sharing the bending, and achromatic combinations.

## Chapter 19: Optical Instruments

How the eye focuses and what sets its near point, apparent size and visual angle, the simple and compound microscopes, astronomical, terrestrial and Galilean telescopes, the resolving power of microscopes and telescopes, and defects of vision with their corrections.

- [The Human Eye: Cornea, Lens, Accommodation, Near Point and Visual Angle](https://physics.sillybits.io/chapters/optical-instruments/the-eye): The parts of the eye and what each does, how most of the focusing happens at the cornea, accommodation by the ciliary muscles, the near point and least distance of clear vision (25 cm), and apparent size set by the visual angle.
- [Simple Microscope: Magnifying Power D/f and 1 + D/f](https://physics.sillybits.io/chapters/optical-instruments/simple-microscope): Angular magnification (magnifying power) of a converging lens used as a magnifier, in normal adjustment (image at infinity, m = D/f) and with the image at the near point (m = 1 + D/f).
- [Compound Microscope: Objective, Eyepiece, Magnifying Power and Tube Length](https://physics.sillybits.io/chapters/optical-instruments/compound-microscope): How the objective forms a magnified real image which the eyepiece views as a magnifier, the magnifying power (v₀/u₀)(D/fₑ) or (v₀/u₀)(1 + D/fₑ), the approximation −(L/f₀)(D/fₑ), and the tube length.
- [Astronomical, Terrestrial and Galilean Telescopes: Magnifying Power f₀/fₑ and Length](https://physics.sillybits.io/chapters/optical-instruments/telescopes): Astronomical (Keplerian) telescopes, magnifying power −f₀/fₑ and −(f₀/fₑ)(1 + fₑ/D), length f₀ + fₑ; the terrestrial telescope with an erecting lens; and the Galilean telescope with a diverging eyepiece.
- [Resolving Power of a Microscope (2μ sin θ/λ) and a Telescope (a/1.22λ)](https://physics.sillybits.io/chapters/optical-instruments/resolving-power): Resolving power of a microscope, R = 2μ sin θ/λ and the oil-immersion objective, and of a telescope, R = a/1.22λ, and why large apertures matter.
- [Defects of Vision: Myopia, Hypermetropia, Presbyopia and Astigmatism with Corrective Lens Power](https://physics.sillybits.io/chapters/optical-instruments/defects-of-vision): The eye as a lens with a fixed image distance, near and far points, myopia corrected by a diverging lens (P = −1/x), hypermetropia and presbyopia corrected by a converging lens (P = 1/25 cm − 1/y), astigmatism and cylindrical lenses, and bifocals.

## Chapter 20: Dispersion & Spectra

How refractive index varies with wavelength, angular dispersion and dispersive power, combining prisms for dispersion without deviation or deviation without dispersion, pure and impure spectra, emission and absorption spectra (continuous, line and band), ultraviolet and infrared, the spectrometer, and the rainbow.

- [Dispersion of Light: Refractive Index vs Wavelength and Cauchy's Relation](https://physics.sillybits.io/chapters/dispersion-and-spectra/dispersion): Why refractive index depends on wavelength (μ = μ₀ + A/λ²), how it varies for common glasses, and how a prism separates white light into colours.
- [Angular Dispersion and Dispersive Power ω = (μv − μr)/(μy − 1)](https://physics.sillybits.io/chapters/dispersion-and-spectra/dispersive-power): Mean deviation and angular dispersion of a thin prism, and the dispersive power ω = (μᵥ − μᵣ)/(μᵧ − 1), including finding ω from a lens's focal lengths for different colours.
- [Dispersion Without Deviation and Deviation Without Dispersion (Achromatic Prisms)](https://physics.sillybits.io/chapters/dispersion-and-spectra/combining-prisms): Two thin prisms placed in opposition: conditions for dispersion without average deviation (direct-vision prism) and deviation without dispersion (achromatic prism).
- [Pure and Impure Spectra, Emission (Continuous, Line, Band) and Absorption Spectra, Ultraviolet and Infrared](https://physics.sillybits.io/chapters/dispersion-and-spectra/spectra): What a spectrum is, conditions for a pure spectrum, continuous, line and band emission spectra, line and band absorption spectra (Fraunhofer lines), and ultraviolet and infrared radiation.
- [Prism Spectrometer: Collimator, Prism Table, Telescope; Measuring Prism Angle and Minimum Deviation](https://physics.sillybits.io/chapters/dispersion-and-spectra/spectrometer): Parts and adjustment of a spectrometer (collimator, prism table, telescope), and using it to measure the angle of a prism, the angle of minimum deviation, and the variation of refractive index with wavelength.
- [How Rainbows Form: Primary (42°) and Secondary (51°) Bows from Raindrops](https://physics.sillybits.io/chapters/dispersion-and-spectra/rainbow): Refraction, internal reflection and dispersion in raindrops, why light concentrates near the angle of maximum deviation, the primary bow (40.6°–42.2°) and the secondary bow (50.5°–54°) with reversed colours.

## Chapter 21: Speed of Light

Why the speed of light is now a defined constant, early attempts by Galileo, Rømer and Bradley, and the terrestrial methods of Fizeau (toothed wheel), Foucault (rotating mirror) and Michelson (rotating polygon).

- [History of Measuring the Speed of Light: Galileo, Rømer, Bradley and the 1983 Definition](https://physics.sillybits.io/chapters/speed-of-light/measuring-c-history): Why c is special, its exact defined value since 1983, Galileo's lantern experiment and why it failed, Rømer's and Bradley's astronomical estimates.
- [Fizeau's Method: Speed of Light with a Toothed Wheel, c = 4Dnν](https://physics.sillybits.io/chapters/speed-of-light/fizeau-method): Fizeau's 1849 terrestrial measurement: the toothed wheel, the eclipse condition, c = 4Dnν, and the method's practical difficulties.
- [Foucault's Rotating Mirror Method and Michelson's Polygon Mirror Method](https://physics.sillybits.io/chapters/speed-of-light/foucault-and-michelson): Foucault's rotating-mirror method, c = 4R²ωa/[s(R + b)], fit for a laboratory and for measuring light in water; and Michelson's rotating polygon, c = DνN.

## Chapter 22: Photometry

Radiant flux, the eye's luminosity curve and the lumen, luminous efficiency, luminous intensity (candela) and solid angle, illuminance (lux), the inverse-square and cosine laws, Lambert's cosine law, and comparing sources with a photometer.

- [Radiant Flux, Relative Luminosity, Luminous Flux and the Lumen](https://physics.sillybits.io/chapters/photometry/luminous-flux): Total radiant flux in watts, the relative luminosity curve peaking at 555 nm, luminous flux and the lumen (1 W at 555 nm = 685 lm).
- [Luminous Efficiency (lm/W), Solid Angle and Luminous Intensity in Candela](https://physics.sillybits.io/chapters/photometry/luminous-intensity): Luminous efficiency and overall luminous efficiency of lamps, solid angle and the steradian, luminous intensity I = dF/dω and the candela.
- [Illuminance in Lux, the Inverse Square Law E = I cos θ / r² and Lambert's Cosine Law](https://physics.sillybits.io/chapters/photometry/illuminance): Illuminance (lux), the formula E = I cos θ/r² with the inverse-square law and the cosine factor, and Lambert's cosine law for perfectly diffusing surfaces.
- [Comparing Light Sources with a Photometer: I₁/I₂ = d₁²/d₂² and Bunsen's Grease-Spot Photometer](https://physics.sillybits.io/chapters/photometry/photometers): The principle of a photometer, I₁/I₂ = d₁²/d₂² for equal illuminance, and Bunsen's grease-spot photometer on an optical bench.

## Chapter 23: Heat & Temperature

Heat as energy transferred without work, thermal equilibrium and the zeroth law, defining temperature scales with mercury and platinum resistance thermometers, constant-volume and constant-pressure gas thermometers, the ideal-gas (kelvin) and Celsius scales, the ideal gas equation, and thermal expansion of solids and liquids, including water's anomaly.

- [What Is Heat? Thermal Equilibrium and the Zeroth Law of Thermodynamics](https://physics.sillybits.io/chapters/heat-and-temperature/heat-and-equilibrium): Internal energy and heat as energy transferred without mechanical work, thermal equilibrium, the zeroth law and the idea of temperature, and adiabatic versus diathermic walls.
- [Temperature Scales: Fixed Points, Celsius, Fahrenheit and the Platinum Resistance Thermometer](https://physics.sillybits.io/chapters/heat-and-temperature/temperature-scales): Defining a temperature scale from a thermometric property and two fixed points (ice and steam points), the mercury centigrade and Fahrenheit scales, the platinum resistance thermometer with its Wheatstone bridge, and why different scales disagree.
- [Constant Volume Gas Thermometer, Triple Point 273.16 K, Ideal Gas Scale, Celsius and pV = nRT](https://physics.sillybits.io/chapters/heat-and-temperature/gas-thermometers): The constant-volume gas thermometer and T = 273.16 K × p/p_tr, the ideal-gas scale as p_tr → 0, the Celsius scale, the ideal gas equation pV = nRT, and Callendar's compensated constant-pressure thermometer.
- [Thermal Expansion: Coefficients of Linear and Volume Expansion and the Anomalous Expansion of Water](https://physics.sillybits.io/chapters/heat-and-temperature/thermal-expansion): The coefficients of linear and volume expansion, L = L₀(1 + αθ), V = V₀(1 + γθ), γ = 3α, the anomalous expansion of water below 4°C and why lakes freeze from the top, and real versus apparent expansion of a liquid.

## Chapter 24: Kinetic Theory of Gases

The kinetic model of an ideal gas, pressure from molecular impacts, rms speed, temperature as molecular kinetic energy, Boltzmann's constant, the gas laws derived from kinetic theory, Maxwell's speed distribution, real gases and van der Waals, Brownian motion, vapour, evaporation, saturated vapour pressure, boiling, dew point, humidity and phase diagrams.

- [Kinetic Theory of Gases: The Assumptions of an Ideal Gas Model](https://physics.sillybits.io/chapters/kinetic-theory/kinetic-model): What kinetic theory sets out to do, its six assumptions, why molecule–molecule collisions can be ignored in steady state, and how realistic the model is.
- [Pressure of an Ideal Gas p = ⅓ρv̄², rms Speed and Translational Kinetic Energy](https://physics.sillybits.io/chapters/kinetic-theory/gas-pressure): Deriving p = ⅓ρv̄² from molecular impacts on a wall, pV = ⅓Nmv̄², the rms speed √(3p/ρ), and the translational kinetic energy K = (3/2)pV.
- [Kinetic Interpretation of Temperature: pV = NkT, v_rms = √(3RT/M) and Average KE (3/2)kT](https://physics.sillybits.io/chapters/kinetic-theory/temperature-and-speed): Why T ∝ v²_rms, equal average kinetic energies in a mixture, Boltzmann's constant and pV = NkT, R = N_A k, v_rms = √(3kT/m), average KE (3/2)kT, internal energy (3/2)nRT, and mean speed √(8RT/πM).
- [Boyle's, Charles's, Avogadro's, Graham's and Dalton's Laws Derived from Kinetic Theory](https://physics.sillybits.io/chapters/kinetic-theory/gas-laws): Deriving Boyle's law, Charles's laws for volume and pressure, Avogadro's law, Graham's law of diffusion and Dalton's law of partial pressures from kinetic theory.
- [Maxwell's Speed Distribution, Most Probable Speed, Van der Waals Equation and Brownian Motion](https://physics.sillybits.io/chapters/kinetic-theory/speed-distribution-and-real-gases): Maxwell's speed distribution and the most probable speed, thermodynamic state and equation of state, the van der Waals equation, and Brownian motion.
- [Critical Temperature, Vapour, Evaporation, Saturated Vapour Pressure and Boiling Point](https://physics.sillybits.io/chapters/kinetic-theory/vapour-and-boiling): Liquefying gases and the critical temperature, vapour versus gas, isotherms, evaporation and its cooling effect, saturated and unsaturated vapour, saturation vapour pressure and its table for water, and boiling when SVP equals the external pressure.
- [Dew Point, Relative Humidity, Regnault's Hygrometer, Phase Diagrams and the Triple Point](https://physics.sillybits.io/chapters/kinetic-theory/humidity-and-phases): Dew point, absolute and relative humidity, Regnault's hygrometer, phase diagrams with the triple points of water and carbon dioxide, dry ice, dew and fog.

## Chapter 25: Calorimetry

Heat as energy in transit, the calorie and the joule, the principle of calorimetry, specific and molar heat capacities, water equivalent, Regnault's method, latent heats of fusion and vaporization and how to measure them, and the mechanical equivalent of heat by Searle's cone.

- [Heat as Energy in Transit, the Calorie (4.186 J) and the Principle of Calorimetry](https://physics.sillybits.io/chapters/calorimetry/heat-and-units): Heat as energy transferred because of a temperature difference, the calorie (1 cal = 4.186 J) and kilocalorie, the simple calorimeter, and the principle of calorimetry.
- [Specific Heat Capacity Q = msΔθ, Molar Heat Capacity, Water Equivalent and Regnault's Method](https://physics.sillybits.io/chapters/calorimetry/specific-heat): Q = msΔθ, a table of specific heats, molar heat capacity, heat capacity and water equivalent, and Regnault's method of mixtures for the specific heat of a solid or a liquid.
- [Latent Heat of Fusion and Vaporization Q = mL and How to Measure Them](https://physics.sillybits.io/chapters/calorimetry/latent-heat): Q = mL for melting and boiling at constant temperature, the molecular reason, ice–water mixing problems, and calorimeter methods for the latent heat of fusion of ice and of vaporization of water.
- [Mechanical Equivalent of Heat J = 4.186 J/cal and Searle's Cone Method](https://physics.sillybits.io/chapters/calorimetry/mechanical-equivalent): W = JH, Searle's cone apparatus, the balance of friction torque against a hanging weight, J = 2πnMgr/[(m₁s₁ + m₂s₂)Δθ], and why 'heat developed' by friction is loose wording.

## Chapter 26: Laws of Thermodynamics

The first law ΔQ = ΔU + ΔW and internal energy as a state function, work done by a gas in isothermal, isobaric and isochoric processes, heat engines and efficiency (steam, petrol and diesel engines), the second law (Kelvin–Planck and Clausius), reversible and quasi-static processes, entropy, the Carnot engine and theorem, and refrigerators.

- [First Law of Thermodynamics ΔQ = ΔU + ΔW, Sign Conventions and Internal Energy as a State Function](https://physics.sillybits.io/chapters/laws-of-thermodynamics/first-law): Changing internal energy by heat and by work, the first law ΔQ = ΔU + ΔW with its sign conventions, and why U is a state function while Q and W depend on the process.
- [Work Done by a Gas W = ∫p dV: Isothermal, Isobaric and Isochoric Processes on a p–V Diagram](https://physics.sillybits.io/chapters/laws-of-thermodynamics/work-done-by-gas): W = ∫p dV as the area under a p–V curve, work in isothermal (nRT ln V₂/V₁), isobaric (pΔV) and isochoric (zero) processes, and net work around a cycle.
- [Heat Engines and Efficiency η = 1 − Q₂/Q₁: Steam Engine, Petrol (Otto) and Diesel Engines](https://physics.sillybits.io/chapters/laws-of-thermodynamics/heat-engines): Cyclic processes, efficiency η = W/Q₁ = 1 − Q₂/Q₁, the steam engine with its D-valve, and the four strokes of petrol and diesel internal combustion engines.
- [Second Law of Thermodynamics (Kelvin–Planck), Quasi-static and Reversible Processes](https://physics.sillybits.io/chapters/laws-of-thermodynamics/second-law): Processes allowed by energy conservation that never run backwards, the Kelvin–Planck statement of the second law, thermodynamic equilibrium, quasi-static and reversible processes, and reversible cycles.
- [Entropy ΔS = ΔQ/T, Disorder and the Entropy Statement of the Second Law](https://physics.sillybits.io/chapters/laws-of-thermodynamics/entropy): Entropy as a state variable, ΔS = ΔQ/T and S_f − S_i = ∫dQ/T for reversible processes, constant entropy in reversible adiabatic processes, entropy and disorder, and the second law as 'isolated systems never lose entropy'.
- [Carnot Cycle, Efficiency 1 − T₂/T₁, Carnot's Theorem, Refrigerators and the Clausius Statement](https://physics.sillybits.io/chapters/laws-of-thermodynamics/carnot-engine): The four steps of the Carnot cycle, its T–S diagram and efficiency η = 1 − T₂/T₁, Carnot's theorem, refrigerators (heat pumps) with minimum work W = Q₂(T₁/T₂ − 1), and the Clausius statement of the second law.

## Chapter 27: Specific Heats of Gases

Specific and molar heat capacities at constant volume and constant pressure, Cp − Cv = R, measuring Cp (Regnault) and Cv (Joly), isothermal and adiabatic processes with pV^γ = constant and adiabatic work, and the equipartition of energy with degrees of freedom.

- [Specific Heat of Gases Depends on the Process: s_V, s_p, C_V and C_p](https://physics.sillybits.io/chapters/specific-heats-of-gases/two-heat-capacities): Why the heat needed to warm a gas depends on the process, the specific heats at constant volume and pressure, and the molar heat capacities C_V and C_p.
- [Mayer's Relation Cp − Cv = R and U = nCvT for an Ideal Gas](https://physics.sillybits.io/chapters/specific-heats-of-gases/cp-minus-cv): Deriving Cp − Cv = R for an ideal gas, Cv = (1/n) dU/dT and U = nCvT, and how heat splits between internal energy and work at constant pressure.
- [Regnault's Method for Cp and Joly's Differential Steam Calorimeter for Cv](https://physics.sillybits.io/chapters/specific-heats-of-gases/measuring-gas-heats): Regnault's flow method for the molar heat capacity at constant pressure, and Joly's differential steam calorimeter for the molar heat capacity at constant volume.
- [Isothermal and Adiabatic Processes: pV^γ = constant, TV^(γ−1) = constant and Adiabatic Work](https://physics.sillybits.io/chapters/specific-heats-of-gases/isothermal-and-adiabatic): Isothermal processes (infinite heat capacity, W = nRT ln V_f/V_i), adiabatic processes (zero heat capacity), deriving pV^γ = constant, the p–T and V–T forms, and the work done in an adiabatic process.
- [Equipartition Theorem, Degrees of Freedom, and Cv, Cp, γ for Monatomic and Diatomic Gases](https://physics.sillybits.io/chapters/specific-heats-of-gases/equipartition): Degrees of freedom of monatomic, diatomic and vibrating molecules, the equipartition theorem, predicted Cv, Cp and γ compared with measurements, and the case of CO₂.

## Chapter 28: Heat Transfer

Thermal conduction and conductivity, thermal resistance in series and parallel, Searle's method, natural and forced convection, thermal radiation and Prévost's theory, blackbodies and Kirchhoff's law, Wien's displacement law, the Stefan–Boltzmann law and emissivity, Newton's law of cooling, and the bolometer and thermopile.

- [Thermal Conduction: Heat Current ΔQ/Δt = KA(T₁ − T₂)/x and Thermal Conductivity](https://physics.sillybits.io/chapters/heat-transfer/conduction): How conduction works by molecular collisions, steady state and heat current, Fourier's law with thermal conductivity, the temperature gradient, a table of conductivities, and why metals conduct best.
- [Thermal Resistance R = x/KA, Rods in Series and in Parallel](https://physics.sillybits.io/chapters/heat-transfer/thermal-resistance): Thermal resistance R = x/KA and the Ohm's-law-like relation i = ΔT/R, with equivalent resistances for rods in series (R₁ + R₂) and in parallel (1/R = 1/R₁ + 1/R₂).
- [Searle's Method for Thermal Conductivity and Natural and Forced Convection](https://physics.sillybits.io/chapters/heat-transfer/searle-and-convection): Searle's apparatus and K = x ms(θ₃ − θ₄)/[A(θ₁ − θ₂)t], natural and forced convection, why ponds freeze from the top, and heat flow in the body.
- [Thermal Radiation, Prévost's Theory, Blackbody and Kirchhoff's Law](https://physics.sillybits.io/chapters/heat-transfer/radiation-and-kirchhoff): Thermal radiation without a medium, Prévost's theory of exchange, blackbodies and cavity radiators, emissive and absorptive power, and the statement and proof of Kirchhoff's law.
- [Wien's Displacement Law λmT = b, Stefan–Boltzmann Law σT⁴, Emissivity and Net Radiation Loss](https://physics.sillybits.io/chapters/heat-transfer/wien-and-stefan): The nature and spectrum of thermal radiation, Wien's displacement law, Planck's quantum hypothesis, the Stefan–Boltzmann law, emissivity equal to absorptivity, and the net loss eσA(T⁴ − T₀⁴).
- [Newton's Law of Cooling dθ/dt = −k(θ − θ₀), the Bolometer and the Thermopile](https://physics.sillybits.io/chapters/heat-transfer/newtons-law-of-cooling): Deriving Newton's law of cooling from radiation and convection for small temperature differences, solving cooling problems with average temperatures, and the bolometer and thermopile detectors.

## Chapter 29: Electric Field & Potential

Electric charge and its quantization and conservation, charging by friction and induction, Coulomb's law, the electric field of point charges and a charged ring, field lines, electric potential energy, potential and its relation to the field, equipotentials, the electric dipole's field, torque and energy, and conductors, insulators and semiconductors.

- [Electric Charge: Two Kinds, Quantization, Conservation, Frictional Electricity and Induction](https://physics.sillybits.io/chapters/electric-field-and-potential/electric-charge): Why particles must carry charge, positive and negative charge, the coulomb and e, quantization and conservation of charge, charging by rubbing, and induction.
- [Coulomb's Law F = q₁q₂/4πε₀r², Permittivity of Free Space and Vector Addition of Forces](https://physics.sillybits.io/chapters/electric-field-and-potential/coulombs-law): Coulomb's law in scalar and vector form, k and ε₀, and adding forces from several charges, including finding a point of zero force.
- [Electric Field E = F/q, Field of a Point Charge and a Ring, and Electric Field Lines](https://physics.sillybits.io/chapters/electric-field-and-potential/electric-field): The electric field as a two-step picture of electric force, its definition with a test charge, the field of a point charge, superposition, the field on the axis of a charged ring, and lines of force.
- [Electric Potential Energy U = q₁q₂/4πε₀r of a System of Charges](https://physics.sillybits.io/chapters/electric-field-and-potential/potential-energy): Defining the change in electric potential energy from the work of electric forces, U = q₁q₂/4πε₀r with zero at infinity, and summing over pairs for many charges.
- [Electric Potential and Potential Difference, the Volt, and Potential of a Point Charge V = Q/4πε₀r](https://physics.sillybits.io/chapters/electric-field-and-potential/electric-potential): Potential difference V_B − V_A = (U_B − U_A)/q, potential at a point relative to a reference, its meaning as external work per unit charge, the potential of point charges, and the electronvolt.
- [Relation Between Electric Field and Potential: dV = −E·dr, E = −∂V/∂x, and Equipotential Surfaces](https://physics.sillybits.io/chapters/electric-field-and-potential/field-and-potential): dV = −E⃗·dr⃗ and V₂ − V₁ = −∫E⃗·dr⃗, the uniform field between plates, Eₓ = −∂V/∂x, the field component −dV/dr, and equipotential surfaces perpendicular to the field.
- [Electric Dipole: Dipole Moment p = qd, Potential, Field, Torque p × E and Energy −p·E](https://physics.sillybits.io/chapters/electric-field-and-potential/electric-dipole): Dipole moment, potential p cos θ/4πε₀r², the generalised definition of a dipole, the radial and transverse field components, axial and equatorial fields, and torque and potential energy in a uniform field.
- [Conductors, Insulators (Dielectrics) and Semiconductors, and Zero Field Inside a Conductor](https://physics.sillybits.io/chapters/electric-field-and-potential/conductors-and-insulators): Conductors with free electrons, insulators (dielectrics), semiconductors, and why the electric field inside a conductor is zero in electrostatics.

## Chapter 30: Gauss's Law

Electric flux and the area vector, solid angle, Gauss's law and its proof from Coulomb's law, the fields of a uniformly charged sphere, a long line and a large sheet, shells and solid spheres and their electrostatic energy, charges on conductors and in cavities, the field σ/ε₀ near a conductor, and earthing.

- [Electric Flux Φ = E·ΔS cos θ, the Area Vector, and Flux Through Closed Surfaces](https://physics.sillybits.io/chapters/gauss-law/electric-flux): The area vector and positive normal, flux of a uniform field through a flat area, flux through curved surfaces as an integral, and closed surfaces with the outward normal positive.
- [Solid Angle Ω = S/r², the Steradian, and the 4π Subtended by a Closed Surface](https://physics.sillybits.io/chapters/gauss-law/solid-angle): Plane angle as arc over radius, solid angle as area over radius squared, the solid angle of a small tilted patch ΔS cos α / r², and why a closed surface subtends 4π inside and zero outside.
- [Gauss's Law ∮E·dS = q_in/ε₀ and Its Proof from Coulomb's Law](https://physics.sillybits.io/chapters/gauss-law/gauss-law): The statement of Gauss's law, what E and q_in each mean, the proof from Coulomb's law using solid angles for inside and outside charges, and worked examples including a cube and a ring.
- [Applying Gauss's Law: Field of a Charged Sphere, an Infinite Line Charge and a Plane Sheet](https://physics.sillybits.io/chapters/gauss-law/using-symmetry): Using symmetry and a well-chosen Gaussian surface to find the field of a uniformly charged sphere inside and outside, a long line charge, and a large plane sheet of charge.
- [Spherical Charge Distributions: Shell Theorem Results and Electrostatic Energy 3Q²/20πε₀R and Q²/8πε₀R](https://physics.sillybits.io/chapters/gauss-law/spherical-distributions): Field and potential of uniformly charged thin shells and solid spheres, inside and outside, by analogy with gravitation, and the electric potential energy of a uniformly charged sphere and a thin shell.
- [Charged Conductors and Gauss's Law: Charge on the Surface, Cavities, E = σ/ε₀, and Earthing](https://physics.sillybits.io/chapters/gauss-law/conductors-and-earthing): Why excess charge on a conductor lies on its surface, induced charge on the walls of a cavity, the field σ/ε₀ just outside a conductor and why it is not σ/2ε₀, and earthing (grounding) a conductor.

## Chapter 31: Capacitors

Capacitors and capacitance, the ideal battery, parallel-plate, spherical and cylindrical capacitors, series and parallel combinations and a general method for networks, the force between plates, energy stored and the energy density of a field, dielectrics and polarization, capacitors filled with dielectrics, Gauss's law with dielectrics and the displacement vector, corona discharge and the van de Graaff generator.

- [Capacitor and Capacitance Q = CV, the Farad, the Ideal Battery, and the Parallel-Plate Capacitor C = ε₀A/d](https://physics.sillybits.io/chapters/capacitors/capacitance): What a capacitor is, the meaning of its charge and potential, capacitance and the farad, properties of an ideal battery, and deriving C = ε₀A/d for parallel plates with Gauss's law.
- [Spherical Capacitor 4πε₀R₁R₂/(R₂ − R₁), Isolated Sphere 4πε₀R, and Cylindrical Capacitor 2πε₀l/ln(R₂/R₁)](https://physics.sillybits.io/chapters/capacitors/spherical-and-cylindrical): Capacitance of concentric spheres and coaxial cylinders from Gauss's law, the isolated sphere as a limit, and the parallel-plate limit of a thin gap.
- [Capacitors in Series and Parallel: Equivalent Capacitance and a General Method for Networks](https://physics.sillybits.io/chapters/capacitors/combinations): Equivalent capacitance, series combination (1/C = Σ1/Cᵢ) with equal charges, parallel combination (C = ΣCᵢ) with equal voltages, and a step-by-step general method for any network.
- [Force Between Capacitor Plates Q²/2Aε₀, Energy Stored ½CV², and Energy Density ½ε₀E²](https://physics.sillybits.io/chapters/capacitors/force-and-energy): The attractive force between parallel plates, the energy stored in a capacitor from the work of separating its plates, and the energy density of an electric field.
- [Dielectrics: Polar and Nonpolar Molecules, Polarization, Bound Charge σp = P, Dielectric Constant and Strength](https://physics.sillybits.io/chapters/capacitors/dielectrics): Polar and nonpolar dielectrics in a field, the polarization vector, induced (bound) surface charge equal to P, the dielectric constant E = E₀/K, dielectric breakdown and a table of dielectric constants and strengths.
- [Capacitor with a Dielectric C = KC₀, Induced Charge Q(1 − 1/K), Gauss's Law in Dielectrics and the Displacement Vector D](https://physics.sillybits.io/chapters/capacitors/capacitor-with-dielectric): Capacitance with a dielectric, the induced charge on the slab, a partially filled capacitor, Gauss's law with K and with the electric displacement D, a point charge in a dielectric and the energy density ½Kε₀E².
- [Charge Density on Curved Conductors, Corona Discharge, and the Van de Graaff High-Voltage Generator](https://physics.sillybits.io/chapters/capacitors/corona-and-van-de-graaff): Why charge density is higher where a conductor is more sharply curved, corona discharge at points, and how a van de Graaff generator works and what limits its voltage.

## Chapter 32: Electric Current in Conductors

Current and current density, drift of free electrons, Ohm's law, resistivity and its temperature dependence, thermistors and superconductors, emf and internal resistance, Joule heating, Kirchhoff's laws, series and parallel resistors and cells, the Wheatstone bridge, ammeters, voltmeters, the potentiometer, charging and discharging capacitors, and atmospheric electricity.

- [Electric Current i = dQ/dt, the Ampere, Direction of Current, and Current Density j = i/S](https://physics.sillybits.io/chapters/electric-current/current-and-density): Electric current as the rate of charge transfer, the ampere, the direction convention for positive and negative carriers, current density as a vector, i = ∫j·dS, and sources of current.
- [Drift Speed of Free Electrons: v_d = eEτ/2m and i = nAev_d](https://physics.sillybits.io/chapters/electric-current/drift-speed): Free electrons and the vibrating lattice, random motion with collisions, the drift caused by a field, the relation between current and drift speed, and why drift speeds are so small.
- [Ohm's Law j = σE and V = Ri, Resistivity, Resistance R = ρl/A, and the Resistor Colour Code](https://physics.sillybits.io/chapters/electric-current/ohms-law): Deriving j = σE from the drift model, conductivity and resistivity, resistance R = ρl/A and conductance, units, limits of Ohm's law, and reading the four-band resistor colour code.
- [Temperature Dependence of Resistivity, Thermistors and Superconductors](https://physics.sillybits.io/chapters/electric-current/temperature-and-resistivity): Why resistivity of metals rises with temperature, ρ(T) = ρ₀[1 + α(T − T₀)], a table of resistivities, conductors, insulators and semiconductors, thermistors and superconductivity.
- [EMF of a Battery, Joule Heating i²Rt, Electric Power, and Internal Resistance V = E − ir](https://physics.sillybits.io/chapters/electric-current/emf-and-energy): How a battery maintains a potential difference, emf as work per unit charge, charging and discharging a battery, thermal energy i²Rt and power, ideal and real batteries and internal resistance.
- [Kirchhoff's Junction Law and Loop Law for Solving Circuits](https://physics.sillybits.io/chapters/electric-current/kirchhoffs-laws): The junction law from charge conservation, the loop law from the conservative electrostatic field, sign conventions, and solving a two-loop circuit.
- [Resistors in Series and Parallel, Current Division, and Grouping of Cells](https://physics.sillybits.io/chapters/electric-current/resistor-combinations): Equivalent resistance, series and parallel combinations, division of current in parallel branches, and the equivalent emf and internal resistance of cells in series and in parallel.
- [Wheatstone Bridge Balance R₁/R₂ = R₃/R₄, Ammeter Shunts, Voltmeter Series Resistance](https://physics.sillybits.io/chapters/electric-current/bridge-and-meters): The Wheatstone bridge and its balance condition, measuring an unknown resistance, the moving-coil ammeter with a shunt, the galvanometer, and the voltmeter with a series resistance.
- [Stretched-Wire Potentiometer: Measuring Voltage, Comparing EMFs and Internal Resistance](https://physics.sillybits.io/chapters/electric-current/potentiometer): How a stretched-wire potentiometer works, balancing and V = V₀l/L, calibration with a standard cell, comparing emfs, and measuring the internal resistance of a cell.
- [RC Circuits: Charging q = EC(1 − e^(−t/CR)), Discharging q = Qe^(−t/CR), and the Time Constant](https://physics.sillybits.io/chapters/electric-current/rc-circuits): Deriving the charge on a capacitor while charging through a resistor and while discharging, the time constant CR, and the 63 %/37 % rule.
- [Atmospheric Electricity: Fair-Weather Field, the Global Circuit, Thunderstorms and Lightning](https://physics.sillybits.io/chapters/electric-current/atmospheric-electricity): The Earth's negative surface charge and fair-weather field, the 400 kV potential of the upper atmosphere, the fair-weather current, and how thunderstorms and lightning keep the global circuit charged.

## Chapter 33: Thermal & Chemical Effects of Current

Joule's laws of heating and their verification, the Seebeck effect, the thermoelectric series, neutral and inversion temperatures, the laws of intermediate metals and temperatures, the Peltier and Thomson effects and their explanation, electrolysis, Faraday's laws, the Faraday constant, voltameters, and primary and secondary cells including the lead accumulator.

- [Joule's Laws of Heating H = i²Rt and Their Verification with Joule's Calorimeter](https://physics.sillybits.io/chapters/thermal-chemical-effects/joules-laws): The heat produced by a current in a resistor, Joule's three laws, verifying them with a Joule calorimeter, power ratings of appliances, and heat inside a shorted cell.
- [Seebeck Effect, Thermoelectric Series, Neutral and Inversion Temperature, and E = aθ + ½bθ²](https://physics.sillybits.io/chapters/thermal-chemical-effects/thermoelectricity): The Seebeck effect in a two-metal loop, the thermoelectric series, neutral and inversion temperatures, the thermo-emf formula and thermoelectric power, and the laws of intermediate metal and intermediate temperatures.
- [Peltier Effect, Thomson Effect, and the Origin of the Seebeck EMF](https://physics.sillybits.io/chapters/thermal-chemical-effects/peltier-and-thomson): The Peltier effect and Peltier emf, how it differs from Joule heating, the Thomson effect and coefficient, and how electron diffusion explains all three thermoelectric effects.
- [Electrolysis, Faraday's Laws m = Zit, Chemical Equivalent, Faraday Constant and Voltameters](https://physics.sillybits.io/chapters/thermal-chemical-effects/electrolysis): How electrolytes conduct, the reactions at the electrodes, Faraday's two laws, electrochemical and chemical equivalents, the ionic explanation, the faraday and Faraday constant, and copper and silver voltameters.
- [Voltaic Cells: Daniell, Leclanché and Dry Cells, and the Lead Accumulator](https://physics.sillybits.io/chapters/thermal-chemical-effects/cells): How a voltaic cell produces an emf, primary versus secondary cells, the Daniell, Leclanché and dry cells, polarization and depolarizers, and the charging and discharging reactions and capacity of a lead accumulator.

## Chapter 34: Magnetic Field

The magnetic field defined through F = qv × B, the tesla and gauss, the unity of electric and magnetic fields, circular and helical motion of charged particles and the cyclotron frequency, the force on a current-carrying wire, and the torque and magnetic dipole moment of a current loop.

- [Magnetic Field B Defined by F = qv × B, the Tesla, and Electric and Magnetic Fields as One Field](https://physics.sillybits.io/chapters/magnetic-field/magnetic-force): Why a neutral current-carrying wire acts on moving but not resting charges, defining B from F = qv × B, its units (tesla, gauss, Wb/m²), and how electric and magnetic fields mix between frames.
- [Motion of a Charged Particle in a Uniform Magnetic Field: r = mv/qB, Cyclotron Frequency and Helical Paths](https://physics.sillybits.io/chapters/magnetic-field/charged-particle-motion): Why the magnetic force does no work, circular motion with r = mv/qB, the speed-independent period and cyclotron frequency, isotope separation, helical paths with radius and pitch, and crossed electric fields.
- [Magnetic Force on a Current-Carrying Wire dF = i dl × B and F = i l × B](https://physics.sillybits.io/chapters/magnetic-field/force-on-current): Deriving the force on a current element from the forces on drifting electrons, the force on a straight wire, zero net force on a closed loop in a uniform field, and a rod on rails.
- [Torque on a Current Loop Γ = μ × B and Magnetic Dipole Moment μ = niA](https://physics.sillybits.io/chapters/magnetic-field/torque-on-loop): The torque on a rectangular loop, the area vector and right-hand rule, magnetic dipole moment niA, the analogy with an electric dipole, and the magnetic moment of an orbiting electron.

## Chapter 35: Magnetic Field due to a Current

The Biot–Savart law and the permeability of vacuum, the field of a straight wire segment and a long wire, the force between parallel currents and the definition of the ampere, the field of a circular loop at its centre and on its axis, magnetic dipoles, Ampère's law, and the fields of solenoids and toroids.

- [Biot–Savart Law dB = (μ₀/4π) i dl × r/r³ and the Magnetic Field of a Straight Wire](https://physics.sillybits.io/chapters/field-due-to-current/biot-savart): The Biot–Savart law, μ₀ = 1/ε₀c², the field of a finite straight segment, on its perpendicular bisector and of a long wire, the right-hand thumb rule, and circular field lines.
- [Force Between Parallel Currents μ₀i₁i₂/2πd and the Definition of the Ampere](https://physics.sillybits.io/chapters/field-due-to-current/parallel-currents): The force per unit length between two long parallel currents, attraction and repulsion, null points of the combined field, and how this force defines the ampere.
- [Magnetic Field of a Circular Current at the Centre μ₀i/2a and on the Axis, and the Magnetic Dipole](https://physics.sillybits.io/chapters/field-due-to-current/circular-current): The field at the centre of a circular loop, its direction by the right-hand rule, the field on the axis, and the far-field magnetic dipole form compared with an electric dipole.
- [Ampère's Circuital Law ∮B·dl = μ₀i and the Field of a Long Wire](https://physics.sillybits.io/chapters/field-due-to-current/amperes-law): Line integrals and circulation, the right-hand sign convention, Ampère's law and its conditions, its equivalence with Biot–Savart, and the long-wire field derived in two lines.
- [Magnetic Field of a Long Solenoid B = μ₀ni and of a Toroid B = μ₀Ni/2πr](https://physics.sillybits.io/chapters/field-due-to-current/solenoid-and-toroid): How a tightly wound solenoid makes a uniform field inside and none outside, Ampère's law for a solenoid, its finite-length behaviour, and the field of a toroid.

## Chapter 36: Permanent Magnets

The magnetic pole model, solenoids and bar magnets as dipoles, torque and energy of a magnet in a field, end-on and broadside-on fields, magnetic scalar potential and the general dipole field, the Earth's magnetism and its elements, the dip circle and apparent dip, neutral points, tangent and moving-coil galvanometers and shunts, deflection and oscillation magnetometers, and Gauss's law for magnetism.

- [Magnetic Pole Model, Pole Strength, Bar Magnets, and Torque and Energy U = −M·B](https://physics.sillybits.io/chapters/permanent-magnets/poles-and-bar-magnets): The analogy between current loops and electric dipoles, magnetic poles and pole strength, a solenoid as two poles, atomic current loops in a bar magnet, magnetic length and moment, and the torque and potential energy of a magnet in a uniform field.
- [Magnetic Field of a Bar Magnet in End-On and Broadside-On Positions, Magnetic Scalar Potential, and the Dipole Field](https://physics.sillybits.io/chapters/permanent-magnets/field-of-a-magnet): Field on the axis (end-on) and perpendicular bisector (broadside-on) of a bar magnet, the magnetic scalar potential of a pole and a dipole, and the radial and transverse components of a dipole's field.
- [Terrestrial Magnetism: Geomagnetic Poles, Declination, Dip, Horizontal Component, Dip Circle and Apparent Dip](https://physics.sillybits.io/chapters/permanent-magnets/earths-magnetism): The Earth's field as a central tilted dipole, geomagnetic poles and reversals, declination, inclination (dip) and horizontal component, measuring dip with a dip circle and correcting its errors, apparent dip, and neutral points.
- [Tangent Law, Tangent Galvanometer i = K tan θ, Moving-Coil Galvanometer and Shunts](https://physics.sillybits.io/chapters/permanent-magnets/galvanometers): The tangent law of perpendicular fields, the tangent galvanometer, its reduction factor and best sensitivity near 45°, the moving-coil galvanometer with radial field and torsion strip, its sensitivity, and protecting it with a shunt.
- [Deflection Magnetometer (Tan-A, Tan-B), Oscillation Magnetometer, and Measuring M and B_H](https://physics.sillybits.io/chapters/permanent-magnets/magnetometers): The deflection magnetometer in Tan-A and Tan-B positions, removing its errors, comparing magnets by the null method, verifying the inverse-square law, comparing B_H at two places, the oscillation magnetometer and its period, and finding M and B_H.
- [Gauss's Law for Magnetism ∮B·dS = 0 and the Absence of Magnetic Monopoles](https://physics.sillybits.io/chapters/permanent-magnets/gauss-law-for-magnetism): Why the absence of isolated magnetic poles makes the magnetic flux through every closed surface zero, and how magnetic and electric field lines differ.

## Chapter 37: Magnetic Properties of Matter

Atomic magnetic moments and electron spin, magnetization, paramagnetism, ferromagnetism and diamagnetism, magnetic domains, magnetic intensity H, susceptibility and permeability, Curie's law and the Curie temperature, a comparison of the three families, hysteresis, and why soft iron and steel are used where they are.

- [Atomic Magnetic Moments, Magnetization I = M/V, and Paramagnetism, Ferromagnetism, Diamagnetism and Domains](https://physics.sillybits.io/chapters/magnetic-properties/magnetization): Orbital and spin magnetic moments, partial alignment and saturation, the intensity of magnetization, why aligned loops strengthen the field, paramagnetism, ferromagnetism with exchange coupling and domains, and diamagnetism.
- [Magnetic Intensity H = B/μ₀ − I, Susceptibility I = χH, and Permeability B = μH](https://physics.sillybits.io/chapters/magnetic-properties/h-susceptibility-permeability): Defining magnetic intensity H, its value in vacuum and in long samples, susceptibility and a table of values, permeability and relative permeability, and cores in solenoids.
- [Curie's Law χ = c/T, Curie Temperature, and Properties of Dia-, Para- and Ferromagnetic Substances](https://physics.sillybits.io/chapters/magnetic-properties/curie-law-and-properties): Curie's law for paramagnets, the Curie point of ferromagnets and the law above it, Curie temperatures, and a comparison of diamagnetic, paramagnetic and ferromagnetic properties including rods in a field.
- [Magnetic Hysteresis Loop, Retentivity and Coercive Force, and Soft Iron versus Steel](https://physics.sillybits.io/chapters/magnetic-properties/hysteresis): Measuring magnetization with a ring and toroid, the hysteresis loop, retentivity, coercive force, energy loss per cycle as loop area, and choosing soft iron or steel.

## Chapter 38: Electromagnetic Induction

Faraday's law and magnetic flux, the direction of induced currents and Lenz's law, motional emf and the energy balance of a moving conductor, the induced electric field, eddy currents and electromagnetic damping, self-inductance and the inductance of a solenoid, growth and decay of current in LR circuits, energy stored in an inductor and magnetic energy density, mutual inductance, and the induction coil.

- [Faraday's Law of Electromagnetic Induction ℰ = −dΦ/dt, Magnetic Flux, and Lenz's Law](https://physics.sillybits.io/chapters/electromagnetic-induction/faradays-law): Faraday's magnet-and-coil experiments, magnetic flux and the weber, Faraday's law and induced current, a procedure for the direction of the induced current, and Lenz's law with examples.
- [Motional EMF ℰ = vBl, a Rod on Rails, and the Energy Balance of Induction](https://physics.sillybits.io/chapters/electromagnetic-induction/motional-emf): How the magnetic force on a moving conductor's free electrons builds up an emf vBl, the equivalent battery, the retarding force on the conductor, and why mechanical power equals the Joule heat.
- [Induced Electric Field ∮E·dl = −dΦ/dt, Eddy Currents and Electromagnetic Damping](https://physics.sillybits.io/chapters/electromagnetic-induction/induced-electric-field): Why a stationary loop's current needs an electric field, the non-conservative induced electric field with closed field lines, its value inside and outside a changing field region, eddy currents, electromagnetic damping and slotted plates.
- [Self-Inductance Φ = Li, Self-Induced EMF −L di/dt, the Henry, and Inductance of a Solenoid](https://physics.sillybits.io/chapters/electromagnetic-induction/self-induction): Self-inductance and the self-induced emf, the henry, finding L from a current reversal, the inductance of a long solenoid, ideal inductors, and how an inductor responds to a changing resistance.
- [Growth and Decay of Current in an LR Circuit, Time Constant L/R, Energy ½Li² and Energy Density B²/2μ₀](https://physics.sillybits.io/chapters/electromagnetic-induction/lr-circuits): Deriving the exponential growth and decay of current in an LR circuit, the time constant L/R, the energy stored in an inductor, and the energy density of a magnetic field.
- [Mutual Inductance Φ = Mi, EMF −M di/dt, Coaxial Solenoids, and Ruhmkorff's Induction Coil](https://physics.sillybits.io/chapters/electromagnetic-induction/mutual-induction): Mutual inductance and its symmetry, the mutual inductance of coaxial solenoids, the charge driven through a coupled coil, and how an induction coil makes high voltage, including the role of its capacitor.

## Chapter 39: Alternating Current

Alternating current and the AC generator, instantaneous, mean and rms values, resistors, capacitors and inductors on AC with reactance and phase, impedance, the phasor method for CR, LR and LCR circuits, resonance and LC oscillations, average power and power factor, the choke coil, hot-wire meters, DC dynamos and motors, and transformers and power transmission.

- [Alternating Current i = i₀ sin(ωt + φ) and the AC Generator ℰ = NBAω sin ωt](https://physics.sillybits.io/chapters/alternating-current/ac-generator): What alternating current is, the parts of an AC generator (magnet, armature, slip rings, brushes), the emf of a rotating coil, and how power stations and gensets use it.
- [Mean, Mean-Square and RMS Current i_rms = i₀/√2, and Why AC Is Rated in RMS](https://physics.sillybits.io/chapters/alternating-current/rms-values): Why alternating current averages to zero, the mean square and root-mean-square current and voltage, the heating equivalence that makes rms useful, and rms of other waveforms.
- [AC Through a Resistor, Capacitor and Inductor: Reactance X_C = 1/ωC, X_L = ωL, Phase and Impedance](https://physics.sillybits.io/chapters/alternating-current/reactance): Current through a pure resistor, capacitor and inductor on AC, capacitive and inductive reactance and their frequency dependence, why a capacitor passes AC but not DC, phase relations, impedance and phase factor.
- [Phasor Method for Series CR, LR and LCR Circuits, Resonant Frequency 1/2π√(LC), and LC Oscillations](https://physics.sillybits.io/chapters/alternating-current/lcr-and-resonance): Using phasor diagrams to find impedance and phase, CR and LR circuits, the series LCR circuit, resonance and sharpness, radio tuning, and LC oscillations.
- [Average Power in AC P = ℰ_rms i_rms cos φ, Power Factor, Choke Coil and Hot-Wire Instruments](https://physics.sillybits.io/chapters/alternating-current/power-and-choke): Deriving average power in an AC circuit, the power factor and wattless current, using a choke coil to reduce voltage without loss, and hot-wire ammeters and voltmeters.
- [DC Dynamo and Commutator, DC Motor and Back EMF, Transformer Ratios and Power Transmission](https://physics.sillybits.io/chapters/alternating-current/dynamos-motors-transformers): How a commutator makes a DC dynamo, smoothing with more coils, the DC motor and back emf, the transformer's voltage and current ratios and power transfer, step-up and step-down, efficiency, and high-voltage transmission.

## Chapter 40: Electromagnetic Waves

Maxwell's displacement current and the generalised Ampère law, continuity of current, Maxwell's equations, plane electromagnetic waves with E₀ = cB₀ and c = 1/√(μ₀ε₀), energy density, intensity, momentum and radiation pressure, the electromagnetic spectrum and its sources, and radiation in the atmosphere.

- [Maxwell's Displacement Current i_d = ε₀ dΦ_E/dt and the Ampère–Maxwell Law](https://physics.sillybits.io/chapters/electromagnetic-waves/displacement-current): Why Ampère's law fails for a charging capacitor, the displacement current, the Ampère–Maxwell law, the magnetic field between capacitor plates, and continuity of current.
- [Maxwell's Equations, Plane Electromagnetic Waves, E₀ = cB₀ and c = 1/√(μ₀ε₀)](https://physics.sillybits.io/chapters/electromagnetic-waves/maxwell-and-em-waves): The four Maxwell equations, their vacuum form, checking that a plane wave satisfies Faraday's and the Ampère–Maxwell laws, the field ratio E₀ = cB₀, and the speed c = 1/√(μ₀ε₀).
- [Energy Density ½ε₀E₀², Intensity ½ε₀E₀²c, Momentum U/c and Radiation Pressure of EM Waves](https://physics.sillybits.io/chapters/electromagnetic-waves/energy-and-momentum): Electric and magnetic energy densities in a wave and their equality on average, intensity, the momentum carried by electromagnetic waves, and radiation pressure on absorbing and reflecting surfaces.
- [The Electromagnetic Spectrum from Gamma Rays to Radio Waves, Their Sources, and Radiation in the Atmosphere](https://physics.sillybits.io/chapters/electromagnetic-waves/em-spectrum): The bands of the electromagnetic spectrum, how each is produced and used, the layers of the atmosphere, and how solar radiation of different wavelengths is absorbed, including by the ozone layer.

## Chapter 41: Electric Current through Gases

How gases are made to conduct, the discharge tube, sparking potential and Paschen's law, the glows and dark spaces at falling pressure and their explanation, cathode rays and canal rays, Thomson's e/m and Millikan's oil-drop experiments, thermionic emission and the Richardson–Dushman equation, the diode valve and rectification, and the triode and its use as an amplifier.

- [Discharge Tube, Sparking Potential, Paschen's Law, and Crookes and Faraday Dark Spaces](https://physics.sillybits.io/chapters/current-through-gases/gas-discharge): Why gases normally insulate, the discharge tube, sparking potential and Paschen's law, the sequence of glows and dark spaces as pressure falls, and how ionization, space charge and mean free path explain them.
- [Properties of Cathode Rays and Canal (Positive) Rays](https://physics.sillybits.io/chapters/current-through-gases/cathode-rays): The experimental properties of cathode rays, their deflection by electric and magnetic fields showing negative charge, their identification as electrons, and canal rays of positive ions.
- [Thomson's e/m Experiment E²/2VB² and Millikan's Oil-Drop Determination of e](https://physics.sillybits.io/chapters/current-through-gases/electron-e-and-m): Thomson's role in discovering the electron, the crossed-field method for e/m, Millikan's oil-drop experiment with balancing and terminal speed, quantization of charge, and the electron's mass.
- [Thermionic Emission, Work Function, Richardson–Dushman Equation, Diode Valve Characteristics and Rectification](https://physics.sillybits.io/chapters/current-through-gases/thermionic-diode): Thermionic emission and the work function, the Richardson–Dushman equation, the diode valve, space charge, the Langmuir–Child law and saturation, dynamic plate resistance, and half- and full-wave rectification.
- [Triode Valve, Plate Resistance, Mutual Conductance, Amplification Factor μ = r_p g_m and Voltage Gain](https://physics.sillybits.io/chapters/current-through-gases/triode): How a grid controls the plate current, anode and mutual characteristics, dynamic plate resistance, transconductance and amplification factor, and the voltage gain of a triode amplifier.

## Chapter 42: Photoelectric Effect & Wave–Particle Duality

Photons and their energy and momentum, photon flux and radiation force, the photoelectric effect and Einstein's equation, work functions and threshold wavelength, the photocell experiment, stopping potential and saturation current, why the wave theory fails, and de Broglie's matter waves.

- [Photon Theory of Light: Energy E = hν = hc/λ and Momentum p = h/λ](https://physics.sillybits.io/chapters/photoelectric-effect/photons): Properties of photons, their energy and momentum, the quantum nature of light energy, counting photons in a beam, and the momentum and force delivered by photons.
- [Photoelectric Effect, Einstein's Equation K_max = hν − φ, Threshold Wavelength and Stopping Potential](https://physics.sillybits.io/chapters/photoelectric-effect/photoelectric-effect): Einstein's photoelectric equation, work functions, threshold wavelength and frequency, the photocell apparatus, saturation current and stopping potential, the V₀–ν graph, and why the wave theory fails.
- [De Broglie Matter Waves λ = h/p and the Wave Nature of Electrons](https://physics.sillybits.io/chapters/photoelectric-effect/matter-waves): Evidence that electrons and other particles behave as waves, de Broglie's relation, wavelengths of accelerated electrons, protons and everyday objects, the electron microscope, and the limits of classical mechanics.

## Chapter 43: Bohr's Model & Physics of the Atom

Thomson's, Lenard's and Rutherford's atomic models and alpha scattering, the hydrogen spectrum and Rydberg formula, why the classical nuclear atom fails, Bohr's postulates, orbit radii and energy levels, ionization and excitation energies, hydrogen-like ions, the wave-function picture and quantum numbers, spin, shells and the Pauli principle, and stimulated emission and the laser.

- [Thomson's, Lenard's and Rutherford's Atomic Models and the Alpha-Scattering Experiment](https://physics.sillybits.io/chapters/bohrs-model/atomic-models): The path from atomism to the electron, Thomson's plum-pudding model, Lenard's empty atom, Rutherford's gold-foil experiment, the size of the nucleus, and the nuclear model of the atom.
- [Hydrogen Line Spectrum, Lyman, Balmer and Paschen Series, the Rydberg Formula, and the Failure of the Classical Atom](https://physics.sillybits.io/chapters/bohrs-model/hydrogen-spectrum): Continuous and line spectra, the series of hydrogen lines and their limits, the Rydberg formula, Balmer's discovery, and why an orbiting electron should radiate and collapse.
- [Bohr's Postulates, Radii r = n²a₀/Z, Energies −13.6Z²/n² eV, Hydrogen Series, Ionization and Excitation](https://physics.sillybits.io/chapters/bohrs-model/bohr-model): Bohr's four postulates, the speed, radius and energy of each orbit for hydrogen-like atoms, ground and excited states, how jumps produce each spectral series, ionization, binding and excitation energies, and the model's limitations.
- [Wave Function of the Electron, Schrödinger Equation for Hydrogen, Quantum Numbers, Spin, Shells and the Pauli Exclusion Principle](https://physics.sillybits.io/chapters/bohrs-model/quantum-atom): The electron wave function and probability, the Schrödinger equation for hydrogen-like atoms, quantum numbers n, l, m_l, the ground-state wave function and radial probability, spin, shells and subshells, and the Pauli exclusion principle.
- [Stimulated Emission, Metastable States, Population Inversion, the He–Ne Laser and Uses of Lasers](https://physics.sillybits.io/chapters/bohrs-model/laser): Absorption, spontaneous and stimulated emission, lifetimes and metastable states, population inversion and pumping, the mirror cavity, how the helium–neon laser works, and uses of lasers.

## Chapter 44: X-rays

Röntgen's discovery and the Coolidge tube, continuous X-rays and the cutoff wavelength, characteristic Kα, Kβ and L lines and the energy levels of an atom with an inner vacancy, intensity versus hardness, Moseley's law and the ordering of the periodic table, Bragg's law and crystal diffraction, and the properties, uses and dangers of X-rays.

- [Coolidge Tube, Continuous and Characteristic X-rays, Cutoff Wavelength λmin = hc/eV, Kα and Kβ Lines, Hard and Soft X-rays](https://physics.sillybits.io/chapters/x-rays/production-and-spectrum): How a Coolidge tube works, why the continuous spectrum ends sharply at λmin = hc/eV, how inner-shell vacancies give characteristic Kα, Kβ and Lα lines, the atom-energy picture, and how filament current and voltage control intensity and hardness.
- [Moseley's Law √ν = a(Z − b), Its Explanation by Bohr's Model, and the Ordering of the Periodic Table](https://physics.sillybits.io/chapters/x-rays/moseley-law): Moseley's straight-line relation between the square root of the Kα frequency and atomic number, its derivation from Bohr's model with screening, the constants a and b, and how it fixed the order of elements and predicted missing ones.
- [Bragg's Law 2d sin θ = nλ, X-ray Crystallography, Properties of X-rays, and Their Uses and Dangers](https://physics.sillybits.io/chapters/x-rays/bragg-and-uses): Why crystals diffract X-rays, Bragg reflection from atomic planes, measuring wavelengths and crystal structure, the properties of X-rays, and their uses in medicine, industry, security and science, with their dangers.

## Chapter 45: Semiconductors & Semiconductor Devices

Energy bands in solids and why metals, insulators and semiconductors differ, holes and intrinsic conduction, n-type and p-type doping, carrier densities, conductivity and temperature, the p-n junction and diode, dynamic resistance, photodiodes, LEDs and Zener diodes, rectifiers, junction transistors and amplifiers, oscillators, and logic gates up to the full adder.

- [Energy Bands, Band Gaps, Valence and Conduction Bands, and Why Metals, Insulators and Semiconductors Differ](https://physics.sillybits.io/chapters/semiconductors/energy-bands): How atomic levels spread into bands when atoms pack together, why only electrons with empty states nearby can conduct, the four band pictures (sodium-type and zinc-type metals, insulators, semiconductors), and valence and conduction bands.
- [Intrinsic Semiconductors, Holes, n-type and p-type Doping, Carrier Density, σ = e(nₑμₑ + nₕμₕ) and Temperature Dependence](https://physics.sillybits.io/chapters/semiconductors/holes-and-doping): Broken bonds and holes, intrinsic conduction, donor and acceptor impurities and their impurity levels, carrier densities and the mass-action rule, mobility and conductivity, and why semiconductor resistivity falls with temperature.
- [p-n Junction, Depletion Layer, Diffusion and Drift Currents, Forward and Reverse Bias, Dynamic Resistance, Photodiode, LED and Zener Diode](https://physics.sillybits.io/chapters/semiconductors/pn-junction): How the depletion layer and barrier form, diffusion and drift currents, forward and reverse bias and the diode characteristic, dynamic resistance, photodiodes and LEDs, avalanche and Zener breakdown, and the Zener voltage regulator.
- [Half-wave and Full-wave Rectification with p-n Junction Diodes](https://physics.sillybits.io/chapters/semiconductors/rectifiers): How a single diode gives half-wave rectification, how two diodes and a centre-tapped transformer give full-wave rectification, and the average output of each.
- [n-p-n and p-n-p Transistors, Biasing, α and β, Common-Emitter Amplifier Gains, Transconductance, Oscillators and Characteristics](https://physics.sillybits.io/chapters/semiconductors/transistors): Transistor structure and doping, biasing and the three configurations, how carriers cross the base, I_E = I_B + I_C, α and β, the common-emitter amplifier and its current, voltage and power gains, transconductance, the LC oscillator, and input and output characteristics.
- [Logical Variables, Truth Tables, AND, OR, NOT, NAND, NOR and XOR Gates, Diode and Transistor Gates, Universal Gates, Half and Full Adders](https://physics.sillybits.io/chapters/semiconductors/logic-gates): Logical variables and truth tables, AND, OR and NOT with switches, gates from diodes and transistors, NAND, NOR and XOR, building any gate from NAND gates, binary arithmetic, and half and full adders.

## Chapter 46: The Nucleus

Protons, neutrons and atomic mass units, isotopes, isobars and isotones, the valley of stability, nuclear radius and density, the nuclear force, binding energy and the liquid-drop model, alpha, beta and gamma decay with Q-values, the radioactive decay law, half-life and dating, fission and nuclear reactors, and fusion in stars and on Earth.

- [Nuclear Constituents, Atomic Mass Unit, Isotopes, Isobars and Isotones, Nuclear Stability, Radius R = R₀A^(1/3) and the Nuclear Force](https://physics.sillybits.io/chapters/the-nucleus/nuclear-properties): Protons and neutrons, masses in MeV/c² and u, Z, N and A, isotopes, isotones, isobars and nuclides, the chart of stable nuclides, nuclear radius and constant density, nuclear spin, and the properties of the nuclear force.
- [Mass Defect, Binding Energy, Binding Energy per Nucleon Curve, the Liquid-Drop Model and Mass Excess](https://physics.sillybits.io/chapters/the-nucleus/binding-energy): How the binding energy follows from the mass defect, the binding-energy-per-nucleon curve and its peak near iron, the volume, surface and Coulomb terms, mass excess, and why fission and fusion both release energy.
- [Alpha Decay, Q-value, Beta-minus and Beta-plus Decay, Neutrinos, Electron Capture, Decay Series, Gamma Decay and Properties of Nuclear Radiation](https://physics.sillybits.io/chapters/the-nucleus/radioactive-decay): Alpha decay and its Q-value, beta-minus and beta-plus decay with neutrinos and antineutrinos, the continuous beta spectrum, electron capture, the uranium-238 decay series, gamma decay from excited nuclei, and the properties of alpha, beta and gamma rays.
- [Decay Constant, N = N₀e^(−λt), Activity, Becquerel and Curie, Half-life, Average Life and Radioactive Dating](https://physics.sillybits.io/chapters/the-nucleus/decay-law): The probabilistic nature of decay, the exponential decay law, activity and its units, half-life and average life, carbon and uranium dating, parallel decay routes and decay chains, and uses of radioactivity.
- [Energy from Fission, Barrier Penetration, Uranium-235 Fission, Chain Reactions, Moderators, Control Rods and Breeder Reactors](https://physics.sillybits.io/chapters/the-nucleus/fission): Why splitting heavy nuclei releases energy, the deformation barrier and tunnelling, fission of uranium-235 by thermal neutrons, fragments and delayed neutrons, chain reactions and criticality, the parts of a nuclear reactor, and breeder reactors.
- [Fusion of Light Nuclei, Thermonuclear Temperatures, the Proton–Proton and Carbon Cycles, D–T Fusion, the Lawson Criterion, Tokamaks and Inertial Confinement](https://physics.sillybits.io/chapters/the-nucleus/fusion): Why fusing light nuclei releases energy, the Coulomb barrier and required temperatures, the proton–proton and carbon cycles in stars, fusion reactions usable on Earth, the Lawson criterion, and magnetic and inertial confinement.

## Chapter 47: The Special Theory of Relativity

The principle of relativity, why the constancy of the speed of light forces new ideas of time and length, Einstein's postulates, time dilation and proper time with the light clock, length contraction, the relativity of simultaneity and clock synchronization, relativistic momentum, E = mc² and E² = m₀²c⁴ + p²c², the ultimate speed, and the twin paradox.

- [The Principle of Relativity, the Constancy of the Speed of Light, the Michelson–Morley Experiment and Einstein's Two Postulates](https://physics.sillybits.io/chapters/special-relativity/postulates): Why no experiment can detect uniform motion, how Maxwell's equations fix the speed of light, the puzzle this creates for distances and times, the Michelson–Morley null result, and Einstein's two postulates.
- [Transverse Lengths, the Light Clock, Time Dilation Δt′ = γΔt, the Lorentz Factor and Proper Time](https://physics.sillybits.io/chapters/special-relativity/time-dilation): Why lengths perpendicular to motion don't change, the light-clock derivation of time dilation, the Lorentz factor γ, proper and improper time intervals, and worked cases from trains to satellites.
- [Length Contraction L′ = L/γ, Rest Length, the Relativity of Simultaneity and Synchronization of Moving Clocks](https://physics.sillybits.io/chapters/special-relativity/length-and-simultaneity): How the light clock moving along its length gives length contraction, rest length, why events simultaneous in one frame are not in another, the time lag Lvγ/c², and why moving clocks appear out of synchronization.
- [Relativistic Momentum p = γm₀v, Motion under a Constant Force, E = mc², Kinetic Energy, E² = m₀²c⁴ + p²c² and the Ultimate Speed](https://physics.sillybits.io/chapters/special-relativity/mass-and-energy): Relativistic momentum and moving mass, why a constant force never reaches c, mass–energy equivalence, rest energy and kinetic energy, the energy–momentum relation and photons, and why nothing travels faster than light.
- [The Twin Paradox Resolved with Time Dilation, Length Contraction and the Relativity of Simultaneity](https://physics.sillybits.io/chapters/special-relativity/twin-paradox): A round trip at 0.6c analysed from Earth and from the traveller's ships, why each twin seems to see the other age slower, how changing frames shifts simultaneity, and the spacetime-diagram view.