Capacitors

Put two conductors close together, push charge onto one and pull the same amount off the other, and you have a device that stores charge and energy: a capacitor.

Capacitor and capacitance#

• The charge on the capacitor means Q, the charge on the positive plate. (The total charge, +Q − Q, is zero.)
• The potential of the capacitor means the potential difference V = V₊ − V₋ between the plates.

For a given capacitor, double the charge and the potential difference doubles too: Q is proportional to V.

The constant C, the capacitance, depends only on the shape, size and arrangement of the conductors and on what fills the space between them. Its unit is the coulomb per volt, called the farad (F). A farad is enormous; everyday capacitors are measured in microfarads (µF), nanofarads (nF) or picofarads (pF).

Charging with a battery#

The easy way to put ±Q on the plates is to connect them to a battery. For now we treat the battery as ideal:

• It has two terminals with a fixed potential difference between them, equal to its emf ℰ. The higher-potential terminal is positive.
• A conductor joined to a terminal takes that terminal’s potential, so a capacitor across the battery ends up with V = ℰ.
• Its net charge stays zero: if the + terminal supplies +Q, the − terminal supplies −Q.
• Moving charge Q through itself from − to +, the battery does work Qℰ.

In circuit diagrams a battery is drawn as a long and a short line (the long line is positive), and a capacitor as two equal parallel lines.

Finding a capacitance#

The recipe is always the same: place +Q and −Q on the plates, find the field between them (usually with Gauss’s law), integrate it to get V, then C = Q/V.

The parallel-plate capacitor#

Two large flat plates, each of facing area A, a small distance d apart, with vacuum between. The charges sit on the facing surfaces with density σ = Q/A. If the plates are large compared with d, the field between them is uniform and perpendicular to the plates (except near the edges).

A pillbox with one face inside the positive plate (no field there) and one face in the gap encloses charge σΔA, so Gauss’s law gives EΔA = σΔA/ε₀, i.e. E = Q/ε₀A. Going across the gap,

The same recipe (charge the plates, find E, integrate to V) works for curved conductors. Spheres and cylinders are the two shapes symmetric enough to do by hand.

Spherical capacitor#

A conducting sphere of radius R₁ sits inside a concentric hollow conducting sphere of radius R₂. Give the inner one +Q and the outer one −Q. Between them, a concentric Gaussian sphere of radius r encloses Q, so E = Q/4πε₀r², radially outward. Integrating from R₂ in to R₁:

Two limits worth knowing#

Isolated sphere. Send the outer sphere off to infinity (R₂ → ∞). Then R₁R₂/(R₂ − R₁) → R₁ and

Parallel-plate limit. Make both radii large but keep the gap d = R₂ − R₁ fixed. Then 4πR₁R₂ ≈ 4πR² = A, the area of either sphere, and C → ε₀A/d: the curved plates look flat from up close.

Cylindrical capacitor#

A conducting cylinder of radius R₁ inside a coaxial hollow cylinder of radius R₂, both of length l, much longer than the gap. With +Q inside and −Q outside, the charge per length is Q/l. A coaxial Gaussian cylinder of radius r and length x gives E · 2πrx = (Q/l)x/ε₀, so E = Q/2πε₀rl. Integrating across the gap:

Real circuits join capacitors together. Any network with two terminals behaves like one capacitor, whose capacitance we can work out.

Equivalent capacitance#

Connect a battery across the two terminals of a network. If it supplies charge Q (+Q from one terminal, −Q from the other) at potential difference V, the network’s equivalent capacitance is C = Q/V. A single capacitor of that value would store the same charge at the same voltage.

Series#

Capacitors joined end to end, one after another. The battery puts +Q on the first plate and −Q on the last. Each inner pair of connected plates is isolated, so if one plate gets −Q, its partner must be left with +Q. Result: every capacitor in series carries the same charge Q. The voltages add:

Parallel#

Capacitors connected across the same two points. All their positive plates are at one potential and all their negative plates at another, so each capacitor in parallel has the same voltage V. The charges add: Q = C₁V + C₂V + C₃V.

A general method#

When a network is not simply series and parallel, go back to first principles:

1. Name the two terminals P and N.
2. Imagine a battery with + at P and − at N, supplying +Q and −Q.
3. Write the charge on every plate. Facing plates carry equal and opposite charge; isolated groups of connected plates keep zero total charge. Use unknowns Q₁, Q₂ … where needed.
4. Set V_N = 0 and V_P = V; connected plates share a potential. Use unknowns V₁, V₂ … where needed.
5. Write Q = CV for each capacitor and eliminate the unknowns to get C = Q/V.

Example of the method

The plates of a charged capacitor pull on each other. Work done against that pull is stored as energy, and the energy turns out to live in the electric field itself.

Force between the plates#

Give a parallel-plate capacitor +Q and −Q. The positive plate on its own makes a field E₊ = σ/2ε₀ = Q/2Aε₀ (a large charged sheet). The negative plate sits in that field and feels F = QE₊:

Energy stored in a capacitor#

Start with the plates touching, give them ±Q, then slowly pull one plate away to a distance d. The force is constant (it doesn’t depend on the gap), so you do work W = Fd = Q²d/2Aε₀ = Q²/2C, where C is the final capacitance. That work is stored as electrostatic potential energy:

The energy is in the field#

Pull the plates from d₁ to d₂ with Q fixed. You do extra work Q²(d₂ − d₁)/2Aε₀. Meanwhile a new region of field appears, of volume A(d₂ − d₁). Dividing, the energy per unit volume of the field is

An insulator has no free electrons, but its charges can still shift a little. In a field those tiny shifts add up to charge on its surfaces, and that charge weakens the field inside.

Polar and nonpolar molecules#

Every atom has a positive nucleus surrounded by electrons. What happens in a field depends on whether the centres of positive and negative charge already sit apart.

• Polar materials (water, for example): each molecule has a permanent dipole moment p⃗. Normally they point every which way, so any sizeable volume has zero net dipole moment. A field twists them toward its own direction, while thermal motion keeps scrambling them, so the alignment is only partial.
• Nonpolar materials: the centres coincide and there is no permanent dipole. A field pulls the electron cloud slightly against itself, inducing a small dipole in every atom or molecule.

Either way, a dielectric in a field acquires dipole moment throughout its volume. This is called polarization.

Bound charge on the surfaces#

Line up a slab’s dipoles from left to right. Inside, the head of each dipole sits next to the tail of the next, so the interior stays neutral. But the left face is left with a layer of negative ends and the right face with positive ends. This surface charge is the induced or bound charge: it comes from charges shifted slightly while still bound to their atoms.

For a slab of length l and face area A with bound charge density σ_p, the dipole moment is (σ_pA)l, and dividing by the volume Al:

Dielectric constant#

The bound charges make their own field E⃗_p, opposite to the applied field E⃗₀ in a uniform (homogeneous, isotropic) material. The total field E⃗ = E⃗₀ + E⃗_p points the same way as E⃗₀ but is weaker:

In a very strong field outer electrons are ripped from their atoms and the insulator starts to conduct: dielectric breakdown. The lowest field that causes it is the material’s dielectric strength.

Slide an insulator between the plates of a capacitor and its capacitance goes up by a factor K. This lesson shows why, and what it does to Gauss’s law and to stored energy.

Filling the gap#

A parallel-plate capacitor (area A, gap d) carries ±Q and its gap is filled with a dielectric of constant K. The plate charges alone would make E₀ = Q/ε₀A. The dielectric cuts this to E = E₀/K = Q/ε₀AK, so the potential difference is V = Ed = Qd/ε₀AK and

This is often used as the definition of K: the factor by which a material multiplies the capacitance when it fills the gap.

How much bound charge?#

The bound charges ±Q_p on the slab faces make a field Q_p/ε₀A opposing E₀, so E = (Q − Q_p)/ε₀A. Setting this equal to Q/ε₀AK:

Gauss’s law with dielectrics#

Draw a Gaussian pillbox around the + plate’s inner face in a filled capacitor. It encloses the free charge Q and bound charge −Q_p, so ∮E⃗·dS⃗ = (Q − Q_p)/ε₀ = Q/ε₀K. Moving K to the left:

This holds for any homogeneous, isotropic dielectric. A version valid for every material uses the polarization. The bound charges make E⃗_p = −P⃗/ε₀, so ε₀E⃗ + P⃗ = ε₀E⃗₀, and since E⃗₀ comes from the free charge alone:

A point charge inside a dielectric#

Surround a charge q in a large dielectric with a sphere of radius r. Using (31.14), KE · 4πr² = q/ε₀:

The field is K times weaker than in vacuum. Physically, the medium’s negative ends crowd round a positive q, putting bound charge −q(1 − 1/K) on the tiny cavity around it, so the field outside sees only q/K.

Energy in a dielectric#

For a filled parallel-plate capacitor, U = ½CV² = ½(Kε₀A/d)V², spread through volume Ad:

For the same field, a dielectric stores K times more energy per volume than vacuum: the interacting dipoles hold the extra.

On a charged conductor, charge piles up where the surface is most sharply curved. Near sharp points the field gets strong enough to tear air apart, and the same physics sets the limit of a van de Graaff generator.

Two spheres joined by a wire#

Spheres A (radius r₁) and B (radius r₂ < r₁) are connected by a wire and share charge Q. Joined, they must be at the same potential:

The smaller sphere ends up with the larger charge density, and since the field just outside a conductor is σ/ε₀, with the stronger field too.

Corona discharge#

Any conductor behaves the same way. Flatter parts are like pieces of a big sphere and get little charge; sharply curved parts (small radius of curvature) get a lot. At a needle point the field can exceed the breakdown strength of air (about 3 × 10⁶ V/m). The air there conducts, charge leaks off, and often a faint glow surrounds the point. This is corona discharge.

The van de Graaff generator#

Designed in 1929 to reach potentials around 10⁷ V, it moves charge onto a large hollow metal dome:

1. An insulating belt runs over two pulleys, one inside the dome and one near the base, driven by a motor.
2. A comb of needles near the bottom is held at about +10⁴ V. Corona at its points makes the air conduct; positive charge settles on the belt.
3. The belt carries this charge up into the dome. Corona at a second comb, connected to the dome, transfers it to the dome.
4. Charge inside a hollow conductor goes straight to its outer surface, so the dome keeps accepting more, whatever its potential.

For a sphere, E just outside = Q/4πε₀R² and V = Q/4πε₀R, so V = ER. The dome stops charging when E reaches air’s dielectric strength: