Electric Field & Potential

Two protons a centimetre apart push each other apart with a force about 10³⁶ times their gravitational pull. Mass can’t explain that. Particles carry a second property: electric charge.

A force much stronger than gravity#

Electrons (9.11 × 10⁻³¹ kg), protons (1.673 × 10⁻²⁷ kg) and neutrons (1.675 × 10⁻²⁷ kg) all attract gravitationally. But two protons 1 cm apart repel with about 2.3 × 10⁻²⁴ N, whereas gravity between them is only about 1.9 × 10⁻⁶⁰ N. Two electrons repel equally strongly. Two neutrons only attract, with exactly the tiny gravitational force. The extra force is the electric force, and the property responsible is charge; neutrons have none.

Two kinds of charge#

A proton and an electron 1 cm apart attract with the same 2.3 × 10⁻²⁴ N. Their charges are equally strong but of opposite kinds, and packed together they cancel. By convention the proton’s charge is called positive and the electron’s negative: the labels mean “opposite”, not “more” and “less”.

Unit, quantization and conservation#

Charge is quantized

A body with n₁ protons and n₂ electrons has charge (n₁ − n₂)e. Every known charged particle carries a whole multiple of e, so charge changes only in steps of e. These steps are tiny: rubbing a glass rod gives charges of about a microcoulomb, roughly 6 × 10¹² electrons, so we usually treat charge as continuous.

Charge is conserved

The total charge of an isolated system never changes. Charged particles can be created or destroyed, but only in ways that keep the net charge the same: in beta decay a neutron becomes a proton and a new electron, zero before and after.

Charging by friction#

The Greeks knew around 600 BC that amber rubbed with wool attracts bits of straw. Combs through dry hair, cars driving through air, paper running through a press all get charged the same way: rubbing transfers some electrons from one body to the other. Rub glass with silk and electrons move from the glass to the silk; the glass is left positive, the silk negative.

Induction

Why does a charged rod pick up uncharged paper? A positive rod pulls the paper’s electrons toward its near edge, leaving the far edge positive. This redistribution is induction. The attracted negative charge is closer than the repelled positive charge, so the attraction wins.

Coulomb measured how the electric force depends on charge and distance. The answer looks just like Newton’s law of gravitation, with charge in place of mass.

The law#

The force acts along the line joining the charges: repulsive for like charges, attractive for unlike. In vector form, with r⃗ the position of the charge feeling the force relative to the one exerting it:

Charges don’t reach across space to tug each other directly. Each fills the space around it with an electric field, and the field pushes on other charges.

The electric field#

A charge produces an electric field around it, which exerts forces on other charges placed there. The field exists whether or not anything is there to feel it, and it takes time to change: move a charge and the field a distance r away changes only after r/c.

Field of a point charge

Continuous charge: a ring

For spread-out charge, add the fields of small pieces dQ by integration. For a ring of radius a with charge Q, at a point on its axis a distance x from the centre, the sideways components cancel by symmetry and the axial parts add:

Lines of force#

Field lines are drawn so that the tangent at each point gives the field direction there. From a positive point charge they spread straight outward; into a negative one they converge. Drawn evenly from each charge, the number of lines crossing a small area at right angles is proportional to the field strength: crowded lines, strong field.

Push two like charges together and you do work against their repulsion. That work is stored as electric potential energy, ready to fling them apart again.

Defining the change#

When charges in a system move, the electric forces between them may do work. The change in electric potential energy is defined as the negative of the work done by the electric forces as the configuration changes.

Hold q₁ fixed and move q₂ from distance r₁ to r₂. In a small step dr the electric force does work (q₁q₂/4πε₀r²) dr; adding up,

Choosing zero at infinity#

Taking U = 0 when the charges are infinitely far apart:

For three or more charges, add the potential energies of every pair.

A field is a vector at every point. Potential describes the same field with just one number at every point, which makes many problems far easier.

Potential difference#

Move a test charge q from A to B while the other charges stay fixed. If the electric potential energy changes by U_B − U_A:

Conversely, taking a charge q through a potential difference V_B − V_A changes its potential energy by q(V_B − V_A).

Potential at a point#

Choose a reference point P where V = 0 (usually infinitely far from all the charges). Then

Potential is a scalar: potentials from several charges simply add.

Potential of a point charge#

Bringing q from infinity to distance r from Q changes the energy by Qq/4πε₀r, so

Field and potential are two descriptions of one thing. Knowing either, you can find the other: integrate the field to get potential, differentiate the potential to get the field.

From field to potential#

Displace a test charge q by dr⃗. The field does work qE⃗·dr⃗, so U changes by −qE⃗·dr⃗ and V by:

From potential to field#

Writing dV = −Eₓdx − E_y dy − E_z dz and changing one coordinate at a time:

More generally, for a step dr at angle θ to the field, dV = −E dr cos θ, so

• −dV/dr gives the component of E in the direction you step.
• Stepping along the field (θ = 0), V falls fastest: the field points the way the potential decreases most steeply.
• Stepping at right angles to the field, V doesn’t change.

Equipotential surfaces#

A surface on which the potential is the same everywhere is an equipotential surface. Since V doesn’t change along it, the field has no component along it: the field is perpendicular to equipotentials everywhere. For a point charge they are concentric spheres.

A water molecule is neutral overall but slightly positive at one end and negative at the other. Such a pair of separated charges, a dipole, has a field of its own and turns in an applied field.

Dipole moment#

Potential of a dipole#

At P, distance r ≫ d from the dipole’s centre, at angle θ to its axis, the distances to the charges are about r ∓ (d/2) cos θ. Adding the two potentials and neglecting d² against r²:

In fact any charge distribution whose potential has this form is called a dipole, with dipole moment p along the direction θ is measured from.

Field of a dipole#

The radial and transverse field components come from differentiating V:

Two special positions

• End-on (on the axis, θ = 0): V = p/4πε₀r², E = 2p/4πε₀r³ along p⃗.
• Broadside-on (on the bisector, θ = 90°): V = 0, E = p/4πε₀r³, opposite to p⃗.

A dipole in a uniform field#

Torque

Forces qE⃗ and −qE⃗ cancel, but they don’t act along the same line. Their torques about the centre add to

Potential energy

Turning the dipole by dθ against the torque changes its energy by pE sin θ dθ. Taking U = 0 when the dipole is perpendicular to the field:

Put a material in an electric field and what happens depends on whether its electrons are free to roam. That single fact divides materials into conductors, insulators and semiconductors.

Three kinds of material#

Semiconductors behave as insulators at 0 K, but at higher temperatures a few electrons break free and respond to a field; the vacancies they leave behind also help conduction. They sit between conductors and insulators (a later chapter covers them).

No field inside a conductor#

A neutral metal plate has free electrons spread evenly. Switch on a field from left to right: the electrons drift left, the left face becomes negative and the right face positive. These charges make an internal field pointing right to left, which grows as more electrons drift. The drift stops when it exactly cancels the applied field. This happens in well under a millisecond.

So, in electrostatics, the electric field inside a conductor is zero, and any extra charge on a conductor sits on its surface. This result will be used again and again in the next chapters.