Electric fields

18.1 Electric fields and field lines

An electric field is a field of force. The electric field strength at a point is the force per unit positive charge:

$E = \frac{F}{q} \qquad F = qE$

The unit is N/C, which equals V/m.

Field lines

Field lines show the direction of the force on a positive charge: away from positive charges and towards negative ones. Closer lines mean a stronger field, and lines never cross. They meet a conductor's surface at right angles.

18.2 Uniform electric fields

Between parallel plates with a p.d. $\Delta V$ and separation $\Delta d$, the field is uniform, except near the edges:

$E = \frac{\Delta V}{\Delta d}$

Charged particles in a uniform field

The force $qE$ is constant and acts parallel to the field lines, so the acceleration is constant: $a = qE/m$. Positive charges accelerate along the field, and negative charges against it.

A particle entering at right angles to the field follows a parabola, just like a projectile: constant velocity along the plates and uniform acceleration across them.

The weight of an electron or a proton is negligible compared with the electric force.

18.3 Electric force between point charges

For a point outside a spherical conductor, the charge on it may be treated as a point charge at its centre.

Coulomb's law, for two point charges in free space:

$F = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$

where $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m is the permittivity of free space. Like charges repel and unlike charges attract. Air can be treated as free space.

18.4 Electric field of a point charge

$E = \frac{Q}{4\pi\varepsilon_0 r^2}$

The field points away from a positive charge and towards a negative one. It is a vector, so fields from several charges add as vectors.

18.5 Electric potential

Definition

The electric potential at a point is the work done per unit positive charge in bringing a small test charge from infinity to the point.

$V = \frac{Q}{4\pi\varepsilon_0 r}$

The unit is the volt. It is positive near a positive charge, because you must do work to push a positive charge in, and negative near a negative charge. Potential is a scalar, so potentials from several charges simply add.

Potential gradient

On a graph of $V$ against $x$, the steeper the slope, the stronger the field. The field points from high to low potential.

Electric potential energy

The potential energy of two point charges is:

$E_P = \frac{Qq}{4\pi\varepsilon_0 r}$

Unlike gravity, it can be positive (like charges) or negative (unlike charges).

Comparing gravitational and electric fields