Electric fields and potential

Coulomb's law says that charges push and pull on each other across empty space. A more powerful way to think about this is that every charge fills the space around it with an electric field, and any other charge placed in that field feels a force from it. The field idea turns out to be essential: it explains how energy travels as light and radio waves, and it makes it easy to describe what happens inside capacitors, cables and electronic devices.

This chapter defines the electric field, shows how to compute it for point charges and simple arrangements, and introduces electric potential, the "voltage" you meet on every battery and power outlet. It ends with the link between field and potential, and with practical limits such as the breakdown of air.

The electric field

The electric field at a point is the force per unit charge that a small positive test charge would feel there:

Once you know the field, the force on any charge \(q\) placed there is simply \(\mathbf{F} = q\mathbf{E}\). A positive charge is pushed along the field; a negative charge (such as an electron) is pushed against it.

For a single point charge \(Q\), dividing Coulomb's law by the test charge gives

directed away from a positive charge and toward a negative one. The field of several charges is the vector sum of their individual fields, by the same superposition principle as in Lesson 1.

Field lines

Field lines are a way to picture a field. They are drawn so that the field at any point is tangent to the line through it, and the lines are closer together where the field is stronger. Their rules:

• Lines start on positive charges and end on negative charges (or go off to infinity).
• The number of lines leaving a charge is proportional to its size.
• Lines never cross, because the field has only one direction at each point.

A pair of equal and opposite charges is called an electric dipole. Dipoles are everywhere: water molecules are dipoles, which is why water dissolves salts so well and why microwave ovens heat it so effectively.

Uniform fields and moving charges

Between two large, parallel, oppositely charged plates the field is nearly uniform: the same size and direction everywhere, except near the edges. Uniform fields are used to accelerate and steer charged particles in X-ray tubes, mass spectrometers, inkjet printers and electrostatic precipitators that clean smokestack gases.

A charge in a uniform field feels a constant force, so it has constant acceleration \(a = qE/m\), and all the kinematics from Mechanics applies. An electron moving sideways through the plates follows a parabola, exactly like a projectile.

Gauss's law in brief

For charge distributions with high symmetry, there is a shortcut to the field. Gauss's law says that the total field passing out through any closed surface (the electric flux) equals the enclosed charge divided by \(\varepsilon_0\). Applying it to symmetric shapes gives some results worth remembering:

• Outside a uniformly charged sphere, the field is the same as if all the charge were at the center.
• Inside a conductor in equilibrium, the field is zero. This is why a metal enclosure (a Faraday cage) shields what is inside from external fields, and why you are safe inside a car struck by lightning.
• Near a large flat sheet with charge density \(\sigma\), the field is \(\sigma/2\varepsilon_0\), independent of distance. Between two oppositely charged plates the fields add to \(\sigma/\varepsilon_0\); outside they cancel.

You will use Gauss's law in full, with surface integrals, in electromagnetics courses later in the program.

Electric potential energy and potential

The electric force is conservative, like gravity, so it has a potential energy. Moving a positive charge against the field takes work, which is stored as electric potential energy; letting it go, the field gives that energy back as kinetic energy.

Just as the field is force per unit charge, the electric potential \(V\) is potential energy per unit charge:

Only differences in potential matter. The potential difference between two points, \(\Delta V\), is what a voltmeter measures and what we commonly call "voltage". When a charge \(q\) moves through a potential difference \(\Delta V\), its potential energy changes by

Positive charges naturally move toward lower potential, and negative charges toward higher potential, both losing potential energy and gaining kinetic energy.

The electron volt

Atomic and particle physics use a convenient energy unit, the electron volt: the energy an electron gains when accelerated through 1 V. \(1 \text{ eV} = 1.602 \times 10^{-19}\) J. Chemical bonds are a few eV; X-ray photons are tens of keV; the Large Hadron Collider reaches several TeV.

Potential of point charges

The potential at distance \(r\) from a point charge \(Q\), taking \(V = 0\) far away, is

Unlike the field, potential is a scalar, and it carries the sign of the charge. The potential of several charges is simply the algebraic sum of their potentials, with no components to worry about. This is the great practical advantage of working with potential.

Equipotentials

An equipotential is a surface on which the potential is constant. Moving a charge along one takes no work. Equipotentials are always perpendicular to field lines, and the field points from high to low potential. Around a point charge they are spheres; between parallel plates they are flat planes parallel to the plates. The surface of any conductor in equilibrium is an equipotential.

Field from potential

Potential and field are two descriptions of the same thing. Moving a distance \(dx\) along the field changes the potential by \(dV = -E\,dx\). So the field is the rate at which potential falls with distance:

This is why the units V/m and N/C are the same. A steep change in potential over a short distance means a strong field.

Dielectric breakdown

Insulators stop conducting only up to a point. If the field is strong enough, it rips electrons from atoms and the material breaks down: a spark jumps through air, or an insulator punctures. Dry air breaks down at about 3 MV/m (3 kV per millimeter). This sets real design limits.