Electric fields

Revision notes

An electric field is a region in which a charge experiences an electric force. Electric field strength is force per unit positive test charge: E = \(\dfrac{F}{Q}\). For a point charge Q in vacuum, E = \(\dfrac{(\frac{1}{4\pi \varepsilon 0})Q}{r^{2}}\), directed away from positive charges and toward negative charges. Coulomb's law gives the magnitude of the force between point charges as F = \(\dfrac{1}{4\pi \varepsilon 0}\)|Qq|/r².

Electric potential V is potential energy per unit charge. A charge q moved through a potential difference ΔV has a change in electric potential energy ΔE = qΔV. Electric field strength is related to potential gradient; in a uniform field between parallel plates, E = \(\dfrac{V}{d}\) when edge effects are neglected. Equipotential surfaces are perpendicular to electric field lines, and no work is done moving a charge along a single equipotential.

A charged particle in an electric field experiences F = qE. Its acceleration follows from Newton's second law, with the force direction reversing for a negative charge.

Electric field strength

Electric field strength is force per unit positive test charge, E = \(\dfrac{F}{Q}\). Field direction is defined as the direction of force on a positive test charge. Around a positive point charge the field points outward; around a negative point charge it points inward.

For a uniform field between large parallel plates, E = \(\dfrac{V}{d}\) where edge effects are neglected. A charge q in an electric field experiences force F = qE, with direction depending on the sign of q.

Coulomb's law and inverse-square fields

The force between point charges has magnitude proportional to \(\dfrac{Qq}{r^{2}}\). The corresponding field of a point charge also follows an inverse-square dependence. Doubling distance reduces field strength and force to one quarter, provided the point-charge approximation remains valid.

Superposition applies: the resultant electric field is the vector sum of the fields produced by individual charges. Symmetry can simplify multi-charge problems.

Electric potential

Electric potential is potential energy per unit charge. Potential difference is work done per unit charge between two points. For a positive point charge, potential is positive and decreases towards zero with distance; for a negative charge it is negative.

Field strength is related to the gradient of potential. In one dimension, the field points in the direction of decreasing potential. Equipotential surfaces are everywhere perpendicular to electric field lines.

Energy changes

When a charge moves through a potential difference, electrical potential-energy change is ΔE = QΔV. Whether potential energy increases or decreases depends on charge sign and direction of motion.

Charged-particle motion

In a uniform electric field, a charged particle can undergo constant acceleration if relativistic effects and other forces are negligible. Motion can be analysed by combining F = qE with F = ma, often treating perpendicular directions separately.

Exam technique

Keep field strength and potential distinct: field is a vector, potential is a scalar. Use signs carefully for negative charges. In inverse-square questions, distinguish distance from the source from separation between two charges.

Radial potential and potential energy

Near a point charge Q, the electric potential is V = \(\dfrac{Q}{4\pi \varepsilon _{0}r}\), positive for a positive charge and zero at infinity. The electric potential energy of a charge q at distance r is E = \(\dfrac{Qq}{4\pi \varepsilon _{0}r}\). Here \(\dfrac{1}{4\pi \varepsilon _{0}}\) ≈ 8.99 × 10⁹ N m² C⁻², with ε₀ = 8.85 × 10⁻¹² F m⁻¹.

Charged particles between plates

A charged particle entering a uniform field at right angles to it has constant acceleration \(\dfrac{qE}{m}\) along the field and constant velocity across it, so it follows a parabolic path, just like a horizontal projectile.

Millikan's oil-drop experiment balanced the electric force \(\dfrac{qV}{d}\) against the weight of a charged drop and showed that charge is quantised in multiples of e.

Comparing gravitational and electric fields

Both point-source fields follow inverse-square laws for force and field strength, and both have potentials proportional to \(\dfrac{1}{r}\) with zero at infinity. Gravitational forces are always attractive, while electric forces can attract or repel.