Result

Uniform field

(E3.2.1) E3.2 · Potential Difference in a Uniform Electric Field →

Δ V = - E d

Statement

The potential falls steadily along the field direction, by E volts per metre.

Symbols and units

$Δ V$	potential difference	V
$E$	uniform field	V/m
$d$	distance moved along the field	m

When it applies

Uniform field, with d measured along the field direction.

Don't use when

d must be measured along the field direction — moving perpendicular to the field produces no change in potential at all.

Derivation

Start from the definition, moving a distance d straight along the field:

Δ V = - \int_{A}^{B} E \cdot d s

E is constant and parallel to the path, so it comes out of the integral:

Δ V = - E \int_{A}^{B} d s

The remaining integral is just the distance:

Δ V = - E d

Turned around, this is how you get the field from plate voltage and spacing:

E = \frac{∣Δ V ∣}{d}

Worked examples

Field between plates

ΔV = 12 V across a 2.0 mm gap:

E = \frac{12}{0.0020} = 6.0 \times 1 0^{3} V/m

Small voltages produce large fields across small gaps, which is why thin insulation fails.

Proton accelerated through 12 V

Δ U = q Δ V = 1.9 \times 1 0^{- 18} J = 12 eV

v = \frac{2Δ U}{m} \approx 4.8 \times 1 0^{4} m/s

Watch out for

d is measured <i>along</i> the field. Move at right angles to E and the potential does not change at all — that is exactly what an equipotential surface is.

Connects to

Potential difference Parallel plates