Result

Torque and energy

(E4.6.1) E4.6 · Electric Dipole in an Electric Field →

τ = p \times E, U = - p \cdot E

Statement

A uniform field twists a dipole into alignment but does not push it anywhere.

Symbols and units

$p$	dipole moment, from − to +	C·m
$τ$	torque on the dipole	N·m
$U$	orientation energy	J
$θ$	angle between p⃗ and E⃗	degrees

When it applies

Uniform field. In a non-uniform field there is also a net force.

Don't use when

Assumes a uniform field — in a non-uniform field there is also a net force on the dipole, since its two ends sit in different field strengths.

Derivation

The two forces are equal and opposite, so there is no net force — but each has a moment arm a sin θ about the centre:

τ = 2 (q E) (a sin θ)

With p = 2aq this is:

τ = pE sin θ ⟹ τ = p \times E

Work done against that torque in rotating the dipole:

U = \int_{9 0^{\circ}}^{θ} pE sin θ^{'} d θ^{'}

Integrating, with the zero chosen at θ = 90°:

U = - pE cos θ = - p \cdot E

Worked examples

Water molecule in a field

p = 6.2 × 10⁻³⁰ C·m in E = 2.5 × 10⁵ V/m:

τ_{m a x} = pE = 1.6 \times 1 0^{- 24} N m

Full energy swing against thermal energy at 300 K:

2 pE = 3.1 \times 1 0^{- 24} J ≪ k T = 4.1 \times 1 0^{- 21} J

Alignment is therefore partial, and κ falls as temperature rises.

Watch out for

In a <i>non-uniform</i> field there IS a net force, because the two ends sit in different field strengths. That is why a charged comb attracts neutral paper.

Connects to

Dielectric constant