Year 1 Sem 2/LC Electromagnetism I/Lecture 08

Lecture 8: Electrostatics VIII: Electric Dipoles in E-Fields
Thu 12 Feb 2026 11:00

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In this lecture:

• Electric dipoles in uniform external E-fields:
  • Torque
  • Potential energy
  • Work done
  • Right-hand rule
  • Examples

Torque

Consider a dipole in a uniform external E-field:

As the two charges have equal and opposite forces applied by the field, they do not have a net force. The torque $\pmb{\tau}$ rotates the whole couple.

The magnitude of this torque is given by:

$$| \pmb{\tau} | = Fd$$

Were $F$ is one of the two forces and $d$ is the perpendicular distance. This is given by:

$$\tau = | \pmb{\tau} | = Fd = qEd = qE a \sin \theta = pE \sin \theta$$

In vector form its given as a cross product:

$$\pmb{\tau} = \pmb{r} \wedge \pmb{F} = q \pmb{a} \wedge \pmb{E} = \pmb{p} \wedge \pmb{E}$$

Potential Energy

The electric potential at the positive charge be $V_+$ and hence the electric potential energy is $qV_+$, and likewise $V_-$ and $qV_-$ at the negative charge.

The net potential energy on the system is therefore given by:

$$U = q(V_+ - V_-) = -q \int_{0}^{d} \pmb{E} \cdot \, d \pmb{x}$$

$$= - qEd$$

$$= -qE a \cos \theta = -qa E \cos \theta = -p E \cos \theta$$

Hence in vector notation:

$$U = - \pmb{p} \cdot \pmb{E}$$

Directionality

What is the direction of this setup:

Here, as the dipole is aligned with the field, there’s no movement. This (as p goes from the negative to positive charge) is the lowest potential energy configuration. The opposite, with p antialigned to the field would still have zero torque, but is the maximum potential energy configuration.

Microwaves

Dipoles explain how microwave ovens work. For foods containing water, the alternating electric field causes the water molecule’s dipole to align itself with the E-field.

Alternating the microwaves cause the water molecules to oscillate in phase. They gain kinetic energy, so heat.

This works for microwaves because a wavelength too low would cause too little a rate of oscillation and a wavelength too high would oscillate the direction of the E-field before the water dipoles have time to fully align. Microwaves fit this region.

Work Done

Over some infinitesimal change in angular displacement, $\delta \theta$:

$$\delta W = \tau \delta \theta$$

And takes place in the direction which minimises the angle, hence:

$$\tau = - pE \sin \theta \implies \delta W = -p E \sin \theta \delta \theta$$

So, over a displacement from $\theta_1$ to $\theta_2$:

$$W = \int dW = \int_{\theta_1}^{\theta_2} -pE \sin \theta \, d \theta$$

$$W = pE \cos \theta_2 - pE \cos \theta_1 = -(U_2 - U_1) = U_1 - U_2$$

Direction of Torque

We can use the Right Hand Rule to determine the direction of the torque:

Here, $\pmb{a}$ represents $\pmb{p}$ and $\pmb{b}$ represents $\pmb{E}$.

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