# Torque on an Electric Dipole in an Uniform Electric Field

Before we understand the properties of the torque acting on an electric dipole in a uniform electric field, let us brush up our understanding of torque and electric dipole clearly.

### Torque:

The measure of force that causes an object to rotate about an axis is known as torque. Torque is a vector quantity and its direction depends on the direction of the force on the axis. The magnitude of the torque vector is calculated as follows:

$\tau =Frsin\Theta$

where r is the length of the moment arm, and θ is the angle between the moment arm and the force vector.

### Electric dipole:

A pair of electric charges with an equal magnitude but opposite charges separated by a distance d is known as an electric dipole. The electric dipole moment for this is defined as the product of the magnitude of these charges and the distance between them. The electric dipole moment is a vector having a defined direction from the negative charge to the positive charge.

$\overrightarrow{p}=q\overrightarrow{d}$

## Derivation of Torque on an Electric Dipole

Now, consider a dipole with charges +q and –q forming a dipole since they are a distance d away from each other. Let it be placed in a uniform electric field of strength E such that the axis of the dipole forms an angle θ with the electric field.

The force on the charges is

$\overrightarrow{F_+}$ = $+q\overrightarrow{E}$

$\overrightarrow{F_-}$ = $-q\overrightarrow{E}$

The components of force perpendicular to the dipole are:

$F_{+}^⊥$ = $+qE~sinθ$

$F_{-}^⊥$ = $-qE~sinθ$

Since the force magnitudes are equal and are separated by a distance d, the torque on the dipole is given by:

$Torque ~(τ)$ = $Force~×~distance~ seperating ~forces$

$τ$ = $d~ qE ~sin~θ$

Since dipole moment is given by

$p$ = $qd$

And the direction of the dipole moment is from the positive to the negative charge; it can seen from the above equation that the torque is the cross product of the dipole moment and electric field. Notice that the torque is in the clockwise direction (hence negative) in the above figure if the direction of Electric field is positive.

Thus,

$τ$ = $-pE~ sin~θ$

Or,

$\overrightarrow{τ}$ = $\overrightarrow{p}~×~\overrightarrow{E}$

$|\overrightarrow{τ}|$ = $|\overrightarrow{p}~×~\overrightarrow{E}|$ = $pE~sin~θ$

## Summary

• The measure of force that causes an object to rotate about an axis is known as a torque.
• A pair of electric charges with an equal magnitude but opposite charges separated by a distance ‘d’ is known as an electric dipole.
• Since the force magnitudes are equal and are separated by a distance d, the torque on the dipole is given by:

Torque (τ) = Force × distance separating forces

• The torque on an electric dipole in a uniform electric field is given by the equation

|→τ| = |→p × →E| = pE sin θ