Torque on Dipole in Uniform Field

Torque on a dipole with moment p in uniform field E, where θ is the angle between p and E. Torque tends to align p along E.

Class 11Class 12

Derivation

Forces on the dipole

A dipole with charges $+q$ and $-q$ separated by $\vec{d}$ is placed in a uniform field $\vec{E}$ .

Force on $+q$ : $\vec{F}_+ = q\vec{E}$

Force on $-q$ : $\vec{F}_- = -q\vec{E}$

Net force: $\vec{F}_{net} = \vec{F}_+ + \vec{F}_- = 0$

A uniform field exerts no net translational force on a dipole. However, the two forces act at different points, forming a couple.

Let the dipole moment $\vec{p}$ make angle $\theta$ with $\vec{E}$ .

The perpendicular distance between the lines of action of the two forces is $d\sin\theta$ .

Torque = Force × perpendicular distance:

\tau = qE \cdot d\sin\theta = pE\sin\theta

In vector form:

\boxed{\vec{\tau} = \vec{p} \times \vec{E}}

The torque acts to align $\vec{p}$ with $\vec{E}$ (reducing $\theta$ ).

$\theta = 0$ : $\tau = 0$ , stable equilibrium ( $\vec{p} \parallel \vec{E}$ )
$\theta = \pi/2$ : $\tau = pE$ , maximum torque
$\theta = \pi$ : $\tau = 0$ , unstable equilibrium ( $\vec{p}$ antiparallel to $\vec{E}$ )

Note

In a non-uniform field, $\vec{F}_{net} \neq 0$. The dipole also experiences a translational force proportional to the gradient of the field.