What is electric dipole in an uniform electric field?

Consider an electric dipole consisting of two charges, -q and +q, separated from each other by a distance 2a and placed in a uniform electric field $overrightarrow{E}$, in the plane of the page. Let its dipole moment $overrightarrow{p}$ make an angle θ with $overrightarrow{E}.$

(a) Force acting on the dipole

Force acting on charge –q at A, i.e., $overrightarrow{{{F}_{1}}}=-qoverrightarrow{E}$ and force acting on charge +q at B, i.e., $overrightarrow{{{F}_{2}}}=+qoverrightarrow{E}$.

Since $overrightarrow{{{F}_{1}}},and,overrightarrow{{{F}_{2}}}$ are equal in magnitude (qE) but act in opposite directions,

Net force acting on the dipole,

${{overrightarrow{F}}_{net}}={{overrightarrow{F}}_{1}}+{{overrightarrow{F}}_{2}}=-qoverrightarrow{E}+qoverrightarrow{E}=overrightarrow{0}$

Thus, net force acting on a dipole in a uniform electric field is zero.

(b) Torque acting on the dipole

Since forces ${{overrightarrow{F}}_{1}},and,{{overrightarrow{F}}_{2}}$ act at different points, they form a couple. The magnitude of the moment of the couple, i.e., torque (t) is given by

t = magnitude of either force × arm of the couple

$=qEtimes BC=qEtimes 2asin theta =left( 2qa right)Esin theta ,or,tau =pEsin theta$

Were, p = 2qa

This torque tends to rotate $overrightarrow{p}$ (hence the dipole) in the direction of $overrightarrow{E},$ thereby reducing θ and producing clockwise rotation. The direction of this torque (according to right hand rule for the cross-product of two vectors) is into the page [figure (b)] and is represented by the symbol $otimes$. Since $overrightarrow{p},and,overrightarrow{E}$ lie in the plane of the page and $overrightarrow{tau }$ is into the page as shown in figure. (c) eqn. (1) can be generalised to vector form as

$overrightarrow{tau }=overrightarrow{p}times overrightarrow{E}$ … (2)

Alternative approach:

The expression $overrightarrow{tau }=overrightarrow{p}times overrightarrow{E}$ can also be obtained as follows.

Torque of ${{overrightarrow{F}}_{1}}$ about O, i.e.,

${{overrightarrow{tau }}_{1}}=overrightarrow{OA}times {{overrightarrow{F}}_{1}}=overrightarrow{OA}times left( -qoverrightarrow{E} right)=-qleft( overrightarrow{OA}times overrightarrow{E} right)$ $=qleft( overrightarrow{AO}times overrightarrow{E} right)left( as,,-overrightarrow{OA}=overrightarrow{AO} right)$

Torque of $overrightarrow{{{F}_{2}}}$ about O, i.e.,

${{overrightarrow{tau }}_{2}}=overrightarrow{OB}times {{overrightarrow{F}}_{2}}=overrightarrow{OB}times left( qoverrightarrow{E} right)=qleft( overrightarrow{OB}times overrightarrow{E} right)$

Net torque acting on the dipole, i.e.,

$overrightarrow{tau }=overrightarrow{{{tau }_{1}}}+overrightarrow{{{tau }_{2}}}=qleft( overrightarrow{AO}times overrightarrow{E} right)+qleft( overrightarrow{OB}times overrightarrow{E} right)$ $=qleft( overrightarrow{AO}+overrightarrow{OB} right)times overrightarrow{E}$ $=qleft( overrightarrow{AB} right)times overrightarrow{E}left( as,overrightarrow{AO}+overrightarrow{OB}=overrightarrow{AB} right)$ $=overrightarrow{p}times overrightarrow{E}$

(as $qtimes AB=qtimes 2a=p$ and the direction of $overrightarrow{p}$ is from A to B, i.e., along $overrightarrow{AB}$)

Thus, $overrightarrow{tau }=overrightarrow{p}times overrightarrow{E}$

Thus, there acts a torque on a dipole in a uniform electric field tending to align the dipole parallel to the field.

Combining the results of (a) and (b), we conclude that:

In a uniform electric field, a dipole experiences a torque but no net force.

Definition of p

Let θ = 90° and E = 1.

As t = pE sin θ, t = p × 1 × sin 90° = p. Thus:

The electric dipole moment of an electric dipole is numerically equal to the torque required to keep the dipole perpendicular to an electric field of unit intensity.

Special cases:

1. When θ = 0°, t = 0 and $overrightarrow{p},and,overrightarrow{E}$ are parallel (to each other) and the dipole is in a position of stable equilibrium.
2. When θ = 180°, t = 0 and $overrightarrow{p},and,overrightarrow{E}$ are antiparallel and the dipole is in a position of unstable equilibrium.
3. When θ = 90°, t = pE sin 90° = pE = maximum. If this value of torque is denoted by ${{tau }_{max }},{{tau }_{max }}=pE,,or,,p=frac{{{tau }_{max }}}{E}.$ Thus, dipole moment of an electric dipole is also defined as the maximum torque acting on it per unit electric field.