91Ó°ÊÓ

The combination of two equal and opposite charges placed at a certain distance is known as an electric dipole. The dipole rotates due to the torque exerted by the electric field on the dipole. This will further result in work done in rotating this dipole.

Formula:

The work done to rotate a dipole in a external electric field,$W=−pE\left(\cos \mathrm{θ}\right)$ (i)

We need to rotate in the clockwise direction to match up with the electric field vector.

Thus, rotating from orientation 1 to 2 clockwise, the work done by the dipole is given using equating (i) as follows:

$\begin{array}{c}{W}_a=W_{orientation2}−W_{orientation1}\\ =−pE\left(\cos \mathrm{θ}\right)−[−pE\left(\cos (\mathrm{π}−\mathrm{θ})\right)]\\ =−pE\left(\cos \mathrm{θ}\right)+pE(−\cos \mathrm{θ})\\ =−2pE\cos \mathrm{θ}\end{array}$

Since, the angle is acute, thus the cosine term will be positive.

Hence, the work done is negative.

Now, rotating from orientation 1 to 4 clockwise, the work done by the dipole is given using equating (i) as follows:

$\begin{array}{c}{W}_a=W_{orientation4}−W_{orientation1}\\ =−pE\left(\cos (2\mathrm{π}−\mathrm{θ})\right)−[−pE\left(\cos (\mathrm{π}−\mathrm{θ})\right)]\\ =−pE\left(\cos \mathrm{θ}\right)−pE\left(\cos \mathrm{θ}\right)\\ =−pE\cos \mathrm{θ}\end{array}$$\begin{array}{c}{W}_a=W_{orientation4}−W_{orientation1}\\ =−pE\left(\cos (2\mathrm{π}−\mathrm{θ})\right)−[−pE\left(\cos (\mathrm{π}−\mathrm{θ})\right)]\\ =−pE\left(\cos \mathrm{θ}\right)−pE\left(\cos \mathrm{θ}\right)\\ =−pE\cos \mathrm{θ}\end{array}$

Hence, the work done in this case will be negative.