Question

An electric dipole is held in a uniform electric field. Show that the net force acting on it is zero. The dipole is aligned parallel to the field. Find the work done in rotating it through the angle of 180 degrees.

Answer 1

Step 1: Dipole in a uniform electric field

1. When two opposite charges of the same value are placed at a distance, then the system is called a dipole.
2. So, any dipole consists of two charges, $+q$ and $-q$.
3. We know in an electric field, the torque on a dipole is $PE\sin \theta$, where P is the dipole moment and E is the uniform electric field.
4. The work done to rotate a dipole at an angle $d\theta$ in a uniform electric field is, $W={\int }_{{\theta }_1}^{{\theta }_2}PE\sin \theta .d\theta$.

Step 2: The net force acting on a dipole

In a uniform electric field $E$, the force exerted on $+q$is $+qE$ and the force exerted on $-q$is $-qE$.

So, the net force exerted on the dipole is,

$$\begin{gathered} F=+qE-qE \\ orF=0. \end{gathered}$$

Diagram

Therefore, the net force acting on a dipole is zero.

Step 3: Work done to rotate a dipole in an electric field

As we know know that the work done to rotate a dipole is at an angle $d\theta$ is $W={\int }_{{\theta }_1}^{{\theta }_2}PE\sin \theta .d\theta$

Now work is done to rotate the dipole at an angle of 180 is,

$$\begin{gathered} W=PE{\int }_0^{180}\sin \theta .d\theta =PE{\left[-\cos \theta \right]}_0^{180} \\ orW=-PE\left[\cos 180-\cos 0\right]=-PE(-1-1) \\ orW=-PE(-2) \\ orW=2PE \end{gathered}$$

Therefore, the work done in rotating it through the angle of 180 degrees is $2PE$.