Question

An electric dipole of dipole moment $$\overrightarrow{p}$$ is placed in an uniform electric field $$\overrightarrow{E}$$ has minimum potential energy when the angle between $$\overrightarrow{p}$$ and $$\overrightarrow{E}$$ is
A. $${\displaystyle \frac{\pi }{2}}$$
B. Zero
C. $$\pi$$
D. $${\displaystyle \frac{3\pi }{2}}$$

Answer 1

Hint: Refer to the formula for the potential energy of the electric dipole. The negative sign of the potential energy of the dipole implies the minimum potential energy.

Formula used:
$$U\left(\theta \right)=-pE\cos \theta$$
Here, $$\theta$$ is the angle between the dipole moment $$\overrightarrow{p}$$ and the electric field $$\overrightarrow{E}$$.

Complete step by step answer:
An electric dipole is a system of two opposite charges $$+q$$ and $$-q$$ separated by the distance $$2a$$ placed in a uniform electric field $$\overrightarrow{E}$$ as shown in the figure below.

The potential energy of the dipole is given by the equation,
$$U\left(\theta \right)=-pE\cos \theta$$
Here, $$\theta$$ is the angle between the dipole moment $$\overrightarrow{p}$$ and the electric field $$\overrightarrow{E}$$.
The potential energy of the dipole is minimum, that is $$-pE$$ when $$\cos \theta =1$$. Therefore, we can substitute $$0^{\circ }$$ for $$\theta$$.
Thus,
$$U\left(\theta \right)=-pE\cos \left(0^{\circ }\right)$$
$$\Rightarrow U\left(\theta \right)=-pE$$
The potential energy of the dipole is zero, when the angle between dipole moment $$\overrightarrow{p}$$ and electric field $$\overrightarrow{E}$$ is $${\displaystyle \frac{\pi }{2}}$$.
Also, the potential energy of the dipole moment is maximum, that is $$+pE$$ when the angle between dipole moment $$\overrightarrow{p}$$ and electric field $$\overrightarrow{E}$$, is $$\pi$$.

So, the correct answer is “Option C”.

Note:
The minimum potential energy of the dipole does not mean the potential energy to be zero. The potential energy of the dipole is minimum when the dipole is parallel to the external electric field. Also, it is the maximum when the dipole is anti-parallel to the external electric field.