Question

An electric dipole of moment $\vec{p}$ is lying along a uniform electric field $\vec{E}$. The work done in rotating the dipole by 90$^{\circ}$ is :-

• A. p E
• B. $ \sqrt 2 $ pE
• C. pE/2
• D. 2pE

Answer 1

The Correct Option is A

The problem involves calculating the work done in rotating an electric dipole within a uniform electric field. An electric dipole consists of two equal and opposite charges separated by a distance, and the dipole moment $\vec{p}$ is a vector quantity pointing from the negative to the positive charge. When placed in a uniform electric field $\vec{E}$, the dipole experiences a torque that tends to align it with the field.

The potential energy $U$ of an electric dipole in a uniform electric field is given by the formula:

$$U = -\vec{p} \cdot \vec{E} = -pE \cos \theta$$

where:

• $p$ is the magnitude of the dipole moment.
• $E$ is the magnitude of the electric field.
• $\theta$ is the angle between the dipole moment and the electric field.

Initially, the dipole is aligned with the field, so $\theta = 0^{\circ}$. The initial potential energy $U_i$ is:

$$U_i = -pE \cos 0^{\circ} = -pE$$

Finally, the dipole is rotated to be perpendicular to the field, making $\theta = 90^{\circ}$. The final potential energy $U_f$ is:

$$U_f = -pE \cos 90^{\circ} = 0$$

The work done $W$ in rotating the dipole is equal to the change in potential energy:

$$W = U_f - U_i$$

Substituting the values of initial and final potential energies, we have:

$$W = 0 - (-pE) = pE$$

Therefore, the work done in rotating the dipole by $90^{\circ}$ is $pE$.

Conclusion: The correct answer is $pE$.