Question

An electric dipole has a fixed dipole moment $\vec{p}$, which makes angles $\theta$ with respect to x-axis. When subjected to an electric field $\vec{E}_1 = E \hat{i}$, it experiences a torque $\vec{T}_1 = \tau \hat{k}$. When subjected to another electric field $\vec{E}_2 = \sqrt{3 E_1} \hat{j}$ it experiences a torque $\vec{T}_2 = - \vec{T}_1$. The angle $\theta$ is

• A. $30^{\circ}$
• B. $45^{\circ}$
• C. $60^{\circ}$
• D. $90^{\circ}$

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the torque experienced by an electric dipole in two different electric fields. The torque experienced by an electric dipole in an electric field is given by the formula:

$\vec{T} = \vec{p} \times \vec{E}$

Where $\vec{p}$ is the dipole moment and $\vec{E}$ is the electric field.

1. **Given**: For the electric field $\vec{E}_1 = E \hat{i}$, the torque is $\vec{T}_1 = \tau \hat{k}$.

The torque can be expressed as: $\vec{T}_1 = |\vec{p}||\vec{E}_1|\sin \theta \hat{k}$

So, $\tau = pE\sin \theta$ (Equation 1).

2. **Given**: For the electric field $\vec{E}_2 = \sqrt{3}E \hat{j}$, the torque is $\vec{T}_2 = - \vec{T}_1 = -\tau \hat{k}$.

The torque can also be expressed as: $\vec{T}_2 = |\vec{p}||\vec{E}_2|\sin(90^\circ - \theta)\hat{k}$

Which simplifies to: $-\tau = p (\sqrt{3} E)\cos \theta$

⇒ $-\tau = \sqrt{3}pE\cos \theta$ (Equation 2).

3. **Equating torques from Equations 1 and 2**:

$pE\sin \theta = \sqrt{3}pE\cos \theta$

Cancel out the common terms, assuming $pE \neq 0$:

$\tan \theta = \sqrt{3}$

The angle that satisfies this condition is $60^\circ$ since

$\tan 60^\circ = \sqrt{3}$.

Thus, the angle $\theta$ is $60^\circ$.