An electric dipole has a fixed dipole moment `vec(p)`,which makes angle with respect to X-axis. When subjected to an electric field `E_1=E_1`, it experience a torpue `vec(tau)_1=tau hatk``.When subjected to another electric field E_2=sqrt3E_1 hatl`experiences a torque `vec(tau)_2=tau_1`. The angle `theta`is:

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. `30^(@)`

Explanation

To solve the problem, we need to analyze the torque experienced by an electric dipole in two different electric fields and relate them to the angles involved. Here's a step-by-step solution:

### Step 1: Understanding the Torque on an Electric Dipole
The torque (\( \vec{\tau} \)) experienced by an electric dipole in an electric field is given by the formula:
\[ \vec{\tau} = \vec{p} \times \vec{E} \]
where \( \vec{p} \) is the dipole moment and \( \vec{E} \) is the electric field.

### Step 2: Define the Electric Fields and Torques
Given:
1. The first electric field \( \vec{E_1} = E_1 \hat{i} \) results in a torque \( \vec{\tau_1} \).
2. The second electric field \( \vec{E_2} = \sqrt{3} E_1 \hat{j} \) results in a torque \( \vec{\tau_2} = \tau_1 \).

### Step 3: Express the Dipole Moment
Assume the dipole moment \( \vec{p} \) makes an angle \( \theta \) with the x-axis. We can express \( \vec{p} \) in component form:
\[ \vec{p} = p \cos \theta \hat{i} + p \sin \theta \hat{j} \]

### Step 4: Calculate the Torque for Each Electric Field
1. **For \( \vec{E_1} \)**:
\[ \vec{\tau_1} = \vec{p} \times \vec{E_1} = (p \cos \theta \hat{i} + p \sin \theta \hat{j}) \times (E_1 \hat{i}) \]
The cross product results in:
\[ \vec{\tau_1} = p \sin \theta E_1 \hat{k} \]

2. **For \( \vec{E_2} \)**:
\[ \vec{\tau_2} = \vec{p} \times \vec{E_2} = (p \cos \theta \hat{i} + p \sin \theta \hat{j}) \times (\sqrt{3} E_1 \hat{j}) \]
The cross product results in:
\[ \vec{\tau_2} = p \cos \theta \sqrt{3} E_1 \hat{k} \]

### Step 5: Relate the Two Torques
According to the problem, \( \vec{\tau_2} = -\vec{\tau_1} \):
\[ p \cos \theta \sqrt{3} E_1 = -p \sin \theta E_1 \]
This implies:
\[ p \cos \theta \sqrt{3} = -p \sin \theta \]

### Step 6: Simplify the Equation
Dividing both sides by \( p E_1 \) (assuming \( p \) and \( E_1 \) are not zero):
\[ \cos \theta \sqrt{3} = -\sin \theta \]
Rearranging gives:
\[ \tan \theta = -\sqrt{3} \]

### Step 7: Find the Angle \( \theta \)
The angle \( \theta \) that satisfies \( \tan \theta = -\sqrt{3} \) is:
\[ \theta = 60^\circ \text{ or } \theta = 240^\circ \]
However, since we are looking for the angle with respect to the x-axis, we take:
\[ \theta = 60^\circ \]

### Final Answer
Thus, the angle \( \theta \) is:
\[ \theta = 60^\circ \]