An electric dipole moment `p` is placed in an electric field of intensity `'E'`. The dipole acquires a position such that the axis of the dipole makes an angle `theta` with the direction of the field. Assuming that the potential energy of the dipole to be zero when `theta= 90^(@)`, the torque and the potential energy of the dipole will respectively be

Options

A. `pE cos theta, - pE sin theta`
B. `pE sin theta, - pE cos theta`
C. `pE sin theta, - 2pE cos theta`
D. `pE sin theta, 2pE cos theta`

Explanation

To solve the problem of finding the torque and potential energy of an electric dipole moment \( p \) placed in an electric field \( E \) at an angle \( \theta \), we will follow these steps:

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

### Step 2: Calculate the Magnitude of Torque
The magnitude of the torque can be expressed as:
\[ \tau = pE \sin \theta \]
Here, \( p \) is the magnitude of the dipole moment, \( E \) is the magnitude of the electric field, and \( \theta \) is the angle between the dipole moment and the electric field.

### Step 3: Understand the Potential Energy of the Dipole
The potential energy \( U \) of an electric dipole in an electric field is given by the formula:
\[ U = -\mathbf{p} \cdot \mathbf{E} \]
This can also be expressed in terms of the angle \( \theta \):
\[ U = -pE \cos \theta \]

### Step 4: Substitute the Values
Since we have assumed that the potential energy is zero when \( \theta = 90^\circ \), we can directly use the formulas derived:
- The torque is:
\[ \tau = pE \sin \theta \]
- The potential energy is:
\[ U = -pE \cos \theta \]

### Final Results
Thus, the torque and potential energy of the dipole are:
- Torque: \( \tau = pE \sin \theta \)
- Potential Energy: \( U = -pE \cos \theta \)