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

Answer

To solve the problem, we need to determine the torque and potential energy of an electric dipole placed in an electric field. Here are the steps to derive the required expressions:

### Step 1: Understand the Electric Dipole in an Electric Field
An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment \( \mathbf{p} \) is defined as:
\[ \mathbf{p} = q \cdot \mathbf{d} \]
where \( q \) is the charge and \( \mathbf{d} \) is the separation vector pointing from the negative charge to the positive charge.

### Step 2: Define the Torque on the Dipole
When an electric dipole is placed in an electric field \( \mathbf{E} \), it experiences a torque \( \mathbf{\tau} \) given by the cross product of the dipole moment and the electric field:
\[ \mathbf{\tau} = \mathbf{p} \times \mathbf{E} \]
The magnitude of the torque can be expressed as:
\[ |\tau| = pE \sin \theta \]
where \( \theta \) is the angle between the dipole moment \( \mathbf{p} \) and the electric field \( \mathbf{E} \).

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

### Step 4: Substitute the Values
Given that the potential energy is zero when \( \theta = 90^\circ \), we can now summarize our findings:
- The torque on the dipole is:
\[ |\tau| = pE \sin \theta \]
- The potential energy of the dipole is:
\[ U = -pE \cos \theta \]

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