The torque acting on an electric dipole of moment p held at an angle `theta` with an electric field E is ………. .

Options

A. p.E
B. `p xx E`
C. zero
D. `E xx p`

Answer

To find the torque acting on an electric dipole of moment \( \mathbf{p} \) held at an angle \( \theta \) with an electric field \( \mathbf{E} \), we can follow these steps:

### Step 1: Understand the System
An electric dipole consists of two equal and opposite charges, \( +Q \) and \( -Q \), separated by a distance \( 2L \). The dipole moment \( \mathbf{p} \) is defined as:
\[ \mathbf{p} = Q \cdot (2L) \cdot \hat{r} \]
where \( \hat{r} \) is a unit vector pointing from the negative charge to the positive charge.

### Step 2: Identify Forces on the Dipole
In a uniform electric field \( \mathbf{E} \):
- The positive charge \( +Q \) experiences a force \( \mathbf{F_1} = +Q \mathbf{E} \) in the direction of the electric field.
- The negative charge \( -Q \) experiences a force \( \mathbf{F_2} = -Q \mathbf{E} \) in the opposite direction.

### Step 3: Calculate the Torque
The torque \( \mathbf{\tau} \) acting on the dipole can be calculated using the formula:
\[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]
where \( \mathbf{r} \) is the position vector from the pivot point to the point of application of the force, and \( \mathbf{F} \) is the force vector.

For the dipole, we can consider the torque due to both forces. The torque due to the force on the positive charge \( +Q \) is:
\[ \mathbf{\tau_1} = \mathbf{r_1} \times \mathbf{F_1} \]
and for the negative charge \( -Q \):
\[ \mathbf{\tau_2} = \mathbf{r_2} \times \mathbf{F_2} \]

### Step 4: Determine the Magnitudes and Directions
Since both charges are at a distance \( L \) from the midpoint (pivot point), the magnitudes of the position vectors \( \mathbf{r_1} \) and \( \mathbf{r_2} \) are both \( L \). The angle \( \theta \) is the angle between the dipole moment \( \mathbf{p} \) and the electric field \( \mathbf{E} \).

The torque due to the positive charge is:
\[ \mathbf{\tau_1} = L \cdot Q \mathbf{E} \cdot \sin(\theta) \hat{k} \]
and for the negative charge:
\[ \mathbf{\tau_2} = -L \cdot Q \mathbf{E} \cdot \sin(\theta) (-\hat{k}) = L \cdot Q \mathbf{E} \cdot \sin(\theta) \hat{k} \]

### Step 5: Sum the Torques
The total torque \( \mathbf{\tau} \) acting on the dipole is the sum of the torques due to both charges:
\[ \mathbf{\tau} = \mathbf{\tau_1} + \mathbf{\tau_2} = 2L \cdot Q \cdot E \cdot \sin(\theta) \hat{k} \]

### Step 6: Relate Torque to Dipole Moment
Since the dipole moment \( \mathbf{p} = Q \cdot (2L) \), we can substitute this into our torque equation:
\[ \mathbf{\tau} = \mathbf{p} \cdot \mathbf{E} \cdot \sin(\theta) \hat{k} \]

### Final Result
Thus, the torque acting on the electric dipole is given by:
\[ \mathbf{\tau} = \mathbf{p} \times \mathbf{E} \]