An electric dipole of length 2 cm is placed with its axis making an angle of `60^(@)` to a uniform electric field of `10^(5)NC^(-1)` if its experiences a torque is `8sqrt(3)`Nm, calculate the
(i). Magnitude of the charge on the dipole and
(ii). potential energy of the dipole.

Options

A. `-10J`
B. `-20J`
C. `-30J`
D. `-40J`

Answer

To solve the problem, we will break it down into two parts: (i) calculating the magnitude of the charge on the dipole and (ii) calculating the potential energy of the dipole.

### Given Data:
- Length of the dipole, \( L = 2 \, \text{cm} = 2 \times 10^{-2} \, \text{m} \)
- Angle with the electric field, \( \theta = 60^\circ \)
- Electric field strength, \( E = 10^5 \, \text{N/C} \)
- Torque experienced by the dipole, \( \tau = 8\sqrt{3} \, \text{Nm} \)

### (i) Magnitude of the charge on the dipole

1. **Formula for Torque**:
The torque \( \tau \) experienced by a dipole in an electric field is given by:
\[ \tau = pE \sin(\theta) \]
where \( p \) is the dipole moment given by \( p = Q \cdot L \).

2. **Substituting for Dipole Moment**:
Substitute \( p \) into the torque equation:
\[ \tau = (Q \cdot L) E \sin(\theta) \]

3. **Rearranging the Equation**:
Rearranging the equation to solve for \( Q \):
\[ Q = \frac{\tau}{L E \sin(\theta)} \]

4. **Substituting Known Values**:
- \( L = 2 \times 10^{-2} \, \text{m} \)
- \( E = 10^5 \, \text{N/C} \)
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

Now substituting these values into the equation:
\[ Q = \frac{8\sqrt{3}}{(2 \times 10^{-2}) (10^5) \left(\frac{\sqrt{3}}{2}\right)} \]

5. **Simplifying the Equation**:
\[ Q = \frac{8\sqrt{3}}{(2 \times 10^{-2}) (10^5) \left(\frac{\sqrt{3}}{2}\right)} = \frac{8 \cdot 2}{2 \times 10^{-2} \cdot 10^5} = \frac{8}{10^{-2} \cdot 10^5} \]
\[ Q = \frac{8}{10^3} = 8 \times 10^{-3} \, \text{C} \]

### (ii) Potential Energy of the Dipole

1. **Formula for Potential Energy**:
The potential energy \( U \) of a dipole in an electric field is given by:
\[ U = -pE \cos(\theta) \]

2. **Substituting for Dipole Moment**:
Substitute \( p = Q \cdot L \):
\[ U = - (Q \cdot L) E \cos(\theta) \]

3. **Substituting Known Values**:
- \( Q = 8 \times 10^{-3} \, \text{C} \)
- \( L = 2 \times 10^{-2} \, \text{m} \)
- \( E = 10^5 \, \text{N/C} \)
- \( \cos(60^\circ) = \frac{1}{2} \)

Now substituting these values into the equation:
\[ U = - (8 \times 10^{-3} \cdot 2 \times 10^{-2}) (10^5) \left(\frac{1}{2}\right) \]

4. **Calculating Potential Energy**:
\[ U = - (8 \times 10^{-3} \cdot 2 \times 10^{-2} \cdot 10^5 \cdot \frac{1}{2}) \]
\[ U = - (8 \times 10^{-3} \cdot 10^5 \cdot 10^{-2}) \]
\[ U = - (8 \times 10^0) = -8 \, \text{J} \]

### Final Answers:
(i) The magnitude of the charge on the dipole is \( 8 \times 10^{-3} \, \text{C} \).

(ii) The potential energy of the dipole is \( -8 \, \text{J} \).