An electric dipole is placed at an angle of `30^(@)` with an electric field intensity `2xx10^(5)N//C`. It experiences a torque equal to `4Nm`. The charge on the dipole, if the dipole is length is `2 cm`, is

Options

A. 8 mC
B. 2 mC
C. 5 mC
D. `7 muC`

Answer

To solve the problem step by step, we will use the formula for torque experienced by an electric dipole in an electric field.

### Step 1: Write down the formula for torque
The torque (\( \tau \)) experienced by an electric dipole in an electric field is given by the formula:
\[ \tau = p \cdot E \cdot \sin \theta \]
where:
- \( \tau \) is the torque,
- \( p \) is the dipole moment,
- \( E \) is the electric field intensity,
- \( \theta \) is the angle between the dipole moment and the electric field.

### Step 2: Substitute the known values
From the question, we know:
- \( \tau = 4 \, \text{Nm} \)
- \( E = 2 \times 10^5 \, \text{N/C} \)
- \( \theta = 30^\circ \)

We also know that \( \sin 30^\circ = \frac{1}{2} \).

Substituting these values into the torque formula:
\[ 4 = p \cdot (2 \times 10^5) \cdot \sin(30^\circ) \]
\[ 4 = p \cdot (2 \times 10^5) \cdot \frac{1}{2} \]

### Step 3: Simplify the equation
This simplifies to:
\[ 4 = p \cdot (10^5) \]

### Step 4: Solve for the dipole moment \( p \)
Rearranging the equation to solve for \( p \):
\[ p = \frac{4}{10^5} = 4 \times 10^{-5} \, \text{C m} \]

### Step 5: Relate dipole moment to charge
The dipole moment \( p \) is also defined as:
\[ p = q \cdot a \]
where:
- \( q \) is the charge,
- \( a \) is the separation between the charges.

Given that the length of the dipole is \( 2 \, \text{cm} = 2 \times 10^{-2} \, \text{m} \), we can substitute this value into the equation:
\[ 4 \times 10^{-5} = q \cdot (2 \times 10^{-2}) \]

### Step 6: Solve for charge \( q \)
Rearranging the equation to solve for \( q \):
\[ q = \frac{4 \times 10^{-5}}{2 \times 10^{-2}} = 2 \times 10^{-3} \, \text{C} \]

### Step 7: Convert to milliCoulombs
To express the charge in milliCoulombs:
\[ q = 2 \, \text{mC} \]

### Final Answer
The charge on the dipole is \( 2 \, \text{mC} \).

---