An electric dipole consists of two equal and opposite charges placed `2 cm` apart. When the dipole is placed in a uniform electric field of strength `10^(5)NC^(-1)`, it experiences a maximum torque of `0.2xx10^(-3)Nm`. Find the magnitude of each charge

Options

A. `20 mu m`
B. `10 muC`
C. `15 muC`
D. `40 muC`

Explanation

To solve the problem, we need to find the magnitude of each charge in the electric dipole given the distance between the charges, the strength of the electric field, and the maximum torque experienced by the dipole.

### Step 1: Understand the relationship between torque, electric dipole moment, and electric field.

The torque (\( \tau \)) experienced by an electric dipole in a uniform electric field (\( E \)) is given by the formula:

\[ \tau = pE \sin(\theta) \]

where:
- \( \tau \) is the torque,
- \( p \) is the electric dipole moment,
- \( E \) is the electric field strength,
- \( \theta \) is the angle between the dipole moment and the electric field.

For maximum torque, \( \sin(\theta) = 1 \) (which occurs when \( \theta = 90^\circ \)), thus:

\[ \tau_{\text{max}} = pE \]

### Step 2: Calculate the dipole moment.

The electric dipole moment (\( p \)) is defined as:

\[ p = q \cdot d \]

where:
- \( q \) is the magnitude of each charge,
- \( d \) is the distance between the charges.

Given that the distance \( d = 2 \, \text{cm} = 0.02 \, \text{m} \), we can rewrite the maximum torque equation as:

\[ \tau_{\text{max}} = q \cdot d \cdot E \]

### Step 3: Substitute the known values.

We know:
- \( \tau_{\text{max}} = 0.2 \times 10^{-3} \, \text{Nm} \)
- \( E = 10^5 \, \text{N/C} \)
- \( d = 0.02 \, \text{m} \)

Substituting these values into the equation gives:

\[ 0.2 \times 10^{-3} = q \cdot (0.02) \cdot (10^5) \]

### Step 4: Solve for \( q \).

Rearranging the equation to solve for \( q \):

\[ q = \frac{0.2 \times 10^{-3}}{0.02 \times 10^5} \]

Calculating the denominator:

\[ 0.02 \times 10^5 = 2 \times 10^3 = 2000 \]

Now substituting back:

\[ q = \frac{0.2 \times 10^{-3}}{2000} \]

Calculating \( q \):

\[ q = \frac{0.2}{2000} \times 10^{-3} = \frac{0.2 \times 10^{-3}}{2000} = \frac{0.2}{2 \times 10^3} \times 10^{-3} = 0.1 \times 10^{-6} = 1 \times 10^{-7} \, \text{C} \]

### Final Answer:

The magnitude of each charge is:

\[ q = 1 \times 10^{-7} \, \text{C} \]