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 length is `2 cm`,-

Options

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

Explanation

To find the charge on the dipole, we will follow these steps:

### Step 1: Understand the torque formula
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:
- \( p \) 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
We know:
- \( \tau = 4 \, \text{Nm} \)
- \( E = 2 \times 10^5 \, \text{N/C} \)
- \( \theta = 30^\circ \)

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

### Step 3: Calculate \( \sin(30^\circ) \)
The value of \( \sin(30^\circ) \) is \( \frac{1}{2} \). Therefore, we can rewrite the equation:
\[ 4 = p \cdot (2 \times 10^5) \cdot \frac{1}{2} \]

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

### Step 5: Solve for \( p \)
Now, we can solve for \( p \):
\[ p = \frac{4}{10^5} = 4 \times 10^{-5} \, \text{C m} \]

### Step 6: Relate dipole moment to charge
The dipole moment \( p \) is also given by the formula:
\[ p = Q \cdot d \]
where \( Q \) is the charge and \( d \) is the distance between the charges.

### Step 7: Substitute the length of the dipole
Given that the length of the dipole \( d = 2 \, \text{cm} = 2 \times 10^{-2} \, \text{m} \), we can substitute this into the equation:
\[ 4 \times 10^{-5} = Q \cdot (2 \times 10^{-2}) \]

### Step 8: Solve for \( Q \)
Now, we can solve for \( Q \):
\[ Q = \frac{4 \times 10^{-5}}{2 \times 10^{-2}} = \frac{4}{2} \times 10^{-5 + 2} = 2 \times 10^{-3} \, \text{C} \]

### Step 9: Convert to milliCoulombs
Finally, we can express the charge in milliCoulombs:
\[ Q = 2 \, \text{mC} \]

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

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