Two charges each of magnitude `0.01 C` and separated by a distance of `0*4mm` constitute an electric dipole . If the dipole is placed in an unifrom electric field 'vecE' of `10` dyne/ C making `30^@ ` angle with `vecE` , the magnitude of torque acting on dipole is :

Options

A. 1*0 timess 10(-8)Nm`
B. 1*5 timess 10(-9)Nm`
C. 4*0 timess 10(-10)Nm`
D. 2*0 timess 10(-10)Nm`

Answer

To find the torque acting on an electric dipole placed in a uniform electric field, we can use the formula for torque (\( \tau \)) given by:

\[ \tau = \vec{p} \times \vec{E} = pE \sin \theta \]

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

### Step 1: Calculate the dipole moment \( p \)

The dipole moment \( p \) is given by:

\[ p = q \cdot d \]

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

Given:
- \( q = 0.01 \, C \)
- \( d = 0.4 \, mm = 0.4 \times 10^{-3} \, m \)

Substituting the values:

\[ p = 0.01 \, C \times 0.4 \times 10^{-3} \, m = 0.004 \times 10^{-3} \, C \cdot m = 4 \times 10^{-6} \, C \cdot m \]

### Step 2: Convert the electric field \( E \)

The electric field is given as \( 10 \, dyne/C \). To convert this to SI units (N/C), we use the conversion factor \( 1 \, dyne = 10^{-5} \, N \):

\[ E = 10 \, dyne/C = 10 \times 10^{-5} \, N/C = 10^{-4} \, N/C \]

### Step 3: Calculate the torque \( \tau \)

Now we can substitute the values into the torque formula. The angle \( \theta \) is given as \( 30^\circ \).

Using the sine of the angle:

\[ \sin(30^\circ) = \frac{1}{2} \]

Now substituting the values into the torque formula:

\[ \tau = pE \sin \theta = (4 \times 10^{-6} \, C \cdot m)(10^{-4} \, N/C) \left(\frac{1}{2}\right) \]

Calculating this gives:

\[ \tau = 4 \times 10^{-6} \times 10^{-4} \times \frac{1}{2} = 2 \times 10^{-10} \, N \cdot m \]

### Final Answer

The magnitude of the torque acting on the dipole is:

\[ \tau = 2 \times 10^{-10} \, N \cdot m \]