The length of a bar magnet is 10 cm and its pole strength is `10^(-3)` Weber. It is placed in a magnetic field of induction `4 pi × 10^(-3)` Tesla in a direction making an angle `30^(@)` with the field direction. The value of torque acting on the magnet will be –

Options

A. (a)`2pi xx10^(-7)`N-m
B. (b)`2pi xx10^(-5)`N-m
C. (c)`0.5 xx10^(2)`N-m
D. (d)0.5N-m

Answer

To find the torque acting on a bar magnet placed in a magnetic field, we can use the formula for torque (\( \tau \)) given by:

\[ \tau = M \cdot B \cdot \sin \theta \]

where:
- \( M \) is the magnetic moment of the magnet,
- \( B \) is the magnetic field induction,
- \( \theta \) is the angle between the magnetic moment and the magnetic field.

### Step 1: Calculate the Magnetic Moment (M)

The magnetic moment \( M \) can be calculated using the formula:

\[ M = m \cdot L \]

where:
- \( m \) is the pole strength,
- \( L \) is the length of the magnet.

Given:
- Pole strength \( m = 10^{-3} \, \text{Wb} \)
- Length of the magnet \( L = 10 \, \text{cm} = 0.1 \, \text{m} \)

Now, substituting the values:

\[ M = 10^{-3} \, \text{Wb} \cdot 0.1 \, \text{m} = 10^{-4} \, \text{Wb m} \]

### Step 2: Identify the Magnetic Field (B)

The magnetic field induction is given as:

\[ B = 4 \pi \times 10^{-3} \, \text{T} \]

### Step 3: Determine the Angle (θ)

The angle \( \theta \) is given as:

\[ \theta = 30^\circ \]

### Step 4: Calculate the Torque (τ)

Now substituting the values into the torque formula:

\[ \tau = M \cdot B \cdot \sin \theta \]

Substituting \( M \), \( B \), and \( \sin 30^\circ \):

\[ \tau = (10^{-4} \, \text{Wb m}) \cdot (4 \pi \times 10^{-3} \, \text{T}) \cdot \sin(30^\circ) \]

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

\[ \tau = (10^{-4}) \cdot (4 \pi \times 10^{-3}) \cdot \frac{1}{2} \]

Calculating this step-by-step:

1. Calculate \( 4 \pi \):

\[ 4 \pi \approx 12.5664 \]

2. Substitute and simplify:

\[ \tau = (10^{-4}) \cdot (12.5664 \times 10^{-3}) \cdot \frac{1}{2} \]

\[ \tau = (10^{-4}) \cdot (6.2832 \times 10^{-3}) \]

\[ \tau = 6.2832 \times 10^{-7} \, \text{N m} \]

### Final Result

Thus, the value of torque acting on the magnet is:

\[ \tau \approx 6.2832 \times 10^{-7} \, \text{N m} \]