A closely wound solenoid of 200 turns and area of cross-section `1.5xx10^(-4)m^(2)` carries a current of 2.0 A It is suspended through its centre and perpendicular to its length, allowing it to turn in a horizontal plane in a uniform magnetic field `5xx10^(-2)`T, making an angle of `30^(@)` with the axis of the solenoid. The torque on the solenoid will be

Answer

To find the torque on the solenoid, we can use the formula for torque (\( \tau \)) in a magnetic field, which is given by:

\[ \tau = N \cdot B \cdot I \cdot A \cdot \sin(\theta) \]

where:
- \( N \) = number of turns in the solenoid
- \( B \) = magnetic field strength
- \( I \) = current through the solenoid
- \( A \) = area of cross-section of the solenoid
- \( \theta \) = angle between the magnetic field and the axis of the solenoid

### Step 1: Identify the values given in the problem
- Number of turns, \( N = 200 \)
- Area of cross-section, \( A = 1.5 \times 10^{-4} \, \text{m}^2 \)
- Current, \( I = 2.0 \, \text{A} \)
- Magnetic field, \( B = 5 \times 10^{-2} \, \text{T} \)
- Angle, \( \theta = 30^\circ \)

### Step 2: Calculate \( \sin(30^\circ) \)
\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 3: Substitute the values into the torque formula
Now we can substitute the values into the torque formula:

\[ \tau = N \cdot B \cdot I \cdot A \cdot \sin(\theta) \]
\[ \tau = 200 \cdot (5 \times 10^{-2}) \cdot 2 \cdot (1.5 \times 10^{-4}) \cdot \left(\frac{1}{2}\right) \]

### Step 4: Simplify the expression
Calculating each part step by step:

1. Calculate \( 200 \cdot (5 \times 10^{-2}) \):
\[ 200 \cdot 5 = 1000 \quad \text{and} \quad 1000 \times 10^{-2} = 10 \]

2. Now multiply by \( 2 \):
\[ 10 \cdot 2 = 20 \]

3. Next, multiply by \( (1.5 \times 10^{-4}) \):
\[ 20 \cdot 1.5 = 30 \quad \text{and} \quad 30 \times 10^{-4} = 3 \times 10^{-3} \]

4. Finally, multiply by \( \frac{1}{2} \):
\[ 3 \times 10^{-3} \cdot \frac{1}{2} = 1.5 \times 10^{-3} \]

### Step 5: Write the final answer
Thus, the torque on the solenoid is:
\[ \tau = 1.5 \times 10^{-3} \, \text{N m} \]