A rectangular coil (Dimension `5 cmxx2.5 cm`) with 100 turns, carrying a current of 3A in the origin and in the X-Z plane. A magnetic field of 1 T is applied along X-axis. If the coil is tilted through `45^(@)` about Z-axis, then the torque on the coil is :

Options

A. `0.38 Nm`
B. `0.55 Nm`
C. `0.42 Nm`
D. `0.27 Nm`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Area of the Coil
The dimensions of the rectangular coil are given as 5 cm and 2.5 cm.

\[ \text{Area} (A) = \text{length} \times \text{width} = 5 \, \text{cm} \times 2.5 \, \text{cm} = 12.5 \, \text{cm}^2 \]

Converting this area into square meters:

\[ A = 12.5 \, \text{cm}^2 = 12.5 \times 10^{-4} \, \text{m}^2 \]

### Step 2: Determine the Magnetic Moment
The magnetic moment (m) of the coil can be calculated using the formula:

\[ m = n \cdot I \cdot A \]

Where:
- \( n = 100 \) (number of turns)
- \( I = 3 \, \text{A} \) (current)
- \( A = 12.5 \times 10^{-4} \, \text{m}^2 \)

Substituting the values:

\[ m = 100 \cdot 3 \cdot (12.5 \times 10^{-4}) = 0.375 \, \text{A m}^2 \]

### Step 3: Calculate the Torque
The torque (\( \tau \)) on the coil in a magnetic field can be calculated using the formula:

\[ \tau = m \cdot B \cdot \sin(\theta) \]

Where:
- \( B = 1 \, \text{T} \) (magnetic field strength)
- \( \theta = 45^\circ \) (angle of tilt)

Since \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \):

Substituting the values:

\[ \tau = 0.375 \cdot 1 \cdot \sin(45^\circ) = 0.375 \cdot 1 \cdot \frac{1}{\sqrt{2}} = \frac{0.375}{\sqrt{2}} \approx 0.265 \, \text{N m} \]

### Step 4: Final Result
The torque on the coil is approximately:

\[ \tau \approx 0.265 \, \text{N m} \]

### Conclusion
Thus, the torque on the coil when tilted at \( 45^\circ \) about the Z-axis is approximately \( 0.27 \, \text{N m} \).

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