A coil is placed in y-z plane making an angle of `30^(@)` with x-axis . The current through coil is I, and number of turns are N. If a magnetic field of strength 'B' is applied in positive x-direction, then find the torque experienced by coil: (Radius of coil is R)
`(N = 100, I = 1A, r = 2m , B = 1/pi T)`

Options

A. `100 N-m`
B. `50 N-m`
C. `200 N-m`
D. `150 N-m`

Explanation

To find the torque experienced by the coil, we can follow these steps:

### Step 1: Understand the Torque Formula
The torque (\(\tau\)) experienced by a coil in a magnetic field is given by the formula:
\[ \tau = \mathbf{M} \times \mathbf{B} \]
where \(\mathbf{M}\) is the magnetic moment vector and \(\mathbf{B}\) is the magnetic field vector.

### Step 2: Calculate the Magnetic Moment
The magnetic moment (\(\mathbf{M}\)) of the coil can be calculated using the formula:
\[ \mathbf{M} = N \cdot I \cdot A \]
where:
- \(N\) = number of turns of the coil
- \(I\) = current through the coil
- \(A\) = area of the coil

### Step 3: Calculate the Area of the Coil
The area \(A\) of a circular coil is given by:
\[ A = \pi R^2 \]
Given that the radius \(R = 2 \, \text{m}\), we can calculate:
\[ A = \pi (2^2) = 4\pi \, \text{m}^2 \]

### Step 4: Substitute Values into the Magnetic Moment Formula
Now substituting the values into the magnetic moment formula:
- \(N = 100\)
- \(I = 1 \, \text{A}\)
- \(A = 4\pi \, \text{m}^2\)

Thus,
\[ \mathbf{M} = 100 \cdot 1 \cdot 4\pi = 400\pi \, \text{A m}^2 \]

### Step 5: Calculate the Torque
The torque can also be expressed as:
\[ \tau = M \cdot B \cdot \sin(\theta) \]
where \(\theta\) is the angle between the magnetic moment and the magnetic field. Here, \(\theta = 30^\circ\) and \(\sin(30^\circ) = \frac{1}{2}\).

Given \(B = \frac{1}{\pi} \, \text{T}\), we can substitute the values into the torque formula:
\[ \tau = (400\pi) \cdot \left(\frac{1}{\pi}\right) \cdot \frac{1}{2} \]

### Step 6: Simplify the Expression
Now simplifying the expression:
\[ \tau = 400 \cdot \frac{1}{2} = 200 \, \text{N m} \]

### Final Answer
The torque experienced by the coil is:
\[ \tau = 200 \, \text{N m} \]

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