A coil of `100` turns and area `2 xx 10^(-2) m^(2)`, pivoted about a vertical diameter in a uniform magnetic field carries a current of `5A`. When the coil is held with its plane in North-South direction, it experiences a torque of `0.3 N//m`. When the plane is in East-West direction the torque is `0.4 Nm`. The value of magnetic induction is (Neglect earth's magnetic field)

Options

A. `0.2 T`
B. `0.3 T`
C. `0.4 T`
D. `0.05 T`

Explanation

To find the value of magnetic induction (B) in the given problem, we can follow these steps:

### Step 1: Understand the Torque Formula
The torque (τ) experienced by a coil in a magnetic field is given by the formula:
\[ \tau = n \cdot I \cdot A \cdot B \cdot \sin(\theta) \]
where:
- \( n \) = number of turns in the coil
- \( I \) = current through the coil
- \( A \) = area of the coil
- \( B \) = magnetic induction (magnetic field strength)
- \( \theta \) = angle between the normal to the coil and the magnetic field direction

### Step 2: Set Up the Equations for Both Positions
1. When the coil is in the North-South direction, the angle \( \theta \) is 90 degrees (since the magnetic field is assumed to be horizontal). Thus, \( \sin(90^\circ) = 1 \):
\[ \tau_1 = n \cdot I \cdot A \cdot B \cdot \sin(90^\circ) = n \cdot I \cdot A \cdot B \]
Given \( \tau_1 = 0.3 \, \text{N m} \):
\[ 0.3 = n \cdot I \cdot A \cdot B \quad \text{(1)} \]

2. When the coil is in the East-West direction, the angle \( \theta \) is 0 degrees (the plane of the coil is perpendicular to the magnetic field). Thus, \( \sin(0^\circ) = 0 \) and \( \cos(0^\circ) = 1 \):
\[ \tau_2 = n \cdot I \cdot A \cdot B \cdot \cos(0^\circ) = n \cdot I \cdot A \cdot B \]
Given \( \tau_2 = 0.4 \, \text{N m} \):
\[ 0.4 = n \cdot I \cdot A \cdot B \quad \text{(2)} \]

### Step 3: Solve the Equations
From equations (1) and (2), we can express them as:
1. \( n \cdot I \cdot A \cdot B = 0.3 \)
2. \( n \cdot I \cdot A \cdot B = 0.4 \)

We can square both equations and add them:
\[ (0.3)^2 + (0.4)^2 = (n \cdot I \cdot A \cdot B)^2 \cdot (\sin^2(\theta) + \cos^2(\theta)) \]
Since \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ 0.09 + 0.16 = (n \cdot I \cdot A \cdot B)^2 \]
\[ 0.25 = (n \cdot I \cdot A \cdot B)^2 \]
Taking the square root:
\[ n \cdot I \cdot A \cdot B = 0.5 \quad \text{(3)} \]

### Step 4: Substitute Known Values
Now we can substitute the known values into equation (3):
- \( n = 100 \) turns
- \( I = 5 \) A
- \( A = 2 \times 10^{-2} \, \text{m}^2 \)

Substituting these values into equation (3):
\[ 100 \cdot 5 \cdot (2 \times 10^{-2}) \cdot B = 0.5 \]
\[ 1000 \cdot (2 \times 10^{-2}) \cdot B = 0.5 \]
\[ 20B = 0.5 \]
\[ B = \frac{0.5}{20} = 0.025 \, \text{T} \]

### Final Answer
The value of magnetic induction \( B \) is:
\[ \boxed{0.025 \, \text{T}} \]