A moving coil galvanometer coil has a coil of area A, number of turns N. The radial magnetic field present is B. the moment of inertia of the coil is I about its rotation axis. The torque is applied by the magnetic field on the coil of the galvanometer when current `I_(0)` passes through it and produces a deflection of `pi//2` of the pointer. Then answer the following questions based on the paragraph.
Then the value of torsionl constant of the spring is

Options

A. `(NBAI_(0))/(pi)`
B. `(2NBAI_(0))/(pi^(2))`
C. `(NBAI_(0))/(2)`
D. `(2NBAI_(0))/(pi)`

Answer

To find the value of the torsional constant of the spring in a moving coil galvanometer, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Torque on the Coil**:
The torque (\( \tau \)) acting on the coil when a current \( I_0 \) passes through it is given by the formula:
\[ \tau = N \cdot I_0 \cdot A \cdot B \]
where:
- \( N \) = number of turns in the coil,
- \( A \) = area of the coil,
- \( B \) = magnetic field strength,
- \( I_0 \) = current flowing through the coil.

2. **Restoring Torque**:
The restoring torque (\( \tau_r \)) exerted by the spring is proportional to the angle of deflection (\( \phi \)):
\[ \tau_r = K \cdot \phi \]
where:
- \( K \) = torsional constant of the spring,
- \( \phi \) = angle of deflection (given as \( \frac{\pi}{2} \)).

3. **Set the Torques Equal**:
At equilibrium, the deflecting torque is equal to the restoring torque:
\[ N \cdot I_0 \cdot A \cdot B = K \cdot \phi \]

4. **Substituting the Deflection Angle**:
Substitute \( \phi = \frac{\pi}{2} \) into the equation:
\[ N \cdot I_0 \cdot A \cdot B = K \cdot \frac{\pi}{2} \]

5. **Solve for the Torsional Constant \( K \)**:
Rearranging the equation to solve for \( K \):
\[ K = \frac{N \cdot I_0 \cdot A \cdot B}{\frac{\pi}{2}} = \frac{2 \cdot N \cdot I_0 \cdot A \cdot B}{\pi} \]

### Final Answer:
The value of the torsional constant \( K \) of the spring is:
\[ K = \frac{2 \cdot N \cdot I_0 \cdot A \cdot B}{\pi} \]