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 in the Galvanometer**:
The torque (\( \tau \)) exerted on the coil by the magnetic field when a current \( I_0 \) flows through it is given by the formula:
\[ \tau = N \cdot I_0 \cdot A \cdot B \]
where \( N \) is the number of turns, \( A \) is the area of the coil, and \( B \) is the magnetic field strength.

2. **Deflection Angle**:
The deflection angle \( \phi \) is given as \( \frac{\pi}{2} \) radians (or 90 degrees).

3. **Restoring Torque**:
The restoring torque (\( \tau_r \)) due to the spring is proportional to the angle of deflection and can be expressed as:
\[ \tau_r = -k \cdot \phi \]
where \( k \) is the torsional constant of the spring.

4. **Equilibrium Condition**:
At equilibrium, the deflecting torque is equal to the restoring torque:
\[ N \cdot I_0 \cdot A \cdot B = k \cdot \phi \]

5. **Substituting the Deflection Angle**:
Substitute \( \phi = \frac{\pi}{2} \) into the equilibrium condition:
\[ N \cdot I_0 \cdot A \cdot B = k \cdot \frac{\pi}{2} \]

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

### Final Result
Thus, the value of the torsional constant \( k \) of the spring is:
\[ k = \frac{2 \cdot N \cdot I_0 \cdot A \cdot B}{\pi} \]