In a moving coil galvanometer, there is a coil of copper having number of insulated turns N, each of area A. The coil is suspended in a radial magnetic field B. The moment of inertia of the coil about its rotational axis is I. The scale divisions in the galvanometer are n and resistance of the coil is R.
The voltage sensitivity of the galvanometer (in rad/volt) is

Options

A. `(pi)/(3R)`
B. `(3i_0R)/(pi)`
C. `(pi)/(3i_0R)`
D. `(pi)/(i_0R)`

Explanation

To find the voltage sensitivity of a moving coil galvanometer, we will follow these steps:

### Step 1: Understanding Voltage Sensitivity
Voltage sensitivity (Vs) of a galvanometer is defined as the angle of deflection (θ) produced per unit voltage (V) applied across the coil. Mathematically, it can be expressed as:
\[ Vs = \frac{\theta}{V} \]

### Step 2: Relating Voltage to Current
The voltage across the galvanometer can be related to the current (I) flowing through it and the resistance (R) of the coil using Ohm's Law:
\[ V = I \cdot R \]

### Step 3: Substituting Voltage in Sensitivity Formula
Substituting the expression for voltage into the voltage sensitivity formula gives:
\[ Vs = \frac{\theta}{I \cdot R} \]

### Step 4: Defining the Deflection Angle
In a galvanometer, the deflection angle (θ) is proportional to the current (I) flowing through the coil. The relationship can be expressed as:
\[ \theta = k \cdot I \]
where k is a constant of proportionality.

### Step 5: Substituting the Deflection Angle
Now, substituting the expression for θ into the voltage sensitivity formula:
\[ Vs = \frac{k \cdot I}{I \cdot R} = \frac{k}{R} \]

### Step 6: Defining k in Terms of N, A, B, and I
The constant k can be expressed in terms of the number of turns (N), area (A), magnetic field (B), and moment of inertia (I). For a moving coil galvanometer, it can be shown that:
\[ k = \frac{N \cdot B \cdot A}{I} \]
Thus, substituting this into the voltage sensitivity formula gives:
\[ Vs = \frac{N \cdot B \cdot A}{I \cdot R} \]

### Final Expression for Voltage Sensitivity
Therefore, the final expression for the voltage sensitivity of the galvanometer is:
\[ Vs = \frac{N \cdot B \cdot A}{I \cdot R} \]