Recall the definitions for a moving coil galvanometer

For a moving coil galvanometer, the deflecting torque is $\tau_d = N B I A$ where:

• $N$ = number of turns,
• $B$ = magnetic field,
• $I$ = current,
• $A$ = area of coil.

At equilibrium, $N B I A = C\theta$ where $C$ is the torsional constant.

Hence, $\frac{\theta}{I} = \frac{N B A}{C}$

This is the current sensitivity: $S_i = \frac{\theta}{I} = \frac{N B A}{C}$

Check Statement I

Statement I says: If the number of turns in the coil is doubled then the current sensitivity becomes double.

Since $S_i \propto N$ doubling $N$ gives $S_i' = 2S_i$

Therefore, Statement I is true.

Now recall voltage sensitivity

Voltage sensitivity is $S_v = \frac{\theta}{V}$

Using Ohm’s law, $V = IR$, so $S_v = \frac{\theta}{V} = \frac{\theta/I}{R} = \frac{S_i}{R}$

For a galvanometer coil, $R \propto N$ because increasing the number of turns increases the length of wire, hence resistance.

Also, $S_i \propto N$

Therefore, $S_v = \frac{S_i}{R} \propto \frac{N}{N} = \text{constant}$

So by only increasing the number of turns, the voltage sensitivity does not increase in the same ratio.

Check Statement II

Statement II says: Increasing current sensitivity by only increasing the number of turns in the coil will also increase its voltage sensitivity in the same ratio.

This is false, because the resistance also increases in the same ratio, making voltage sensitivity unchanged.

Conclusion

• Statement I: True
• Statement II: False

Therefore, the correct option is: $\boxed{\text{A}}$