Answer

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**Hint:**We know that the current sensitivity of a moving coil galvanometer is defined as the current in a microampere needed to consume a deflection of one millimeter at a scale 1m away from the mirror.

**Formula Used:**

Current Sensitivity $ = \dfrac{1}{{{\text{number of divisions}}}}$

${I_g} = \dfrac{V}{{R + G}}$

**Complete step-by-step solution**

Let full scale deflection of current = 1

As students know that, the voltmeter is a device, which is used to measure the potential difference between the two ends of a current-carrying conductor, and by connecting a high resistance in series with a galvanometer it can be converted into a voltmeter.

When a resistance ‘R’ is connected with the galvanometer in series, the current through the galvanometer is given as, ${I_g} = \dfrac{V}{{R + G}}$

In case 1, using the above relation when R = 2400 $\Omega $ and deflection of 40 divisions is present.

So, $\dfrac{{40}}{{50}}I = \dfrac{V}{{G + R}}$

$ \Rightarrow \dfrac{4}{5}I = \dfrac{2}{{G + 2400}}$ - (1)

Similarly, in case 2, when R = 4900 $\Omega $ and deflection of 20 divisions is present.

So, $\dfrac{{20}}{{50}}I = \dfrac{V}{{G + R}}$

$ \Rightarrow \dfrac{2}{5}I = \dfrac{V}{{G + 4900}}$ - (2)

From equation (1) and (2) we get,

$\dfrac{4}{2} = \dfrac{{G + 4900}}{{G + 2400}}$

$ \Rightarrow 2G + 4800 = G + 4900$

$ \Rightarrow G = 100\Omega $

Substituting G in equation (1) we will get,

$\dfrac{4}{5}I = \dfrac{2}{{100 + 2400}}$

$ \Rightarrow I = 1mA$

Current Sensitivity $ = \dfrac{1}{{{\text{number of divisions}}}}$

$ = \dfrac{1}{{50}}$ = 0.02mA/division$ = 20\mu A/division$

Resistance required for deflection of 10 divisions,

$\dfrac{{10}}{{50}}I = \dfrac{V}{{G + R}}$

$ \Rightarrow \dfrac{1}{5} \times 1 \times {10^{ - 3}} = \dfrac{2}{{100 + R}}$

$ \Rightarrow R = 9900\Omega $

**Hence, option (A) is the correct answer.**

**Additional Information:**

A galvanometer is a device used to measure the magnitude of the small electric current. The current and its strength is determined by the movement of a magnetic needle or that of a coil in a magnetic field. A moving coil galvanometer is a very sensitive device. The principle of the moving coil galvanometer is that when placed in an external magnetic field the current-carrying coil experiences magnetic torque.

**Note:**Current sensitivity of a moving coil galvanometer is defined as the deflection produced in the galvanometer when a unit current flows through it. If $d\theta $ is the change in the deflection produced by a small change in the current $dI$

So, the sensitivity of the galvanometer is given by

$S = \dfrac{{d\theta }}{{dI}}$

The current flowing through the moving coil galvanometer is given by

$I = \dfrac{k}{{nAB}}\theta $

where ‘$\theta $’ expresses the angle of deflection, ‘n’ is the number of turns, ‘A’ is the area, ‘B’ is the magnetic induction and ‘k’ is the couple per unit twist.

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