The galvanometer deflection,
Question:

The galvanometer deflection, when key $\mathrm{K}_{1}$ is closed but $\mathrm{K}_{2}$ is open, equals $\theta_{0}$ (see figure). On closing $\mathrm{K}_{2}$ also and adjusting $R_{2}$ to $5 \Omega$, the deflection in galvanometer becomes $\frac{\theta_{0}}{5}$. The resistance of the galvanometer is, then, given by [Neglect the internal resistance of battery]:

1. (1) $5 \Omega$

2. (2) $22 \Omega$

3. (3) $25 \Omega$

4. (4) $12 \Omega$

Correct Option: , 2

Solution:

(2) When key $\mathrm{K}_{1}$ is closed and key $\mathrm{K}_{2}$ is open

$\mathrm{i}_{\mathrm{g}}=\frac{\mathrm{E}}{220+\mathrm{R}_{\mathrm{g}}}=\mathrm{C} \theta_{0}$             …(1)

When both the keys are closed

$\mathrm{i}_{\mathrm{g}}=\left(\frac{\mathrm{E}}{220+\frac{5 \mathrm{R}_{\mathrm{g}}}{5+\mathrm{R}_{\mathrm{g}}}}\right) \times \frac{5}{\left(\mathrm{R}_{\mathrm{g}}+5\right)}=\frac{\mathrm{C} \theta_{0}}{5}$

$\Rightarrow \frac{5 \mathrm{E}}{225 \mathrm{R}_{\mathrm{g}}+1100}=\frac{\mathrm{C} \theta_{0}}{5}$            …(2)

$\frac{\mathrm{E}}{220+\mathrm{R}_{\mathrm{g}}}=\mathrm{C} \theta_{0}$           …(1)

Dividing (i) by (ii), we get

$\Rightarrow \frac{225 \mathrm{R}_{\mathrm{g}}+1100}{1100+5 \mathrm{R}_{\mathrm{g}}}=5$

$\Rightarrow 5500+25 R_{\mathrm{g}}=225 R_{\mathrm{g}}+1100$

$200 R_{\mathrm{g}}=4400$

$R_{\mathrm{g}}=22 \Omega$