In a circuit for finding the resistance of a galvanometer by half deflection method, a $6\,V$ battery and a high resistance of $11\,k\Omega $ are used. The figure of merit of the galvanometer $60\,\mu A/$ division. In the absence of shunt resistance, the galvanometer produces a deflection of $\theta = 9$ divisions when current flows in the circuit. The value of the shunt resistance that can cause the deflection of $\theta /2 ,$ is closest to .................. $\Omega$
JEE MAIN 2018, Diffcult
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Figure of merit of a galvanometer is the current required to produce a deflection of one division in the galvancmeter i.e., figure of merit $=\frac{\mathrm{I}}{\theta}$
$\mathrm{I}=\frac{\varepsilon}{\mathrm{R}+\mathrm{G}} \quad \mathrm{G}=\frac{1}{9}\, \mathrm{K}\, \Omega$
$\frac{1}{2}=\frac{\varepsilon}{R+\frac{G S}{G+\bar{S}}} \times \frac{S}{S+G} \Rightarrow \frac{1}{2}=\frac{\varepsilon S}{R(S+G)+G S}$
$S=\frac{R G \times \frac{1}{2}}{\varepsilon-\frac{(R+G) I}{2}}$
$S=\frac{11 \times 10^{3} \times \frac{1}{2} \times 10^{2} \times 270 \times 10^{-6}}{6-\left(\frac{6}{2}\right)}=110\, \Omega$
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