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A galvanometer of resistance G is shunted by a resistance S ohm. To keep the main current in the circuit unchanged, the resistance to be put in series with the galvanometer is
A
$\frac{G^2}{(S + G)}$
B
$\frac{G}{(S + G)}$
C
$\frac{S^2}{(S + G)}$
D
$\frac{SG}{(S + G)}$
Detailed Solution
Before shunting, the resistance offered by the galvanometer to the circuit is G.
After the shunt S is connected in parallel, the resistance of the combination is $\frac{GS}{G + S}$, which is less than G.
For the main current to remain unchanged, the total resistance must again be G. Let R be the resistance added in series with the shunted galvanometer.
$G = \frac{GS}{G + S} + R$
$R = G - \frac{GS}{G + S} = \frac{G^2 + GS - GS}{G + S}$
$R = \frac{G^2}{S + G}$
After the shunt S is connected in parallel, the resistance of the combination is $\frac{GS}{G + S}$, which is less than G.
For the main current to remain unchanged, the total resistance must again be G. Let R be the resistance added in series with the shunted galvanometer.
$G = \frac{GS}{G + S} + R$
$R = G - \frac{GS}{G + S} = \frac{G^2 + GS - GS}{G + S}$
$R = \frac{G^2}{S + G}$