When only galvanometer G is present with the resistance R,

JEE Main 2016 (Online) 9th April Morning Slot Physics - Magnetic Effect of Current Question 169 English Explanation 1

JEE Main 2016 (Online) 9th April Morning Slot Physics - Magnetic Effect of Current Question 169 English Explanation 1

Here I_G = $${{{V_E}} \over {R + G}}$$

When shunt of resistance S is connected parallel to galvanometer,

JEE Main 2016 (Online) 9th April Morning Slot Physics - Magnetic Effect of Current Question 169 English Explanation 2

JEE Main 2016 (Online) 9th April Morning Slot Physics - Magnetic Effect of Current Question 169 English Explanation 2

Here I = $${{{V_E}} \over {R + {{GS} \over {G + S}}}}$$

As deflection is half, here current through galvanometer,

I_G' = $${{{{\rm I}_G}} \over 2}$$

As both Galvanometer and shunt are parallel then potential are parallel then potential difference same.

$$ \therefore $$   I_G' (G) = (I $$-$$ I_G^')S

$$ \Rightarrow $$   I'_G (G + S) = IS

$$ \Rightarrow $$   $${{{{\rm I}_G}} \over 2}$$ = $${{{\rm I}S} \over {G + S}}$$

$$ \Rightarrow $$   $${{{V_E}} \over {2\left( {R + G} \right)}}$$ = $${{{V_E}} \over {R + {{GS} \over {G + S}}}}$$ $$ \times $$ $${S \over {\left( {G + S} \right)}}$$

$$ \Rightarrow $$   $${1 \over {2\left( {R + G} \right)}}$$ = $${{G + S} \over {R(G + S) + GS}}$$ $$ \times $$ $${S \over {\left( {G + S} \right)}}$$

$$ \Rightarrow $$   RG + RS + GS = 2RS + 2GS

$$ \Rightarrow $$   RG = RS + GS

$$ \Rightarrow $$    S(R + G) = RG