JEE MAIN - Physics (2005 - No. 51)
Two voltmeters, one of copper and another of silver, are joined in parallel. When a total charge $$q$$ flows through the voltmeters, equal amount of metals are deposited. If the electrochemical equivalents of copper and silver are $${Z_1}$$ and $${Z_2}$$ respectively the charge which flows through the silver voltmeter is
$${q \over {1 + {{{Z_2}} \over {{Z_1}}}}}$$
$${q \over {1 + {{{Z_1}} \over {{Z_2}}}}}$$
$$q{{{Z_2}} \over {{Z_1}}}$$
$$q{{{Z_1}} \over {{Z_2}}}$$
Explanation
Mass deposited
$$m = Zq \Rightarrow Z \propto {1 \over q} \Rightarrow {{{Z_1}} \over {{Z_2}}} = {{{q_2}} \over {{q_1}}}\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$
Also $$q = {q_1} + {q_2}\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$
$$ \Rightarrow {q \over {{q_2}}} = {{{q_1}} \over {{q_2}}} + 1\,\,\,\,\,\,\,\,\,\,$$ $$(\,Dividing\,\,\left( {ii} \right)$$ by $$\left. {{q_2}\,} \right)$$
$$ \Rightarrow {q_2} = {q \over {1 + {{{q_1}} \over {{q_2}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {iii} \right)$$
From equations $$(i)$$ and $$(iii),$$ $${q_2} = {q \over {1 + {{{Z_2}} \over {{z_1}}}}}$$
$$m = Zq \Rightarrow Z \propto {1 \over q} \Rightarrow {{{Z_1}} \over {{Z_2}}} = {{{q_2}} \over {{q_1}}}\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)$$
Also $$q = {q_1} + {q_2}\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)$$
$$ \Rightarrow {q \over {{q_2}}} = {{{q_1}} \over {{q_2}}} + 1\,\,\,\,\,\,\,\,\,\,$$ $$(\,Dividing\,\,\left( {ii} \right)$$ by $$\left. {{q_2}\,} \right)$$
$$ \Rightarrow {q_2} = {q \over {1 + {{{q_1}} \over {{q_2}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {iii} \right)$$
From equations $$(i)$$ and $$(iii),$$ $${q_2} = {q \over {1 + {{{Z_2}} \over {{z_1}}}}}$$
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