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  4. What equal length of an iron wire and a copper-nickel alloy wire, each of 2 mm diameter connected parallel to give an equivalent resistance of 3 Ω ? (Given resistivities of iron and copper-nickel alloy wire are 12 μ Ω cm and 51 μ Ω cm respectively)

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Q. What equal length of an iron wire and a copper-nickel alloy wire, each of $2\, mm$ diameter connected parallel to give an equivalent resistance of $3 \,\Omega$ ?
(Given resistivities of iron and copper-nickel alloy wire are $12\, \mu \Omega\, cm$ and $51\, \mu \Omega\, cm$ respectively)

JEE MainJEE Main 2021Current Electricity

Solution:

$\frac{ R _{1} R _{2}}{ R _{1}+ R _{2}}=3$
$\frac{\frac{\left(12 \times 10^{-6} \times 10^{-2}\right) \ell \times 4}{\pi(2)^{2} \times 10^{-6}} \times \frac{\left(51 \times 10^{-6} \times 10^{-2}\right) \ell \times 4}{\pi(2)^{2} \times 10^{-6}}}{\frac{63 \times 10^{-6} \times 10^{-2} \times \ell \times 4}{\pi(2)^{2} \times 10^{-6}}}$
$\Rightarrow \ell=97 \,m$