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 $12 \mu \Omega \mathrm{~cm}$ and $51 \mu \Omega \mathrm{~cm}$ respectively)

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Current Electricity

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

3Ω

?
(Given resistivities of iron and copper-nickel alloy wire are

12 μ Ω cm

and

51 μ Ω cm

respectively)

Answer: (D)

Both wires have the same length

L

and the same cross-section

A = π r^{2} = π (0.1)^{2} = 0.01 π c m^{2}

, so

R_{1} = \frac{ρ _{1} L}{A}

and

R_{2} = \frac{ρ _{2} L}{A}

. In parallel

R = \frac{R _{1} R _{2}}{R _{1} + R _{2}} = \frac{L}{A} \cdot \frac{ρ _{1} ρ _{2}}{ρ _{1} + ρ _{2}}

With

ρ_{1} = 12 μ Ω cm

and

ρ_{2} = 51 μ Ω cm

\frac{ρ _{1} ρ _{2}}{ρ _{1} + ρ _{2}} = \frac{12 \times 51}{63} μ Ω cm = 9.71 \times 1 0^{- 6} Ω cm

Setting

R = 3 Ω

L = \frac{R A}{9.71 \times 1 0 ^{- 6}} = \frac{3 \times 0.01 π}{9.71 \times 1 0 ^{- 6}} \approx 9.7 \times 1 0^{3} cm

so each wire is about

97

m long.

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Single correct option