Two wires of same length and thickness having specific resistance $6 \Omega \mathrm{~cm}$ and $3 \Omega \mathrm{~cm}$ respectively are connected in parallel. The effective resistivity is $\rho \Omega \mathrm{cm}$. The value of $\rho$, to the nearest integer, is ____

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Current Electricity

Two wires of same length and thickness having specific resistance

6Ω cm

and

3Ω cm

respectively are connected in parallel. The effective resistivity is

ρ Ω cm

. The value of

ρ

, to the nearest integer, is ____

Answer: 4

Each wire has length

l

and cross-section

A

, so

R_{1} = \frac{6 l}{A}

and

R_{2} = \frac{3 l}{A}

, and in parallel

R_{eq} = \frac{R _{1} R _{2}}{R _{1} + R _{2}} = \frac{6 \times 3}{6 + 3} \cdot \frac{l}{A} = \frac{2 l}{A}

The two wires side by side act as one conductor of the same length

l

but of cross-section

2 A

, so its resistivity follows from

R_{eq} = \frac{ρ l}{2 A}

ρ = \frac{R _{eq} ( 2 A )}{l} = \frac{2 l}{A} \cdot \frac{2 A}{l} = 4

The effective resistivity is

4 Ω cm

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Numerical answer