A cylindrical wire of radius 0.5 mm and conductivity $5 \times 10^{7} \mathrm{~S} / \mathrm{m}$ is subjected to an electric field of $10 \mathrm{mV} / \mathrm{m}$. The expected value of current in the wire will be $x^{3} \pi \mathrm{~mA}$. The value of $x$ is \_\_\_\_

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Current Electricity

A cylindrical wire of radius 0.5 mm and conductivity

5 \times 1 0^{7} S / m

is subjected to an electric field of

10 mV / m

. The expected value of current in the wire will be

x^{3} π mA

. The value of

x

is
____

Answer: 5

Inside the wire the current density follows the microscopic form of Ohm's law,

J = σ E = (5 \times 1 0^{7}) (10 \times 1 0^{- 3}) = 5 \times 1 0^{5} A m^{- 2}

The cross-section is

A = π r^{2} = π (0.5 \times 1 0^{- 3})^{2} = 0.25 π \times 1 0^{- 6} m^{2}

, so the current is

I = J A = (5 \times 1 0^{5}) (0.25 π \times 1 0^{- 6}) = 0.125 π A = 125 π mA

Comparing with

x^{3} π

mA gives

x^{3} = 125

, so

x = 5

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Integer answer