Question

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

Hint:

Ohm's law in vector form is $\vec{J} = \sigma \vec{E}$. It's the most direct way to find current when the electric field is known.

Answer 1

Correct Answer: 5

Step 1: Current $I = J \cdot A = (\sigma E) \cdot (\pi r^2)$.
Step 2: $\sigma = 5 \times 10^7 \text{ S/m}$, $E = 10 \times 10^{-3} \text{ V/m}$, $r = 0.5 \times 10^{-3} \text{ m}$.
Step 3: $I = (5 \times 10^7 \times 10^{-2}) \times \pi \times (0.25 \times 10^{-6}) = 5 \times 10^5 \times 0.25 \pi \times 10^{-6} = 0.125 \pi \text{ A} = 125 \pi \text{ mA}$.
Step 4: $x^3 = 125 \Rightarrow x = 5$.