Question

Two coaxial non-conducting cylinders of radius \(a\) and \(b\) are separated by a material of conductivity \(\sigma\) and a constant potential difference \(V\) is maintained between them by a battery. The current per unit length flowing from one cylinder to the other is:

• A. \(\dfrac{4\pi\sigma V}{\ln(b/a)}\)
• B. \(\dfrac{4\pi\sigma V}{b+a}\)
• C. \(\dfrac{2\pi\sigma V}{\ln(b/a)}\)
• D. \(\dfrac{2\pi\sigma V}{b+a}\)

Hint:

For cylindrical symmetry, always use logarithmic potential variation.

Answer 1

The Correct Option is C

Step 1: Electric field between coaxial cylinders: \[ E = \frac{V}{r\ln(b/a)} \]
Step 2: Current density: \[ J = \sigma E \]
Step 3: Current per unit length: \[ I = \int J\, dA = 2\pi r J \] \[ I = \frac{2\pi\sigma V}{\ln(b/a)} \]