Question

A long charged cylinder of linear charge density \(\lambda\) is surrounded by a hollow coaxial conducting cylinder. The magnitude of electric field in the space between the two cylinders at a distance \(R\) from the common axis of the cylinders is

• A. \(\dfrac{\lambda}{4\pi\varepsilon_0R}\)
• B. \(0\)
• C. \(\dfrac{\lambda}{2\pi\varepsilon_0R}\)
• D. \(\dfrac{\lambda}{\pi\varepsilon_0R}\)

Hint:

For an infinitely long line charge, \[ \boxed{ E=\frac{\lambda}{2\pi\varepsilon_0r}. } \] The surrounding conducting cylinder does not alter the electric field in the empty region between the two cylinders; only the enclosed charge matters in Gauss' law.

Answer 1

The Correct Option is C

Step 1: Choose a Gaussian surface. Consider a cylindrical Gaussian surface of radius \[ R \] and length \[ L, \] where the surface lies between the charged cylinder and the conducting cylinder. The enclosed charge is \[ q=\lambda L. \]

Step 2: Apply Gauss' law. According to Gauss' law, \[ \oint\vec E\cdot d\vec A = \frac{q}{\varepsilon_0}. \] Since the electric field is radial and constant over the curved surface, \[ E(2\pi RL) = \frac{\lambda L}{\varepsilon_0}. \] Hence, \[ E = \frac{\lambda}{2\pi\varepsilon_0R}. \]

Step 3: Write the answer. Therefore, \[ \boxed{ E = \frac{\lambda}{2\pi\varepsilon_0R}. } \] Thus, \[ \boxed{(C)} \] is the correct answer.