Question

A current is flowing along the surface of a long solid cylinder of radius \(R\). Select the correct statement(s):

[(A)] Magnetic field (\(B\)) is minimum along the axis of cylinder
[(B)] Magnetic field (\(B\)) is minimum at the surface of cylinder
[(C)] Magnetic field (\(B\)) is maximum at the surface of cylinder
[(D)] Magnetic field (\(B\)) is maximum along the axis of cylinder
[(E)] Magnetic field (\(B\)) is same all over the cross section of cylinder

• A. A, D
• B. A, C
• C. A, B, C, D
• D. A, C, D

Hint:

A cylinder carrying current only on its surface behaves like a {hollow conductor}: magnetic field is zero inside and maximum at the surface.

Answer 1

The Correct Option is B

Concept:
When current flows only along the surface
of a long cylinder, it behaves like a hollow cylindrical conductor
. Magnetic field is determined using Ampere’s circuital law
.

Step 1: Magnetic Field Inside the Cylinder (\(r<R\))
For any Amperian loop inside the cylinder: \[ I_{\text{enclosed}} = 0 \] By Ampere’s law: \[ \oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enclosed}} = 0 \] \[ \Rightarrow B = 0 \] Thus, magnetic field is zero everywhere inside
, including along the axis. \(\Rightarrow\) Statement (A) is correct
\(\Rightarrow\) Statement (D) is incorrect
\(\Rightarrow\) Statement (E) is incorrect

Step 2: Magnetic Field at the Surface (\(r = R\))
Just outside the surface: \[ B = \frac{\mu_0 I}{2\pi R} \] This is the maximum
value of magnetic field. \(\Rightarrow\) Statement (C) is correct
\(\Rightarrow\) Statement (B) is incorrect

Final Conclusion:
Correct statements are \(\boxed{A \text{ and } C}\).