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

To solve this problem, we need to understand how the magnetic field behaves around a long straight conductor carrying current. The magnetic field due to a long straight wire can be analyzed using Ampere's Law or the Biot-Savart Law. However, considering a cylindrical conductor with surface current, the behavior is slightly modified as follows: 

1. Magnetic Field Inside the Cylinder: Inside a conductor, if we consider a radius \(r < R\), the magnetic field due to circulating currents is generally negligible. Since the current flows on the surface, there is no enclosed current inside the cylinder.

2. Magnetic Field on the Surface of the Cylinder: At \(r = R\), the magnetic field is strongest because all the current is effectively at this radius. This can be estimated using Ampere's Circuital Law, where: \(B = \frac{\mu_0 I}{2 \pi r}\) When \(r = R\), it simplifies to a non-zero maximum value at the surface.

3. Magnetic Field Along the Axis of the Cylinder: The axis of the cylinder is essentially a point where no current circulates around it, thus rendering the magnetic field minimum or effectively zero at the axis.

Now, let us analyze the provided statements:

• (A) Magnetic field (\(B\)) is minimum along the axis of cylinder: This is correct since no current circulates around the axis.
• (B) Magnetic field (\(B\)) is minimum at the surface of cylinder: This is incorrect as the field is maximized at the surface.
• (C) Magnetic field (\(B\)) is maximum at the surface of cylinder: This is correct because it experiences the full effect of surface current.
• (D) Magnetic field (\(B\)) is maximum along the axis of cylinder: Incorrect, as the field is zero along the axis.
• (E) Magnetic field (\(B\)) is same all over the cross section of cylinder: Incorrect, as it changes depending on the radius \(r\).

Based on the explanations above, the correct statement options are A and C.