Question

A cylindrical conductor of radius R is carrying a constant current. The plot of the magnitude of the magnetic field, B with the distance, d, from the centre of the conductor, is correctly represented by the figure :

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is C

 To solve this problem, we need to determine the plot of the magnetic field magnitude, \( B \), as a function of the distance, \( d \), from the center of a cylindrical conductor carrying a constant current. The behavior of the magnetic field around the conductor can be explained by Ampere's Circuital Law.

Concept Explanation:

The magnetic field inside and outside a cylindrical conductor depends on the distribution of current and is determined using Ampere’s Circuital Law:

\(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}\),

where:

• \(\mathbf{B}\) is the magnetic field,
• \(\mu_0\) is the permeability of free space,
• \(I_{\text{enc}}\) is the enclosed current.

Analysis and Solution:

1. For the region inside the conductor (\(0 \leq d \leq R\)), the current enclosed by a loop of radius \(d\) is proportional to the square of the distance \(d\). Hence, the magnetic field \(B\) is directly proportional to \(d\), given by:

\(B = \frac{\mu_0 I}{2 \pi R^2} d\).

1. For the region outside the conductor (\(d > R\)), the entire current \(I\) is enclosed. Thus, the magnetic field decreases with an increase in distance \(d\) and is inversely proportional to it:

\(B = \frac{\mu_0 I}{2 \pi d}\).

1. The plot will thus show a linear increase in \(B\) from \(0\) to \(R\), followed by a hyperbolic decrease beyond \(R\).

Correct Option:

The plot that matches this behavior is the third option, as depicted below:

This image correctly shows a linear rise in magnetic field within the conductor and a decrease outside it, matching our analysis.