Question

Figure A and B show two long straight wires of circular cross-section (a and b with a<b), carrying current I which is uniformly distributed across the cross-section. The magnitude of magnetic field B varies with radius r and can be represented as : 

• A. A
• B. B
• C. C
• D. D

Hint:

For a current-carrying wire, the magnetic field is maximum at its surface. For the same total current, a wire with a smaller radius will have a stronger maximum magnetic field at its surface.

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the magnetic field produced by long straight wires with current flowing through them. In this situation, the current is uniformly distributed across the circular cross-section of each wire.

Concept: The magnetic field inside a current-carrying wire varies with the distance from the center of the wire. For a point inside a long straight wire, the magnetic field at a distance \( r \) from the center of the wire is given by:

\(B = \frac{\mu_0 I r}{2 \pi a^2}\) for \( r \leq a \) 

where \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( a \) is the radius of the wire.

Outside the wire, the field is determined only by the total current \( I \) and the distance from the center:

\(B = \frac{\mu_0 I}{2 \pi r}\) for \( r > a \)

Analysis:

1. For wire in Figure A with radius \( a \):
   • Inside the wire (\(0 < r < a\)): The magnetic field increases linearly with \( r \).
   • Outside the wire (\(r > a\)): The magnetic field decreases as \(1/r\).
2. For wire in Figure B with radius \( b \) (\(a < b\)):
   • Inside the wire (\(0 < r < b\)): The magnetic field increases linearly with \( r \), but since \( b > a \), the rate of increase in Figure B's wire is less steep compared to Figure A.
   • Outside the wire (\(r > b\)): The magnetic field decreases as \(1/r\).

Conclusion: In the option, we are given graphs showing these characteristics. The correct graph will show the specified linear increase inside the wires and a \(1/r\) decrease outside. Option A aligns with this reasoning for both wires given their size differences. Therefore, the correct answer is:

A