Question

The figure given below shows a long straight solid wire of circular cross-section of radius ‘a’ carrying steady current I. The current I is uniformly distributed across its cross-section. The plot which correctly represents the variation of magnetic field (B) with distance (r) from the axis of the conductor in the region is: ____.

• A. A
• B. B
• C. C
• D. D

Hint:

This is a very common graph in physics. Remember: inside the "source" (wire/sphere) it's usually linear ($r$), and outside it's always an inverse law ($1/r$ or $1/r^2$).

Answer 1

The Correct Option is B

Step 1: Understanding the Topic:
This problem is from "Moving Charges and Magnetism," specifically applying "Ampere’s Circuital Law." It examines how the magnetic field strength changes as we move from the center of a thick wire to its surface, and then into the surrounding space.
Step 2: Key Formulas and Approach:
For a solid wire of radius '$a$' carrying current '$I$':
Inside ($r<a$): $B_{in} = \frac{\mu_0 I r}{2\pi a^2} \implies B \propto r$.
Outside ($r>a$): $B_{out} = \frac{\mu_0 I}{2\pi r} \implies B \propto \frac{1}{r}$.

Step 3: Detailed Explanation:

Inside the wire ($r<a$): As you move out from the center, the amount of current enclosed by an Amperian loop of radius '$r$' increases with the area ($I_{encl} \propto r^2$). According to Ampere's Law ($B \cdot 2\pi r = \mu_0 I_{encl}$), the magnetic field increases linearly with distance. This results in a straight line starting from the origin $(0,0)$.
At the surface ($r = a$): The magnetic field reaches its maximum value, $B_{max} = \frac{\mu_0 I}{2\pi a}$.
Outside the wire ($r>a$): The entire current $I$ is now enclosed regardless of how much further you move. The field now decreases as $1/r$ (an inverse relationship), which is represented by a rectangular hyperbola curve that approaches but never touches the x-axis.
Comparing these behaviors to the provided plots, Plot (2) correctly shows the linear rise followed by the hyperbolic decay.
Step 4: Final Answer:
The correctly representing plot is Plot (B).