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: ____.
Show 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 $1/r^2$).
Step 1: Understanding the Concept:
According to Ampere's Circuital Law, the magnetic field produced by a long straight wire depends on whether the observation point is inside or outside the conductor. Step 2: Key Formula or Approach:
1. Inside the wire ($r < a$): $B_{in} = \frac{\mu_0 I r}{2\pi a^2} \implies B \propto r$ (Linear)
2. Outside the wire ($r \geq a$): $B_{out} = \frac{\mu_0 I}{2\pi r} \implies B \propto \frac{1}{r}$ (Hyperbolic) Step 3: Detailed Explanation:
1. Internal Region: At the axis ($r=0$), the enclosed current is zero, so $B=0$. As $r$ increases toward the surface, the enclosed current increases with the area ($\pi r^2$), resulting in a linear increase in $B$.
2. At the Surface: $B$ reaches its maximum value at $r=a$.
3. External Region: Once outside the wire, the total current $I$ is constant. As distance $r$ increases, the magnetic field strength drops following an inverse relationship ($1/r$). Step 4: Final Answer:
The correct plot shows a straight line from the origin to the surface, followed by a rectangular hyperbola outside.
