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 :
Certainly! Here’s how to respond to the question by explaining the magnetic field variation in a long straight solid wire carrying a steady current.
To find the variation of the magnetic field \( B \) with distance \( r \) from the axis of a solid wire carrying a steady current \( I \), we can use Ampère's Law and the concept of current distribution.
The given wire has a circular cross-section with radius \( a \), and the current is uniformly distributed across this cross-section. Let's consider two cases based on the distance \( r \) from the axis:
For \( r \leq a \) (inside the wire):
By symmetry, the magnetic field \( B \) at a distance \( r \) from the axis will be in the form of concentric circles. Using Ampère's Law:
\[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enclosed}}\]Here, \( I_{\text{enclosed}} = I \frac{\pi r^2}{\pi a^2} = I \left( \frac{r^2}{a^2} \right) \), since the current is uniformly distributed.
Therefore, \(B(2\pi r) = \mu_0 I \frac{r^2}{a^2}\).
Solving for \( B \):
\(B = \frac{\mu_0 I r}{2\pi a^2}\), which shows that \( B \) increases linearly with \( r \).
For \( r > a \) (outside the wire):
For any point outside the conductor, the magnetic field is determined by the total current \( I \) enclosed, irrespective of how it is distributed inside.
\[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I\]Thus, \(B (2\pi r) = \mu_0 I\).
Solving for \( B \):
\(B = \frac{\mu_0 I}{2\pi r}\), which shows that \( B \) decreases with \( \frac{1}{r} \).
The graph of \( B \) with respect to \( r \) will initially increase linearly inside the conductor (for \( r \leq a \)) and then decrease inversely outside the conductor (for \( r > a \)).
This corresponds to the correct representation of the magnetic field variation. Therefore, the correct option is represented by the image above.
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