Question

A long, cylindrical tube of inner and outer radii a and b carries a current i distributed uniformly over its cross section. Find the magnitude of the magnitude filed at a point (a) just inside the tube (b) just outside the tube.

Answer 1

a) The magnetic field inside any conducting tube is always zero.
∴ Magnetic field just inside the tube is zero.

(b) Let the point outside the tube with distance b be P.
Consider an Amperian loop, as shown in the figure.
Length of the loop, l = $2\mathrm{\pi }\times b=2\mathrm{\pi }b$
Current enclosed in the loop = i
On applying Ampere's law, we get
$$\begin{gathered} \int B.dl={\mu }_{0}i \\ \Rightarrow B\times 2\mathrm{\pi }b={\mu }_{0}i \\ \Rightarrow B=\frac{{\mu }_{0}i}{2\mathrm{\pi }b} \end{gathered}$$