Question

A thin but long, hollow, cylindrical tube of radius r carries i along its length. Find the magnitude  of the magnetic field at a distance r/2 from the surface (a) inside the tube (b) outside the tube.

Answer 1

(a) The magnetic field inside any conducting tube is always zero.
∴ Magnetic field inside the tube at a distance r/2 from the surface = 0 

(b) Let the point outside the tube with distance \[\frac{r}{2}\]  be P.

∴ Net distance from centre, r' = \[r + \frac{r}{2} = \frac{3r}{2}\]

Consider an Amperian loop, as shown in the figure.
Length of the loop, l = \[2\pi \times \frac{3}{2}r = 3\pi r\]

Current enclosed in the loop = i
On applying Ampere's law, we get

\[\int B . dl = \mu_0 i\]
\[ \Rightarrow B \times 3\pi r = \mu_0 i\]
\[ \Rightarrow B = \frac{\mu_0 i}{3\pi r}\]