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 = $2\mathrm{\pi }\times \frac{3}{2}r=3\mathrm{\pi }r$
Current enclosed in the loop = i
On applying Ampere's law, we get
$$\begin{gathered} \int B.dl={\mu }_{0}i \\ \Rightarrow B\times 3\mathrm{\pi }r={\mu }_{0}i \\ \Rightarrow B=\frac{{\mu }_{0}i}{3\mathrm{\pi }r} \end{gathered}$$