Question

A hollow tube is carrying an electric current along its length distributed uniformly over its surface. The magnetic field
(a) increases linearly from the axis to the surface
(b) is constant inside the tube
(c) is zero at the axis
(d) is zero just outside the tube.

Answer 1

(b) is constant inside the tube
(c) is zero at the axis

A hollow tube is carrying uniform electric current along its length, so the current enclosed inside the tube is zero.
According to Ampere's law,
$$\begin{gathered} \oint \overrightarrow{B}.d\overrightarrow{l}={\mathrm{\mu }}_{\mathrm{o}}{i}_{\mathrm{inside}} \\ \mathrm{Inside}\mathrm{the}\mathrm{tube}, \\ \oint \overrightarrow{B}.d\overrightarrow{l}=0,r<R \\ \Rightarrow B_{\mathrm{inside}}=\mathrm{Constant} \\ \Rightarrow B_{\mathrm{axis}}=0 \end{gathered}$$

The magnetic fields from points on the circular surface will point in opposite directions and cancel each other.
$$\begin{gathered} \mathrm{Outside}\mathrm{the}\mathrm{tube}, \\ B\times 2\mathrm{\pi }r={\mathrm{\mu }}_{\mathrm{o}}i \\ \Rightarrow B_{\mathrm{outside}}=\frac{{\mathrm{\mu }}_{\mathrm{o}}i}{2\mathrm{\pi }r},r>R \end{gathered}$$