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

Explanation:

a) Inside any conductive tube the magnetic field is always zero.

∴ Magnetic field at $\frac{r}{2}$ distance from the surface inside the tube = 0

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

∴ Net distance to the middle, $r^{\prime}=r+\frac{r}{2}=\frac{3 r}{2}$

Consider an Amperian loop, as the figure shows.

Loop Length, $1=2 \pi \times \frac{3}{2} r=3 \pi r$

Enclosed current in the loop = I

If we apply the law of Ampere, we obtain

$\int B . d l=\mu_{0} i$

$B \times 3 \pi r=\mu_{0} i$

$B=\frac{\mu_{0} i}{3 \pi r}$