Question

Only the current inside the Amperian loop contributes in

• A. finding magnetic field at any point on the Ampere's loop
• B. line integral of magnetic field
• C. in both of the above
• D. in neither of them

Answer 1

Answer: (B)

line integral of magnetic field

Magnetic field at any point on Ampere's lop can be due to all currents passing through inside or outside the loop. But net contribution in the left hand side will come form inside current only.
For $r<a$ current passing through within the cylinder of radius $r$ is given by

${\int }_0^{r}JdA={\int }_0^{r}k{r}^{2}2\pi rdr=2\pi k{\int }_0^r{r}^{3}dr=k\pi r^4/2$

Now using Ampere's law:

$B\times 2\pi r={\mu }_{0}I={\mu }_{0}k\pi r^4/2\Rightarrow B=\frac{{\mu }_{0}k{r}^3}{4}$