Question

Using Ampere's circuital law, obtain an expression for the magnetic flux density 'B' at a point 'X' at a perpendicular distance 'r' from a long current-carrying conductor.
(Statement of the law is not required).

Answer 1

Let AB, a long straight wire is carrying current i. The magnetic flux density `bar(B)` is to be required at point X at a distance r.
Let us draw a circle of radius r with centre O as shown in above figure. The point X lies on the circumference of the circle.
According to Ampere's circuital law,
                      `oint  bar(B). bar(dl)= μ_0i. "where " bar(dl)` = infinitesimally small portion XY in the circle.

For complete circle, `bar(dl)= 2pir`
∴                        `bar(B). 2pir = μ_0i`

⇒                               `bar (B) = (mu_0)/(2pi).i/r NA^(-1)m^(- 1)`

⇒                               `bar (B) = (mu_0)/(4pi).(2i)/r`

∴                               `bar (B) = 10^-7 .(2i)/r NA^(-1)m^(- 1)`