State Biot- Savart law. Use to obtain the magnetic field at a point due to a long, staright current carrying wire.

Explanation

For statement of Biot - Savart law, refer to point Number 6 under the heading "Chapter At A Glance". Let CD be a long, staright current carrying wire and P be a point situated at a normal distance R from the wire. Consider a current element `I vecdl` situated at a distance OA = I. If AP = r, then due to
`dB = (mu_0)/(4pi) (I dl sin theta)/(r^2)`
As per figure `sin theta = sin (90^@ - phi) = cos theta`
`:. I = R tan phi`
`:. dl = R sec^2 phi cdot d phi`
and `r = R sec phi`
`:. dB = (mu_0)/(4pi)cdot ((R sec^2 phi_2.d phi)cos phi)/((R sec phi^2)) = (mu_0)/(4pi) cdot (I cos phi cdot d phi)/(R)`
Therefore, the magnetic field due to whole wire CD will be
`B = (mu_0 I)/(4 pi R) int_(-phi_1)^(phi_2) cos phi . d phi = (mu_0 I)/(4 pi R) [sin phi_2 - sin (-phi_1)] = (mu_0 I)/(4 pi R) [ sin phi_2 + sin phi_2]`
Here `phi_2` has been taken +ve but `phi_1` has been taken -ve because it is in a direction opposite to that of `phi_2`
For a long long wire (or infinitely long wire) `phi_1 = phi_2 = 90^@`, therefore
`B = (mu_0 I)/(2 pi R)`