Derive the expression for magnetic field at a point on the axis of a circular current loop.

Explanation

Statement for Biot-Savart Law : The magnitude of magnetic field `vec(dB)` due to current element is directly proportional to the current I, the elements length |dl| and inversely proportional to the square of the distance r of the field point. Its direction is perpendicular to the plane containing `vec(dl)` and `vec(r)`.
`vec(dB) alpha ( I vec(dl)xx vec(r))/(r^(3))`
or `vec(dB)= (mu_(0))/(4 pi) * ( I vec(dl) xx vec(r))/(r^(3))`

The magnetic field due to `vec(dl)` is given by Biot Savart law as
`dB = (mu_(0))/(r pi) * ( I|vec(dl) xx vec(r)|)/(r^(3))`
Now `dB_(x) = dB cos theta = (mu_(0))/(4 pi) * ( I dl)/((x^(2)+R^(2))) cos theta`
`= (mu_(0))/(4 pi) * ( I dl)/((x^(2)+R^(2))) (R)/((x^(2)+R^(2))^(1//2))`
So, `B_(x)= int d B_(s) = (mu_(0))/(4 pi) ( IR)/((x^(2)+R^(2))^(3//2)) int dl`
`= (mu_(0))/(4 pi) (IR)/((x^(2)+R^(2))^(3//2)) 2 pi R`
`= ( mu_(0)IR^(2))/(2 (x^(2)+R^(2))^(3//2))`
(The y-compoents, of the field, add up to zero, due to symmetry)
`:.` Magnetic field at P due to a circular loop
`= B = B_(x) vec(i) = ( mu_(0)IR^(2))/(2(x^(2)+R^(2))^(3//2)) vec(i)`
Explanation : A circular current loop produces magnetic field and its magnetic moment is the product of current and its area `vec(M) = vec(I A)`