Give the expression for magnetic field at a point on the axis of a short magnetic dipole.

Explanation

I `to` Current in the loop
`R to ` Radius of the loop
X-axis `to` Axis of the loop
x `to` Distance OP
dl `to` Conducting element of the loop
According to Biot-Savart.s law, the magnetic field at P is
`db=(mu_0I|dlxxr|)/(4pir^3)`
`r^2=x^2+R^2`
`|dl xx r | =rdl` (`because` they are perpendicular )
`therefore dB=mu_0/(4pi). (Idl)/(x^2+R^2)`
i.e., dB has two components `-dB_x` and `dB_1, dB_1` is cancelled out and only the x-component remains.
`therefore dB_x=dBcos theta`
`cos theta =R/(x^2+R^2)^(1//2)`
`therefore dBx=(mu_0Idl)/(4pi). d/(x^2+R^2)^(3//2)`
Summation of dl over the loop is given by `2piR`.
`therefore B=(mu_0 IR^2)/(2(x^2+R^2)^(3//2))hati`