A circular coil of radius R carries a curretn. Find an expression for the magnetic field due to this coil at its centre. Also find the direction of field.

Explanation

Consider a circular loop of radius R carrying a current I in clockwise direction. The loop may be considered to be divided into a large number of current elements.
Consider one such current element of length dl as shown in fig. Magnetic field at the centre point O of the loop due to this element, as per Bio-Sarvart.s law is
`dB = (mu_0)/(4pi) cdot (I dl sin 90^(@))/(R^2) = (mu_0)/(4 pi) cdot (I dl)/(R^2) " [ Here " = theta = 90^(@)]`
The total magnetic field due to current loop
`B = (mu_0)/(4 pi) cdot (I)/(R^2) sum dl = (mu_0 I)/(4 pi R^2) cdot (2 pi R) = (mu_0 I)/(2 R) [ :. sum dl = 2 pi R]`
If instead of a single loop we have a circular coil of N turns, then
`B = (mu_0 N I)/(2 R)`
The magnetic field is directed perpendicular to the plane of paper directed inward.