Write using Biot-Savart law, the expression for the magnetic field `vecB` due to an element `vecdl` carrying current I at a distance `vecr` from it in vector form. Hence, derive the expression for the magnetic field due to a current loop of radius R at a point P distant x from its centre along the aixs of the loop.

Explanation

Consider a circular loop of radius R, carrying a current l. Plane of the coil is Y - Z plane (i.e, perpendicular to the plane of paper), while the axis of the loop OX lies in the plane of paper. Consider the circular loop to be divided into large number of current elements `I vecdl`. Two such elements `N_1 M_1` and `N_2M_2`, diametrically opposite to each other, produce the magnetic fields `vecdB_1` and `dvecB_2` perpendicular to both `vecr` and `vecdl` and given by right hand rule as,
`|dvecB_1| = |dvecB_2| = (mu_0)/(4pi) cdot (I d l sin 90^@)/(r^2) = (mu_0)/(4pi) (I dl)/((R^2 + x^2))`
These field may be resolved into components along x-aix and y - axis. Obviously components along y-axis balance each other but components along x-axis and y-axis. Obviously components along y-axis balance each other but components along x-axis are all summed up. Hence, magnetic field due to whole current loop will be :
`B = oint dB sin phi = oint (mu_0)/(4pi) (I dl)/((R^2 + x^2)) cdot (R)/(sqrt((R^2 + x^2))) = (mu_0 I R)/(4pi (R^2+ x^2)^(3//2)) oint dl`
`= (mu_0 IR)/(4pi (R^2 + x^2)^(3//2)) cdot 2 pi R = (mu_0 I R^2)/(2(R^2 + x^2)^(3//2))`