Use Biot-Savart law to derive the expression for the magnetic field on the axis of a current carrying circular loop of radius *R*.

Draw the magnetic field lines due to a circular wire carrying current *I*.

#### Solution

*I* = Current in the loop

*R* = Radius of the loop

*X*-axis = Axis of the loop

*X* = Distance between O and P

*dl* = Conducting element of the loop

According to the Biot**–**Savart law, the magnetic field at P is

`dB=(mu_0)/(4pi) (I|dlxxr|)/r^3`

*r*^{2} = *x*^{2} + *R*^{2}

|*dl* × *r*| = *rdl* (Because they are perpendicular)

`:.dB=mu_0/(4pi) (Idl)/((x^2+R^2))`

*dB* has two components: *dB*_{x} and *dB*_{⊥}. *dB*_{⊥} is cancelled out and only the *x*-component remains.

∴ *dB*_{x}= *d*Bcos *θ*

`costheta= R/(x^2+R^2)^(1/2)`

`:.dB_x=(mu_0Idl)/(4pi) R/(x^2+R^2)^(3/2)`

Summation of *dl* over the loop is given by 2π*R*.

`:.B=B=B_xhati=(mu_0IR^2)/(2(x^2+R^2)^(3/2))hati`