Question

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.

Answer 1

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`

r² = x² + R²

|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= dBcos 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 2πR.

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