Question

Using Biot-Savart law, deduce the expression for the magnetic field at a point (x) on the axis of a circular current carrying loop of radius R. How is the direction of the magnetic field determined at this point?

Answer 1

Magnetic field on the axis of a circular current loop

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 =(μ_0 I | dl xxr)/(4π   r^3)`

r² = x² + R²

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

`dB =(μ_0      I  dl )/(4π   (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

`cos θ= R/ (x^2 + R^2)^(3/2) hati` 

Summation of dl over the loop is given by 2πR.

∴ `B = B= B_x hati = (μ_0   I R^2)/(2(x^2 + R^2)^(3/2))  hati `

The direction of

\[d B^\rightharpoonup\] is in the plane perpendicular to \[{dl}^\rightharpoonup \text { and } r^\rightharpoonup\]

and is directed as given by right handed screw rule, i.e. the direction is along the axis and away from the loop.