Question

Using Biot-Savart law, derive expression for the magnetic field \( \vec{B} \) due to a circular current carrying loop at a point on its axis and hence at its center.

Hint:

The magnetic field due to a circular current-carrying loop is derived by integrating the magnetic field contributions of all current elements using Biot-Savart's law.

Answer 1

The magnetic field along the axis of a circular current-carrying loop is determined via the Biot-Savart law: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \]. Here, \( r \) represents the distance from a loop element to the calculation point on the axis. Integrating these contributions across the entire loop yields the magnetic field intensity \( B \) on the axis: \[ B = \frac{\mu_0 I}{2R} \left( \frac{1}{1 + (z/R)^2} \right)^{3/2} \]. At the loop's center (where \( z = 0 \)): \[ B = \frac{\mu_0 I}{2R} \]. Consequently, the magnetic field at the center is \( \frac{\mu_0 I}{2R} \), with \( R \) denoting the loop's radius.