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 at a point on the axis of a circular current-carrying loop can be derived using the Biot-Savart law: \[ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \] For a point on the axis, \( r \) is the distance from the element of the loop to the point where the field is calculated. By integrating the contributions from all current elements on the loop, we get the expression for the magnetic field at a point on the axis of the loop: \[ B = \frac{\mu_0 I}{2R} \left( \frac{1}{1 + (z/R)^2} \right)^{3/2} \] At the center of the loop (when \( z = 0 \)): \[ B = \frac{\mu_0 I}{2R} \] Thus, the magnetic field at the center of the loop is \( \frac{\mu_0 I}{2R} \), where \( R \) is the radius of the loop.