Biot Savart Law

The law of Biot Savart allows the calculation of the magnetic field around arbitrary currents. Each element of a conductor at a point \(\vec{r}^{\prime}\) will contribute to the magnetic field at a point \(\vec{r}\) an amount \(\mathrm{d} \vec{B}\) according to

\[ \mathrm{d} \vec{B}(\vec{r})=\frac{\mu_0}{4 \pi} I \mathrm{~d} \vec{l} \times \frac{\vec{r}-\vec{r}^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|^3} \]

which will then lead to the total magnetic field one integrated over the whole volume.

\[ \vec{B}(\vec{r})=\frac{\mu_0}{4 \pi} \int_V \vec{\jmath}\left(\vec{r}^{\prime}\right) \times \frac{\vec{r}-\vec{r}^{\prime}}{\left|\vec{r}-\vec{r}^{\prime}\right|^3} \mathrm{~d} V^{\prime} \]

We would like to explore this relation by calculating the magnetic field around a circular loop of radius \(R\) centered at the origin.

Calculating the Magnetic Field of a Circular Current Loop

We’ll use the Biot-Savart law to calculate the magnetic field of a circular current loop of radius R in the xy-plane, centered at the origin. We’ll divide the loop into small segments and sum their individual contributions to find the total magnetic field.

This simulation divides a circular current loop into small segments and applies the Biot-Savart law to each segment. The first plot shows the magnetic field vectors in the xz-plane, while the second plot shows how the z-component of the field varies along the z-axis.

For the field along the z-axis, the numerical results can be compared with the analytical formula:

\[B_z = \frac{\mu_0 I R^2}{2(R^2+z^2)^{3/2}}\]

The characteristic “dipole” pattern is clearly visible in the vector plot, with field lines emerging from the center of the loop and curving around to form closed loops.