Biot-Savart Law

According to the Biot-Savart law, the magnetic field generated at position vector ${\bf r}$ by a current $I_1$ circulating around a thin loop, an element of which is located at position vector ${\bf r}_1$ , is

$${\bf B}({\bf r}) = \frac{\mu_0\,I_1}{4\pi}\oint_1 \frac{d{\bf r}_1\times ({\bf r}-{\bf r}_1)}{\vert{\bf r}-{\bf r}_1\vert^{\,3}}.$$ (613)

Suppose that a second current loop carries the current $I_2$ . The net magnetic force exerted on an element, $I_2\,d{\bf r}_2$ , of this loop, located at position vector ${\bf r}_2$ , is

$$d{\bf F}_{21} = I_2\,d{\bf r}_2\times {\bf B}({\bf r}_2).$$ (614)

Hence, the net magnetic force exerted on loop 2 by loop 1 is

$${\bf F}_{21} = \frac{\mu_0\,I_1\,I_2}{4\pi}\oint_1\oint_2\frac{d{\bf r}_2\times (d{\bf r}_1\times {\bf r}_{12})}{\vert{\bf r}_{12}\vert^{\,3}},$$ (615)

where ${\bf r}_{12} = {\bf r}_2-{\bf r}_1$ .