PlanetPhysics/Quarter Loop Example of Biot Savart Law

A simple example of the Biot-Savart law is given with a current carrying loop for a quater of a circle as shown in Figure 1.

\begin{figure} \includegraphics[scale=.8]{QuarterCurrentLoop.eps} \vspace{10 pt} \caption{Figure 1: Quarter Current Loop} \end{figure}

The differential line element d{\mathbf l} is perpendicular to ${\hat {\mathbf {r} }}$, so the Biot-Savart law simplifies from the definition of the cross product $d{\mathbf {B} }=-{\frac {\mu _{0}}{4\pi }}{\frac {Idl\,sin(\pi /2)}{r^{2}}}{\hat {\mathbf {k} }}$

Note that the direction is into the web browser, so we have a $-{\hat {\mathbf {k} }}$. For the differential dl of a circular arc

$dl=rd\theta$

plugging this in and setting up the integral gives

${\mathbf {B} }=-{\hat {\mathbf {k} }}{\frac {\mu _{0}I}{4\pi r}}\int _{0}^{\frac {\pi }{2}}\,d\theta$

which simply yields

${\mathbf {B} }=-{\frac {\mu _{0}I{\hat {\mathbf {k} }}}{8r}}$