Here we will examine two examples of the Biot-Savart law, one simple and the other more challenging. To begin we will find the magnetic field at the center of a current carrying loop as shown in figure 1
\begin{figure}
\includegraphics[scale=.8]{CurrentLoop.eps}
\vspace{10 pt}
\caption{Figure 1: Current Loop}
\end{figure}
this setup is the same as the quater loop example of Biot-Savart law where we have
giving us a similar integral, except from 0 to 2 and the lack of a minus sign since the current is going in the opposite direction so the magnetic field will be out of the web browser
taking the integral gives the magnetic field at the center of the loop
The second more challenging example is the magnetic field at a point z above the loop as shown in figure 2
\begin{figure}
\includegraphics[scale=.8]{AxisCurrentLoop.eps}
\vspace{10 pt}
\caption{Figure 2: Current Loop}
\end{figure}
The not so obvious hint is the direction of . The cross product of with leads to a vector perpendicular to both of them and as you go around the loop, will always be off the z axis by an angle . This makes all the horizontal components of cancel leaving just the vertical so
once again the differential is given as , so the integral to get the magnetic field is
From the geometry of the problem we see that
this leads to
substituting these relations into the integral
Finally, taking the integral gives us the magnetic field