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.

The differential line element dl is perpendicular to r, so the Biot-Savart law simplifies from the definition of the cross product

Note that the direction is into the web browser, so we have a −k. For the differential dl of a circular arc

plugging this in and setting up the integral gives

which simply yields