A straight section of wire of length L carries a current . (a) We have to show that the magnetic field associated with this segment at P, a perpendicular distance D from one end of the wire (see figure), is given by

(b) We have to show that the magnetic field is zero at point Q, along the line of the wire.

Solution:             Click For PDF Version

Using the Biot-Savart law we write the expression for the magnetic field at the point P due to an infinitesimal segment of wire of length carrying current . We have

where is a unit vector perpendicular to the plane of the figure.

But

Therefore,

We make the following substitution in the integral

We get

where we have used

(b)

As the point Q is along the line segment carrying the current , from the Biot-Savart law, we note that as the vectors and are parallel, their cross product is zero. Hence the magnetic field due to the line segment will be zero at Q.