- Oersted (1820) discovered that an electric current of
magnitude flowing in a closed
loop of area, , where is normal to the plane of
the loop, produces the same magnetic field as a magnetic dipole of
moment

This is shown is Figure 3.7 *Outside regions where currents flow*we have a potential magnetic field. Thus,

Hence, the magnetic field can be expressed in terms of a magnetic potential as and we have

In cylindrical polars (3.20) becomes

A particularly simple solution is

for which

The field lines are circles on cylinders about the z-axis. The total current through a circle of radius , that is an open surface, is

on using Stokes' theorem. is a closed curve around the open surface . Thus,

Thus, the total current is simply and it flows along the z-axis. This is a*line current*.*Inside regions where currents flow.*Here we must satisfy the equations

where is the current density. Thus, we can solve (3.21) by writing , where is the*vector potential*. If we have , (3.22) becomes

The left hand side may be simplified by using a vector identity

Therefore,

This is a vector form of Poisson's equation. In the same way that the potential due to a volume charge satisfying is

where

so the general solution of (3.23) is

For a wire carrying a current I in an element , this reduces to

Therefore, the magnetic field is given by

Thus, we have

(3.25) is called the*Biot-Savart Law*.

__Example 3. .4__For example, consider the magnetic field due to a current flowing
in an infinite straight wire. Using the Biot-Savart law, (3.25), in
cylindrical coordinates, we have

Finally, we have

Hence, using (3.27), we have

Therefore, using the Biot-Savart law we have

Integrating we obtain

as obtained before.