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## Magnetic effects of currents

1. 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
 (3.19)

This is shown is Figure 3.7

2. 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
 (3.20)

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.

3. Inside regions where currents flow. Here we must satisfy the equations
 (3.21) (3.22)

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,
 (3.23)

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
 (3.24)

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)

(3.25) is called the Biot-Savart Law.

Example 3. .4For 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

 (3.26)

The situation is shown in Figure 3.8.

The field at the general point is obtained by relating , and with held fixed. From Figure 3.8 we have
 (3.27)

Finally, we have

Hence, using (3.27), we have

Therefore, using the Biot-Savart law we have

Integrating we obtain

as obtained before.

Next: Summary Up: Magnetic Fields Previous: Magnetic Fields
Prof. Alan Hood
2000-11-06