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- 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
Figure 3.7:
A current loop with area
produces the same
magnetic field as a dipole.
|
- 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.
- 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,
 |
(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.
Figure 3.8:
The general point
in relation to a current element
of an infinite straight wire.
|
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