Digital Signal Processing Reference
In-Depth Information
2.5 MAGNETOSTATICS
In Section 2.4 we discussed the problem of classical electrostatics, where we
defined the electric field in terms of the force that a collection of charges exert
on each other. In the case of electrostatics, we considered only cases where the
charges are at rest. Now it is time to consider the forces that exist between
charges in motion.
To begin this discussion, let's consider an experiment that most people per-
formed in high school physics class. If you recall, direct current (dc) (from a
battery) driving through a coil will produce an electromagnet. The magnetic field
produced by such a configuration can be descried by Ampere's law,
∇×
H
=
J
.
Note that the time dependence of the electric field
(∂D/∂t)
has been elimi-
nated because we are considering only a dc flow. Ampere's law tells us that
a steady-state current
H
that circulates around
the wire. As described in Example 2-2, the direction of the circulation can be
determined using the right-hand rule. If the thumb points in the direction of the
current flow, the fingers of the right hand will curl around in the direction of the
magnetic field. Subsequently, it is easy to imagine the form of the magnetic field
from a single loop of current in our electromagnet, as shown in Figure 2-15a.
Now consider a tiny elemental loop of current circulating around a point that
will induce a small magnetic field as shown in Figure 2-15b. This small current
loop will produce a small magnetic field that is analogous to an electric charge. In
fact, historically, scientists initially speculated that there was a magnetic charge
analogous to the electric charge described earlier. However, experimental evi-
dence suggests overwhelmingly that magnetic charges do not exist. Magnetic
fields are not generated by the forces that magnetic charges exert on each other;
rather, magnetic fields are generated by current loops.
Similar to our description of the forces between charges in electrostatics,
consider an isolated tiny current loop
(l
0
)
which will induce a magnetic field
and behave like a small electromagnet. Now, bring another tiny current loop
(l
1
)
from infinity into the magnetic field of
l
0
. If the orientations of the current
loops are similar, it will take work to push them together because magnets exert
forces on one another similar to electric charges. Like poles will repel each other,
and unlike poles will attract. Each “electromagnet” has its own north and south
J
will induce a magnetic field
B
B
l
dl
B
(a)
(b)
Figure 2-15
(a) Magnetic field generated by a loop of current; (b) elemental current
loop analogous to an electric charge.
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