Digital Signal Processing Reference
In-Depth Information
Simply put, if the curl is finite, a field will be induced that possesses circulation.
This allows us intuitively to understand Faraday's and Ampere's laws:
∂ B
∂t
=
∇×
E
+
0
(
Faraday's law
)
∂ D
∂t
∇×
H
=
J
+
(
Ampere's law
)
Faraday's law says that a time-varying magnetic field will induce an electric field
that possesses circulation around
B
. More intuitively, if we examine Ampere's
law for a steady-state current, it reduces to
∇×
H
J
=
(2-25)
Equation (2-25) implies that a current flowing in a wire will induce a magnetic
field that circulates around the wire, which is consistent with Gauss's law for
magnetism (2-4), which implies that the flux lines of a magnetic field must consist
of closed lines.
Example 2-2
Calculate the magnetic field of a current
I
flowing through an
infinitely long wire of radius
a
. Show that the current flowing in the wire induces
a magnetic field that circulates around the
z
-axis. See Figure 2-9.
SOLUTION To solve this problem it is necessary to present the integral form
of Ampere's law for static fields:
B
µ
0
·
S
J
·
dl
=
ds
=
i
(2-26)
l
B
=
a
φ
B
φ
and
dl
=
a
φ
rdφ
, yield-
Switching to a cylindrical coordinate system,
ing
2
π
B
φ
µ
0
rdφ
=
2
πrB
φ
µ
0
=
i
0
iµ
0
2
πr
B
φ(r>a)
=
for
r>a
To calculate the magnetic field inside the conductor, only the amount of current
passing through a percentage of the wire area must be considered. This is achieved
by expressing the current in terms of an area ratio:
2
π
=
i
πr
2
πa
2
B
φ
µ
0
rdφ
=
2
πrB
φ
µ
0
0
iµ
0
r
2
πa
2
B
φ(r<a)
=
for
r<a
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