Digital Signal Processing Reference
In-Depth Information
I
B
f
(
r
<
a
)
a
z
z
B
f
(
r
<
a
)
a
r
B
f
(
r
>
a
)
B
f
(
r
>
a
)
B
f
(
r
>
a
)
B
f
(
r
>
a
)
B
f
(
r
<
a
)
r
0
a
Figure 2-9
How the magnetic field will rotate around a wire carrying current.
The analysis above shows that the magnetic field has only a
φ
-component that
is perpendicular to the current flow, proving that the magnetic field will wrap
around the wire. Since the current flow is inducing the magnetic field, its intensity
will increase until
r
becomes greater than the wire radius
a
. When examining
fields outside the wire radius (where no current is flowing), the magnetic fields
will decrease, as shown in Figure 2-9.
We can confirm that
B
circulates around the wire by calculating the curl. The
curl of the magnetic field inside the wire can be calculated using the differential
form of Ampere's law for the static case:
B
µ
0
∇×
H
=
J
=∇×
F
The curl of
in cylindrical coordinate is (from Appendix A)
=
a
r
1
r
+
a
φ
∂F
r
+
a
z
1
r
∂F
z
∂φ
−
∂F
φ
∂z
∂F
z
∂r
∂(rF
φ
)
∂r
1
r
∂F
r
∂φ
∇×
F
∂z
−
−
The solution of the integral form of Ampere's law shows that the only component
of the magnetic field in the
φ
-direction is a function of
r
. Consequently, the curl
Search WWH ::
Custom Search