Digital Signal Processing Reference
In-Depth Information
the magnetic field. Therefore, the flux lines of a magnetic field consist of closed
lines.
2.2.8 Curl
Historically, the concept of
curl
comes from a mathematical model of hydro-
dynamics. Early work by Helmholtz studying the vortex motion of fluid led
ultimately to Maxwell's and Faraday's conceptions of electric fields induced by
time-varying magnetic fields, which is shown in equation (2-1) [Johnk, 1988]. To
visualize the concept of the curl, consider a paddle wheel immersed in a stream
of water, with a velocity field as shown in Figure 2-8. In Figure 2-8a, the paddle
is oriented along the
z
-axis perpendicular to the water flow, and since the velocity
of the fluid is larger on the top of the paddle, the paddle will rotate clockwise,
and therefore has a finite curl along the
z
-axis, with a direction pointing into the
page as determined using the right-hand rule. Similarly, if the paddle is rotated so
that it is oriented along the
x
-axis, as in Figure 2-8b, the paddle will not rotate,
and the curl is zero.
The curl of
F
(
x
,
y
,
z
,
t
) in determinant form (in rectangular coordinates) is
shown as
a
x
a
y
a
z
∂
∂x
∂
∂y
∂
∂z
∇×
F(x,y,z,t)
=
(2-23)
F
x
F
y
F
z
which simplifies to
=
a
x
∂F
z
+
a
y
∂F
x
+
a
z
∂F
y
∂F
y
∂z
∂F
z
∂x
∂F
x
∂y
∇×
F
∂y
−
∂z
−
∂x
−
(2-24)
y
y
n
n
x
x
z
z
(a)
(b)
Figure 2-8
(a) The fluid velocity field causes the paddle wheel to rotate when it is
oriented orthogonal to the field, giving it a nonzero curl with a direction pointing into the
page (
z
); (b) when the paddle wheel is parallel to the field it has a zero curl because
the field will not make it rotate.
−
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