Probability and Stochastic Processes 2nd Roy D Yates and David J Goodman
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Transcript of Probability and Stochastic Processes 2nd Roy D Yates and David J Goodman
Lecture 7 Curl and Laplacian Operators
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Mathematic Operators in EM study
• Gradient
• Divergence
• Curl
• Laplacian
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Highlights
• Concept of Circulation
• Derivation of x B
• Stoke’s theorem
• Definition of 2V
• Definition of 2E
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What is Curl? • The curl of a vector is a measure of the circulation of
the vector field per unit area s, with the orientation of the unit area s chosen such that the circulation is maximum.
• The curl of a vector field B describes the rotational property.
• For a closed contour C, circulation=
• The curl is defined as
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Cd B l
0 max
1ˆcurl lim
Csd
s
B B n B l
The direction of the unit vector n is along the thumb when the other 4 fingers of the right hand follow dl
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Right-hand rule:
Curl In Cartesian Coordinate
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ˆ ˆ ˆ
ˆ ˆ ˆ
x y z
x y z
xB yB zB
x y z
x y z
B B B
B
B =
For Vector B,
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Curl is zero for uniform field
since there is no circulation.
Curl is non-zero for the
azimuthal field
Properties of the Curl
• x(A + B) =x A+x B ; A and B are vectors
• •( x A) = 0, the divergence of the curl of a vector field vanishes.
• x(V) = 0, the curl of the gradient of a scalar field vanishes.
• The curl of a vector field is another vector field;
• The curl of a scalar filed V, makes no sense.
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Stokes’s Theorem
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(s c
d d s B lx )B
If X B = 0, the field B is conservative, or
irrotational (as its circulation = 0)
Example 3-4: Verification of Stoke’s Theorem
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(s c
d d s B lx )B
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What is a Laplacian?
• Laplacian of a scalar function is defined as the divergence of the gradient of that function.
• Or we also can say: The divergence of the gradient of a scalar function is called the Laplacian.
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Laplacian Operator
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2 2 2
2 2 2
2 2 22
2 2 2
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
( )
( )
x y z
yx z
V V VV grad V x y z
x y z
A A A
x y z
AA AV
x y z
V V V
x y z
V V VV V
x y z
x y z A
x y z
A =
Recall the
gradient
And
Now take a
divergence
Laplacian is
defined as
Laplacian of a scalar function is defined as the
divergence of the gradient of that function.
For a vector field E
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2 2 22 2 2 2
2 2 2
2
ˆ ˆ ˆx y zE E E
x y z
E = E = x y z
E = ( E) - ( E)
With identity:
The definition of a scalar Laplacian can be used to
define a Laplacian of a vector filed
2 can also write as , call “del square”
or Laplacian.
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