vector and scalar potentials - south dakota school of...
TRANSCRIPT
Hao Mei
The University of South Dakota
Wednesday, Nov. 05, 2014
Vector and Scalar Potentials
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Maxwell Equations
2
๐ป ยท ๐ท = ฯ Coulombโs Law
๐ป ร ๐ป = ๐ฝ +๐๐ท
๐๐ก Ampereโs Law
๐ป ยท ๐ต = 0 Absence of free magnetic poles
๐ป ร ๐ธ +๐๐ต
๐๐ก= 0 Faradayโs Law
Potentials(ฮฆ and ๐ด )
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If a curl of a vector field (๐น ) vanishes(everywhere), then ๐น can be written as the gradient of a scalar potentials (ฮฆ):
๐ป x ๐น = 0 โบ ๐น = โ ๐ปฮฆ
If a divergence of a vector field (๐น ) vanishes(everywhere),
then ๐น can be written as the curl of a vector potentials (๐ด ):
๐ป ยท ๐น = 0 โบ ๐น = ๐ป x ๐ด
Potentials(ฮฆ and ๐ด )
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Since ๐ป ยท ๐ต = 0, we can still define ๐ต in terms of a vector potential:
๐ต = ๐ป x ๐ด (1)
Then the Faradayโs law can be written:
๐ป ร ๐ธ +๐๐ต
๐๐ก= ๐ป ร ๐ธ +
๐(๐ป x ๐ด )๐๐ก
= ๐ป ร (๐ธ +๐๐ด
๐๐ก)
= 0
Potentials(ฮฆ and ๐ด )
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Recall the definition of scalar potentials:
๐ป x ๐น = 0 โบ ๐น = โ ๐ปฮฆ
here we have
๐ป ร ๐ธ +๐๐ด
๐๐ก= 0
The vanishing curl means that we can define a scalar potential ฮฆ satisfying:
โ๐ปฮฆ = ๐ธ +๐๐ด
๐๐ก
or ๐ธ = โ๐ปฮฆ โ๐๐ด
๐๐ก (2)
Maxwell equations in terms of Vector and Scalar Potentials
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Combining Equations (1) and (2),
๐ต = ๐ป x ๐ด , ๐ธ = โ๐ปฮฆ โ๐๐ด
๐๐ก
These two equations, which is the definitions of ๐ต and ๐ธ in
terms of ฮฆ and ๐ด , automatically satisfy the two homogeneous Maxwell equations.
This reduces the number of equations from 4 to 2.
Then the dynamic behavior of ฮฆ and ๐ด will be determined by the two inhomogeneous equations.
Maxwell equations in terms of Vector and Scalar Potentials
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At this stage we restrict our considerations to the vacuum form of the Maxwell equations.
Recall that ฯต0ฮผ0 =1
๐2 , and ๐ป =๐ต
ฮผ0, ๐ท = ฯต0๐ธ.
Then the two inhomogeneous equations become
๐ป ยท ๐ท = ฯ ๐ป ยท ๐ธ =ฯ
ฯต0
๐ป ยท (โ๐ปฮฆ โ๐๐ด
๐๐ก) =
ฯ
ฯต0
๐ป2ฮฆ +๐(๐ปยท๐ด )
๐๐ก= โ
ฯ
ฯต0 (3)
Maxwell equations in terms of Vector and Scalar Potentials
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๐ป ร ๐ป = ๐ฝ +๐๐ท
๐๐ก, ๐ป =
๐ต
ฮผ0, ๐ท = ฯต0๐ธ
๐ป ร๐ต
ฮผ0= ๐ฝ +
๐(ฯต0๐ธ)
๐๐ก
๐ป ร ๐ต = ฮผ0๐ฝ + ฯต0ฮผ0๐๐ธ
๐๐ก
๐ป ร ๐ป x ๐ด = ฮผ0๐ฝ + ฯต0ฮผ0๐
๐๐ก(โ๐ปฮฆ โ
๐๐ด
๐๐ก)
= ฮผ0๐ฝ โ ฯต0ฮผ0๐
๐๐ก(๐ปฮฆ +
๐๐ด
๐๐ก)
Maxwell equations in terms of Vector and Scalar Potentials
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Using the identity( in Jackson cover):
๐ป ร ๐ป x ๐ด = ๐ป ๐ปยท๐ด โ ๐ป2๐ด
๐ป2๐ด โ1
๐2
๐2๐ด
๐๐ก2 โ ๐ป(๐ปยท๐ด +1
๐2
๐ฮฆ
๐๐ก) = โฮผ0๐ฝ (4)
The four first order coupled differential equations (Maxwell equations) reduce to two second order differential equations, but they are still coupled.
๐ป2ฮฆ +๐(๐ป ยท ๐ด )
๐๐ก= โ
ฯ
ฯต0
๐ป2๐ด โ1
๐2
๐2๐ด
๐๐ก2 โ ๐ป(๐ปยท๐ด +1
๐2
๐ฮฆ
๐๐ก) = โฮผ0๐ฝ
Gauge Transformation
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Since ๐ต = ๐ป x ๐ด , the vector potential is arbitrary to the extent that the gradient of some scalar function ษ can be added.
๐ต is unchanged by the transformation:
๐ด ๐ดโฒ = ๐ด + ๐ปษ
โข For ๐ธ to remain unchanged as well, we require
ฮฆฮฆโฒ = ฮฆ โ๐ษ
๐๐ก
Quick check: ๐ธ = โ๐ปฮฆ โ๐๐ด
๐๐ก
โ๐ปฮฆโฒ โ๐๐ดโฒ
๐๐ก= โ๐ป ฮฆ โ
๐๐ด
๐๐กโ
๐ ๐ด + ๐ปษ
๐๐ก
= โ๐ปฮฆ +๐
๐๐ก๐ปษ โ
๐๐ด
๐๐กโ
๐
๐๐ก๐ปษ
= ๐ธ
Lorenz condition and wave equations
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We can use gauge freedom to specify useful conditions on
ฮฆ, ๐ด .
Until now only ๐ป x ๐ด has been specified; the choice of ๐ปยท๐ด is still arbitrary. Imposing the so-called Lorenz condition:
๐ปยท๐ด +1
๐2
๐ฮฆ
๐๐ก= 0
Applying the Lorenz condition, it will decouples Eqs.(3) and (4), and results in a considerable simplification:
Lorenz condition and wave equations
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For Equation (3): ๐ป2ฮฆ +๐(๐ปยท๐ด )
๐๐ก= โ
ฯ
ฯต0
applying the Lorenz condition:
๐ปยท๐ด +1
๐2
๐ฮฆ
๐๐ก= 0
So, ๐ป2ฮฆ โ๐(
1
๐2๐ฮฆ
๐๐ก)
๐๐ก= โ
ฯ
ฯต0
๐ป2ฮฆ โ1
๐2
๐2ฮฆ
๐๐ก2 = โฯ
ฯต0 (5)
Lorenz condition and wave equations
13
For Equation (4): ๐ป2๐ด โ1
๐2
๐2๐ด
๐๐ก2 โ ๐ป(๐ปยท๐ด +1
๐2
๐ฮฆ
๐๐ก) = โฮผ0๐ฝ
applying the Lorenz condition:
๐ปยท๐ด +1
๐2
๐ฮฆ
๐๐ก= 0
So, ๐ป2๐ด โ1
๐2
๐2๐ด
๐๐ก2 โ ๐ป(0) = โฮผ0๐ฝ
๐ป2๐ด โ1
๐2
๐2๐ด
๐๐ก2 = โฮผ0๐ฝ (6)
Equations (5) and (6), form a set of equations equivalent in all respects to the Maxwell Equations in vacuum. Thus, the Lorentz condition makes and satisfies inhomogeneous wave equations of similar forms.
Questions and Discussion
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Thanks!