Type of potential in electrodynamics
In electrodynamics , the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light c , so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.[ 1]
In the Lorenz gauge [ edit ] Position vectors r and r′ used in the calculation The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge :
◻
φ
=
ρ
ϵ
0
,
◻
A
=
μ
0
J
{\displaystyle \Box \varphi ={\dfrac {\rho }{\epsilon _{0))}\,,\quad \Box \mathbf {A} =\mu _{0}\mathbf {J} }
where φ(r , t ) is the electric potential and A (r , t ) is the magnetic vector potential , for an arbitrary source of charge density ρ(r , t ) and current density J (r , t ), and
◻
{\displaystyle \Box }
is the D'Alembert operator .[ 2] Solving these gives the retarded potentials below (all in SI units ).
For time-dependent fields [ edit ] For time-dependent fields, the retarded potentials are:[ 3] [ 4]
φ
(
r
,
t
)
=
1
4
π
ϵ
0
∫
ρ
(
r
′
,
t
r
)
|
r
−
r
′
|
d
3
r
′
{\displaystyle \mathrm {\varphi } (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0))}\int {\frac {\rho (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|))\,\mathrm {d} ^{3}\mathbf {r} '}
A
(
r
,
t
)
=
μ
0
4
π
∫
J
(
r
′
,
t
r
)
|
r
−
r
′
|
d
3
r
′
.
{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0)){4\pi ))\int {\frac {\mathbf {J} (\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|))\,\mathrm {d} ^{3}\mathbf {r} '\,.}
where r is a point in space, t is time,
t
r
=
t
−
|
r
−
r
′
|
c
{\displaystyle t_{r}=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c))}
is the retarded time , and d3 r' is the integration measure using r' .
From φ(r , t) and A (r , t ), the fields E (r , t ) and B (r , t ) can be calculated using the definitions of the potentials:
−
E
=
∇
φ
+
∂
A
∂
t
,
B
=
∇
×
A
.
{\displaystyle -\mathbf {E} =\nabla \varphi +{\frac {\partial \mathbf {A} }{\partial t))\,,\quad \mathbf {B} =\nabla \times \mathbf {A} \,.}
and this leads to Jefimenko's equations . The corresponding advanced potentials have an identical form, except the advanced time
t
a
=
t
+
|
r
−
r
′
|
c
{\displaystyle t_{a}=t+{\frac {|\mathbf {r} -\mathbf {r} '|}{c))}
replaces the retarded time.
In comparison with static potentials for time-independent fields [ edit ] In the case the fields are time-independent (electrostatic and magnetostatic fields), the time derivatives in the
◻
{\displaystyle \Box }
operators of the fields are zero, and Maxwell's equations reduce to
∇
2
φ
=
−
ρ
ϵ
0
,
∇
2
A
=
−
μ
0
J
,
{\displaystyle \nabla ^{2}\varphi =-{\dfrac {\rho }{\epsilon _{0))}\,,\quad \nabla ^{2}\mathbf {A} =-\mu _{0}\mathbf {J} \,,}
where ∇2 is the Laplacian , which take the form of Poisson's equation in four components (one for φ and three for A ), and the solutions are:
φ
(
r
)
=
1
4
π
ϵ
0
∫
ρ
(
r
′
)
|
r
−
r
′
|
d
3
r
′
{\displaystyle \mathrm {\varphi } (\mathbf {r} )={\frac {1}{4\pi \epsilon _{0))}\int {\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|))\,\mathrm {d} ^{3}\mathbf {r} '}
A
(
r
)
=
μ
0
4
π
∫
J
(
r
′
)
|
r
−
r
′
|
d
3
r
′
.
{\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0)){4\pi ))\int {\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|))\,\mathrm {d} ^{3}\mathbf {r} '\,.}
These also follow directly from the retarded potentials.
In the Coulomb gauge [ edit ] In the Coulomb gauge , Maxwell's equations are[ 5]
∇
2
φ
=
−
ρ
ϵ
0
{\displaystyle \nabla ^{2}\varphi =-{\dfrac {\rho }{\epsilon _{0))))
∇
2
A
−
1
c
2
∂
2
A
∂
t
2
=
−
μ
0
J
+
1
c
2
∇
(
∂
φ
∂
t
)
,
{\displaystyle \nabla ^{2}\mathbf {A} -{\dfrac {1}{c^{2))}{\dfrac {\partial ^{2}\mathbf {A} }{\partial t^{2))}=-\mu _{0}\mathbf {J} +{\dfrac {1}{c^{2))}\nabla \left({\dfrac {\partial \varphi }{\partial t))\right)\,,}
although the solutions contrast the above, since A is a retarded potential yet φ changes instantly , given by:
φ
(
r
,
t
)
=
1
4
π
ϵ
0
∫
ρ
(
r
′
,
t
)
|
r
−
r
′
|
d
3
r
′
{\displaystyle \varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi \epsilon _{0))}\int {\dfrac {\rho (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|))\mathrm {d} ^{3}\mathbf {r} '}
A
(
r
,
t
)
=
1
4
π
ε
0
∇
×
∫
d
3
r
′
∫
0
|
r
−
r
′
|
/
c
d
t
r
t
r
J
(
r
′
,
t
−
t
r
)
|
r
−
r
′
|
3
×
(
r
−
r
′
)
.
{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\dfrac {1}{4\pi \varepsilon _{0))}\nabla \times \int \mathrm {d} ^{3}\mathbf {r'} \int _{0}^{|\mathbf {r} -\mathbf {r} '|/c}\mathrm {d} t_{r}{\dfrac {t_{r}\mathbf {J} (\mathbf {r'} ,t-t_{r})}{|\mathbf {r} -\mathbf {r} '|^{3))}\times (\mathbf {r} -\mathbf {r} ')\,.}
This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but A is not so easily calculable from the current distribution j . However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields:
φ
(
r
,
t
)
=
1
4
π
∫
∇
⋅
E
(
r
′
,
t
)
|
r
−
r
′
|
d
3
r
′
{\displaystyle \varphi (\mathbf {r} ,t)={\dfrac {1}{4\pi ))\int {\dfrac {\nabla \cdot \mathbf {E} (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|))\mathrm {d} ^{3}\mathbf {r} '}
A
(
r
,
t
)
=
1
4
π
∫
∇
×
B
(
r
′
,
t
)
|
r
−
r
′
|
d
3
r
′
{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\dfrac {1}{4\pi ))\int {\dfrac {\nabla \times \mathbf {B} (\mathbf {r} ',t)}{|\mathbf {r} -\mathbf {r} '|))\mathrm {d} ^{3}\mathbf {r} '}
In linearized gravity [ edit ] The retarded potential in linearized general relativity is closely analogous to the electromagnetic case. The trace-reversed tensor
h
~
μ
ν
=
h
μ
ν
−
1
2
η
μ
ν
h
{\textstyle {\tilde {h))_{\mu \nu }=h_{\mu \nu }-{\frac {1}{2))\eta _{\mu \nu }h}
plays the role of the four-vector potential, the harmonic gauge
h
~
μ
ν
,
μ
=
0
{\displaystyle {\tilde {h))^{\mu \nu }{}_{,\mu }=0}
replaces the electromagnetic Lorenz gauge, the field equations are
◻
h
~
μ
ν
=
−
16
π
G
T
μ
ν
{\displaystyle \Box {\tilde {h))_{\mu \nu }=-16\pi GT_{\mu \nu ))
, and the retarded-wave solution is[ 6]
h
~
μ
ν
(
r
,
t
)
=
4
G
∫
T
μ
ν
(
r
′
,
t
r
)
|
r
−
r
′
|
d
3
r
′
.
{\displaystyle {\tilde {h))_{\mu \nu }(\mathbf {r} ,t)=4G\int {\frac {T_{\mu \nu }(\mathbf {r} ',t_{r})}{|\mathbf {r} -\mathbf {r} '|))\mathrm {d} ^{3}\mathbf {r} '.}
Using SI units, the expression must be divided by
c
4
{\displaystyle c^{4))
, as can be confirmed by dimensional analysis.
Occurrence and application [ edit ] A many-body theory which includes an average of retarded and advanced Liénard–Wiechert potentials is the Wheeler–Feynman absorber theory also known as the Wheeler–Feynman time-symmetric theory.
The potential of charge with uniform speed on a straight line has inversion in a point that is in the recent position. The potential is not changed in the direction of movement.[ 7]
^ Rohrlich, F (1993). "Potentials" . In Parker, S.P. (ed.). McGraw Hill Encyclopaedia of Physics (2nd ed.). New York. p. 1072. ISBN 0-07-051400-3 . ((cite encyclopedia ))
: CS1 maint: location missing publisher (link )
^ Garg, A., Classical Electromagnetism in a Nutshell , 2012, p. 129
^ Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-92712-9
^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
^ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
^ Sean M. Carroll, "Lecture Notes on General Relativity" (arXiv:gr-qc/9712019 ), equations 6.20, 6.21, 6.22, 6.74
^ Feynman, Lecture 26, Lorentz Transformations of the Fields