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Electrodynamics-II Mr. Abdissa. Tassama
April/2020
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Outlines
2
Maxwellâs Equations
Conservation Laws
Potential and Fields
Radiation
Covariant Formulation of Electrodynamics
March, 2020
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Maxwellâs Equations
March 2020 3
Electrodynamics before Maxwellâs
But, there is a fatal inconsistency
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Ampereâs law is bound to fail nonsteady currents
March, 2020
4
Applying divergence to eqn. (iii), then
ð» â ðµ = 0
Applying divergence to eqn. (iv)
0
ð» â ðœ = 0 For steady current
Note: divergence of a curl vanishes: it's a vector identity.
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Contâ
March 2020 5
For non-steady currents eqn.(iv)
ð» â ðœ = ðððð¡ â 0
Ampereâs law cannot be right for non-steady currents!
There's another way to see that Ampere's law is bound to fail
for non-steady current.
Consider the process of charging up a capacitor.
Ampereâs law reads,
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Contâ
March, 2020 6
We want to apply it to the Amperian loop or Amperian surface.
How do we determine ð°ððð? ðð ðð
For the surface ð 1, ðŒððð = I
For the surface ð 2, ðŒððð = 0(No current passes through it)
The conflict arises only when charge is piling up somewhere (in this
case, on the capacitor plates).
For nonsteady currents, "the current enclosed by a loop" is an ill defined notion, since it depends entirely on what surface you use.
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How Maxwell fixed ampereâs law
April 2020
7
The inconsistent problem arose on Ampereâs law when;
ð» â (ð»ððµ) â ð». ðœ for nonsteady currents
Applying continuity equation and Gaussâs law
The inconsistency in Ampereâs law is now cured.
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Contâ
March 2020 8
Ampereâs law can generally be expressed as
A changing electric field induces a magnetic field
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Maxwellâs equation
9
âŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠEqn.3
Together with the force law
they summarize the entire theoretical content of classical electrodynamics.
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Magnetic Charge?
10
The symmetry between E and B is spoiled
by the charge term and the current.
If we had ðð (the density of magnetic charge) and ðœð (the current of magnetic charge),
If we replace E B, B -ðð ðð ðž
There could be a pleasing symmetry between E and B.
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Maxwellâs equation in matter
11
It would be nice to reformulate Maxwellâs eqn. in the form (3)
when you are working with material that are subjected to electric
and magnetic polarization
For inside polarized matter there will be accumulation of bound
charges and current over which you exert no direct control.
ðð = ð». PâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ..(4)
Magnetic polarization M results in a bound current is
ðœð = ð»ððŽâŠâŠâŠâŠâŠâŠâŠâŠâŠâŠ . (5)
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Maxwell's Equations in Matter
March, 2020 12
ðð = ð». P
If P is time-varying, we expect there to be a current ðœð associated
with the resulting changes in ðð. In fact, the above expression
suggests a good definition of ðœð :
That is, the definition on the left naturally gives the continuity
relation between ðœð and ðð one would like.
Time dependence of M yields time dependence of ðœð, which
produces time dependence of B and H
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Contâ
13
⢠The charge and current densities have the following parts
⢠âŠâŠâŠâŠ.6
⢠Using Gauss's law,
⢠íðð». ðž = ðð â ð». ð ððð ððððððð¡ððð ðð the displacement field
D= íððž+P we obtain ð». ð· = ðð âŠâŠâŠâŠâŠâŠâŠ7
Ampere's Law with the displacement current term is
ð»ððµ = µð(ðœð + ð» Ãð +ðð
ðð¡)+µððð
ððž
ðð¡âŠâŠâŠ8
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Contâ
14
We use B=µð(H+M) as well as D= ðððž+P we obtain
ð» à ð» = ðœð+ðð·
ðð¡.................................9
Faraday's Law and ð» â ðµ = 0 are not affected since they do not depend on the free and bound currents.
Thus, Maxwell's Equations in matter are (again, putting all the fields on the left sides and the sources on the right):
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Boundary Conditions for Maxwell's Equations
15
In general the fields E,B,D & H will be discontinuous at a
boundary between two different media or at a surface that carries
charge density Ï or current density k.
The integral form of Maxwellâs equations can deduct the boundary conditions
March, 2020
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Contâ
March, 2020
16
Applying (i) to a tiny, wafer-thin Gaussian pillbox extending just slightly into the material on either side of the boundary we obtain
ð·1ð â ð·2ð=ÏððâŠâŠâŠâŠâŠ(12)
Thus the component of D that is perpendicular to the interface is discontinuous in the amount
ð·1â¥-ð·2
⥠=Ïð âŠâŠâŠâŠâŠâŠâŠâŠ.(13)
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contâ
March, 2020 17
Identical reasoning applied to (ii) yields
ðµ1â¥-ðµ2
⥠=0 âŠâŠâŠâŠâŠâŠâŠâŠ.(14)
Turning to (iii) a very thin Amperian loop straddling the surface
(fig.) yields
ðž1ðŒ â ðž2ðŒ=ð
ðð¡ ðµ â ððð
But in the limit as width of the loop goes to zero the flux vanishes.
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Contâ
March, 2020
18
Therefore
ðž1â-ðž2
â=0 âŠâŠâŠâŠâŠâŠâŠâŠâŠ.(15)
That is the component of E parallel to the interface are
continuous across the boundary.
By the same token (iv) implies
ð»1ðŒ â ð»2ðŒ=ðŒðððð
Where ðŒðððð is free current passing through Amperian loop
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Cont;
March, 2020 19
ðŒðððð= ðŸð â ð à ðŒ = ðŸð à ð â ðŒ
And hence
ð»1ââð»2
â= ðŸð à ð â ðŒâŠâŠâŠâŠâŠâŠâŠâŠ.16
So the parallel components of H are discontinuous by an
amount proportional to the free surface current density.
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Contâ
March, 2020
20
Equation 13-16 are the general boundary conditions for electrodynamics
In the case of linear media,
If there is no free charge or free current at the interface
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Conservation Laws
March, 2020
21
Under this we will study conservation of
Charge
Energy
Momentum
Not only is there no creation or destruction of charge over the whole
universe, there is also no creation or destruction of charge at a given
point.
Charge cannot jump from one place to another without a current owing
to move that charge.
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Poynting's Theorem: Conservation of Energy
March, 2020 22
Work necessary to assemble charge distribution (againist coloumbs repulsion of like charge)
âŠâŠâŠâŠâŠâŠâŠâŠ17
Likewise, the work required to get currents going (againist the back emf)
âŠâŠâŠâŠâŠâŠâŠâŠâŠ.18
Therefore the total energy stored in EMF is
âŠâŠâŠâŠ19
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cont;
March, 2020
23
We will prove this by considering the work done to move charges as currents
Given a single particle of charge q acted on by the electromagnetic field, the work done on it as it moves by ðð is
âŠâŠâŠâŠ.20
Now, q= ððð ððð ðð£ = ðœ so the rate at which work is done on all the charges in volume ð is
âŠâŠâŠâŠ..21
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contâ
March, 2020 24
Let's manipulate the integrand using Ampere's Law:
From product rule ð» â ðž Ã ðµ = ðµ â ð» Ã ðž â ðž âð» Ã ðµ
Invoking Faradayâs law
ðž â ð» Ã ðµ = âðµ âððµ
ðð¡â ð» â ðž Ã ðµ
Meanwhile âðµ âððµ
ðð¡=
1ð
2ðð¡(ðµ2)
So Eâ ðœ =1ð
2ðð¡ðððž
2 +1
ðððµ2 â
1
ððð» â ðž à ðµ âŠâŠâŠâŠâŠ..22
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Contâ
March, 2020 25
Putting this into equation (21) and applying divergence theorem
to the 2nd term
ðð
ðð¡= â
ð
ðð¡ 1
2
ð
ðð¡ðððž
2 +1
ðððµ2 ðð â
1
ðð ðž à ðµ â ðð⊠.23
Where ð is the surface bounding ð.
This poyntingâs theorem; it is the âwork-energy theoremâ of
electrodynamics.
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Contâ
March, 2020 26
The first integral on the right is the total energy stored in the
fields. The 2nd term evidently represent the rate at which energy
is carried out of ð, across its boundary surface by the EMF.
Poyntingâs theorem state that, the work done on the charges by
the electromagnetic force is equal to the decrease in energy
stored in the fields, less the energy that flowed out through the
surface.
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March, 2020
27
The energy per unit time, per unit area, transported
by fields is called the poynting vector:
ð =1
ðððž à ðµ âŠâŠâŠ..24
where S is the energy flux density
ðð
ðð¡= â
ðððð
ðð¡â
1
ðð ð â ððð
âŠâŠ.25
Another useful form is given by putting the eld energy density term on the left side:
Contâ
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Conservation of Momentum
March, 2020 28
According to Newtonâs second law
F =ðððððâ
ðð¡
âŠâŠâŠâŠâŠâŠ26
Where ððððâ is the total momentum of particle contained in the volume
V and T is the Maxwell Stress Tensor
âŠâŠâŠ.27
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Contâ
March, 2020 29
The indices I and j refer to the coordinates x,y and z, so the stress
tensor has a total of nine components (ðð¥ð¥, ððŠðŠ, ðð§ð§ and so on).
The kronecker delta ð¿ðð ðð 1 ðð ððð ð¡âð ð ððð ððð ð§ððð ðð ððð¡
Therefore
and so on
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Contâ
March, 2020 30
Eqn. (26) states that the rate of change of the mechanical momentum in a volume V is equal
to the integral over the surface of the volume of the stress tensor's flux through that surface
minus the rate of change of the volume integral of the Poynting vector.
Let ððð ðð the density of momentum in fields and
ððððâ ðððð ðð¡ðŠ ðð ðððâðððððð ðððððð¡ð¢ð
ððð=ððððð
ð
ðð¡ððð + ððððâ =ð» â ð
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Potential and Fields
March, 2020 31
Potential formulation
If you are asking your self how the sources (ð ððð ðœ) generate
electric field and magnetic fields; so this topic answer for your
question.
RECALL MAXWELLâS EQUATIONâŠâŠâŠâŠ..
Recall how we arrived at Maxwell's Equations. We first developed
Faraday's Law by incorporating both empirical information and the
requirement of the Lorentz Force being consistent with Galilean
relativity.
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Contâ
March, 2020 32
âŠâŠâŠâŠâŠâŠ..1
However, Faraday's Law implies that ð» Ã ðž â 0 when B has time
dependence.
Therefore, we cannot assume E = ð»ð However, using B = ð» à ðŽ ,
we see that
Thus, the Helmholtz Theorem implies
âŠâŠâŠâŠ..2
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Contâ
March, 2020 33
Eqn. (1) & (2) fulfill the homogenous Maxwell equation (ii) & (iii)
what about Gauss (i) & Ampereâs-Maxwell (iv) law. Putting (2) in
(i) ; âŠâŠâŠâŠâŠâŠ..3
Putting (1) & (2) in (iv) yields
âŠâŠâŠâŠâŠâŠâŠ..4
Or using the vector identity and
some rearranging terms a bit
âŠâŠâŠ..5
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Contâ
March, 2020 34
Eqn. (3) & (5) contain all the information in Maxwellâs
equations.
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Coulombs and Lorentz gauges
March, 2020 35
Coulombs gauges
As in magnetostatics we pick ;
ð» â ðŽ = 0âŠâŠâŠâŠâŠâŠâŠ6
This eqn. (3) yields ;
âŠâŠâŠâŠâŠ7
That is, the charge density sets the potential in the same way as in electrostatics, so changes in charge density propagate into the potential instantaneously.
Of course, you know from special relativity that this is not possible.
We will see that there are corrections from ððŽ ðð¡ that prevent E from responding instantaneously to such changes.
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Contâ
36
⢠The differential equation for A becomes
âŠâŠâŠâŠâŠâŠ..8
Advantage is that the scalar potential is particularly simple to calculate;
Disadvantage is that A is particularly difficult to calculate.
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37
The Lorenz Gauge
In Lorentz gauge we pick
ð» â ðŽ = âðððððð
ðð¡âŠâŠâŠâŠâŠâŠâŠâŠâŠ.9
This is designed to eliminate the middle term in Eqn. 5.with this
âŠ.. âŠâŠâŠâŠ10
Meanwhile , the differential eqn. for V, (3) becomes
âŠâŠâŠâŠâŠâŠâŠ..11
The virtual of Lorentz gauge is that it treats V and A on equal footing:
the same differential operator
âŠâŠâŠâŠâŠâŠ.13
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Contâ
38
âŠâŠâŠâŠâŠ.14
where â¡2 called the dâ Alembertian
V and A have the same differential operator of lâAlembertian (4-dim.
operator)
Under the Lorentz gauge, the whole of electrodynamics reduces to the
problem of solving the inhomogeneous wave equation for specified
sources.
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Continuous charge distributions
39
⢠Retarded Potentials
⢠In static case Eqs. 14 reduce to poissonâs eqs.
⢠âŠâŠâŠ.15
⢠With the familiar solution
⢠âŠâŠâŠâŠ16
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Contâ
40
⢠In non-static case, therefore itâs not the status of the sources
right now that matters, but rather its condition at some earlier
time ð¡ð(called retarded time) when the message left.
⢠Since this message must travel a distance r, the delay is r
ð
ð¡ð â¡ ð¡ â r
ð âŠâŠâŠ . . 17
⢠The natural generalization of eqn. (16) for non-static sources
is therefore
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Contâ
41
âŠ(18)
Here ð(ðâ², ð¡ð) is the charge density that prevailed at point
ðâ² ðð¡ ð¡âð ððð¡ððððð ð¡ððð . Because the integrand are
evaluated at the retarded time, these called âRetarded
potentialsâ.
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Jefimenko's Equations
42
Time-varying charges and currents generate retarded scalar potential,
retarded vector potential.
Potentials at a distance r from the source at time t depend on
the values of ð and J at an earlier time (t - r/u)
Retarded in time(ð¡ð = ð¡ â r
ð= ð¡ â
r
ð in vacuum)
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Contâ
43
â¢
⢠time-dependent generalization of Coulomb's law
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Contâ
44
⢠Similarly
time-dependent generalization of the Biot Savart law
Taking the divergence
The retarded potential also satisfies the inhomogeneous wave equation.
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Lienard-Wiechert Potentials
45
Lienard-Wiechert potentials for a moving point
charge
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The Fields of a Moving Point Charge
46
â¢
Letâs begin with the gradient of V:
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Radiation
47
⢠How accelerating charges and changing currents produce
electromagnetic waves, how they radiate?
⢠Assume a radiation source is localized near the origin. Total power
passing out through a spherical shell is the integral of the Poynting
vector
⢠The total power radiated from the source is the limit of this
quantity as r goes to infinity:
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Contâ
48
Since the area of the sphere is 4ðð2, so for radiation to occur (for Prad not to be zero), the
Poynting vector must decrease (at large r) no faster than 1/ð2.
But, according to Coulombâs law and Bio-Savart law, S ~ 1/ð4 for static
configurations.
Static sources do not radiate!
Jefimenko's Equations indicate that time dependent fields include
terms that go like 1/ r;(ð & ðœ) it is these terms that are responsible for
electromagnetic radiation.
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Electric Dipole Radiation
49
Suppose the charge back and forth through the wire, from one end to
the other, at an angular frequency Ï:
Dipole charge:
Current:
Electric dipole:
The retarded scalar and vector potentials at P are
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Retarded scalar potential
50
⢠Approximation 1 : d « r To make a perfect dipole,
assume d to be extremely small
â¢
Approximation 2 : d « λ = 2ðc/Ï Assume d to be
extremely smaller than wavelength
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51
⢠In the static limit (ð 0)
⢠Approximation 2 : d >> λ = 2ðc/Ï Assume r to be larger than wavelength (far-field radiation)
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Retarded vector potential:
52
⢠d « λ « r
Retarded potentials:
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Contâ
53
â¢
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Contâ
54
⢠E and B are in phase, mutually perpendicular, and transverse; the ratio
of their amplitudes is ðžð
ðµð =c.
⢠These are actually spherical waves, not plane waves, and their
amplitude decreases like 1/r.
⢠The energy radiated by an oscillating electric dipole is determined by
the Poynting vector:
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Contâ
55
⢠Intensity obtained by averaging
⢠total power radiated
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Magnetic Dipole Radiation
56
⢠Magnetic dipole moment of an oscillating loop current:
⢠where
The loop is uncharged, so the scalar potential is zero.
The retarded vector potential is
For a point r directly above the x axis, A must aim in the y direction, since the x components from symmetrically placed points on either side of the x axis will cancel.
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Contâ
57
⢠( ððð ð serves to pick out the y- c
omponent of dI').
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Contâ
58
Approximation 1 : b « r For a "perfect" dipole, the loop
must be extremely small
Approximation 2 : b « λ= 2ð c/Ï Assume b to be
extremely smaller than wavelength
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Contâ
59
In general A points in the Ï-direction.
In the static limit (ð= 0),
Approximation 3 : r » λ= 2ð c/Ï Assume r to be larger than wavelength (far-field
radiation)
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Contâ
60
These fields are in phase, mutually perpendicular, and transverse
to the direction of propagation (r) and the ratio of their amplitudes is Eo/Bo = c, all of which is as expected for electromagnetic waves.
Energy flux:
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Contâ
61
⢠Total power radiated:
⢠One important difference between electric and magnetic
dipole radiation is that for configurations with
comparable dimensions, the power radiated electrically
is enormously greater.
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Radiation from an Arbitrary Source
62
⢠Consider a configuration of charge and current that is entirely arbitrary
⢠The retarded scalar potential is
⢠Approximation 1 : râ « r (far field)
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Contâ
63
⢠Expanding ð as a Taylor series in t about the retarded time at the origin
⢠Approximation 2 : râ « ð = 2ðc/ð
⢠râ << ð ⪠r
In the static case, the first two terms are the monopole and dipole contributions
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Contâ
64
⢠Now, consider the vector potential:
⢠To first order in r' it suffices to replace by r in the integrand:
(Ignore the effect of magnetic dipole moment)
⢠Approximation 3 : r » ð = 2ðc/ð (discard 1/r2 terms in E and B)
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Contâ
65
â¢
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Contâ
66
In particular, if we use spherical polar coordinates, with the z axis in the direction of
Notice that E and B are mutually perpendicular, transverse to the direction of propagation (r) and in the ratio E/B = c, as always for radiation fields.
Poynting vector:
Total radiated power
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Contâ
67
If the electric dipole moment should happen to vanish (or, at any rate, if its
second time derivative is zero), then there is no electric dipole radiation, and one
must look to the next term: the one of second order in r'.
As it happens, this term can be separated into two parts, one of which is related to
the magnetic dipole moment of the source, the other to its electric quadrupole
moment (The former is a generalization of the magnetic dipole radiation).
If the magnetic dipole and electric quadrupole contributions vanish, the (r')3
term must be considered.
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Relativistic Electrodynamics
68
Magnetism as a Relativistic Phenomenon
In the reference frame where q is at rest,
The line charge sets up an electric field
The force can be transformed into in S
But, in the wire frame (S) the total charge is neutral !
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Contâ
69
⢠Electrostatics and relativity imply the existence of another force in
view point of S frame magnetic force.
⢠and
⢠One observerâs electric field is anotherâs magnetic field!
⢠Therefore, the relativistic force F is the Lorentz force in system S, not Minkowski!
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Field transformation
70
⢠Letâs find the general transformation rules for electromagnetic
fields:
⢠Given the fields in a frame (s ), what are the fields in another frame
( sâ)?
⢠consider the simplest possible electric field in a large parallel-plate
capacitor in ð ð frame.
⢠In the system S, moving to the right at speed v0,
⢠the plates are moving to the left with the different surface charge :
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Field transformation
71
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Contâ:
72
Two special cases:
⢠(1) If B = 0 in S frame, (E â 0);
⢠(2) If E =0 in S frame, (B â 0);
If either E or B is zero (at a particular point) in one system, then in any
other system the fields (at that point) are very simply related.
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Electromagnetic field tensor
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⢠The components of E and B are stirred together when you go from
one inertial system to another.
⢠What sort of an object is this, which has six components and
transforms according to the above relations? It's an antisymmetric,
second-rank tensor.
⢠and a second rank tensor is
⢠Where Lorentz transformation matrix
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THE END
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