SR Theory of Electrodynamics for Relative Moving Charges By James Keele, M.S.E.E. October 27, 2012.
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Transcript of electrodynamics of moving charges
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8/10/2019 electrodynamics of moving charges
1/46
On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
On The Radiation Of Moving Charges
Mathematical Physics Research Project
Lewis Proctor
University of Sussex
2014
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Outline
1 TheoryPotentialsGauge Freedom And The Greens FunctionDeriving The Fields From Given Charge And CurrentDensitiesTensors And RelativityPoynting And Energy
2 ApplicationsLinear AcceleratorsCircular AcceleratorsThe Hydrogen Atom
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Potentials
Maxwells equations:
E= 4 (1)
E= 1
c
B
t (2)
B= 0 (3)
B= 1cEt +4c j (4)
From these we can acquire equations for potentials
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Potentials
with use of v= 0 (5)
and knowingv= A (6)
We define the magnetic potential as
B=A (7)
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Potentials
Substituting the magnetic potential into (2) andre-arranging we have
(E + At
) = 0. (8)
Using the condition that a vector whose curl is 0 can berepresented as a grad of a scalar, we end up with the finalpotential equation
E= At
(9)
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Potentials
Substituting these into equation (4) gives us
(A) = 1
c
t(
1
c
A
t ) +
4
c j (10)
using (A) =( A) 2A (11)
We then end up with
( A) 2A= 1
c
2A
t2 (
1
c
t) +
4
c j (12)
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
The gauge structure of electrodynamics is down to the freedomto choose an arbitary reference frame in quantum mechanics.There are two gauges to think about, Lorenz and Coulomb.With the coulomb gauge, there is no instantaneous propagation
of observable quantities involved. The advantage of the Lorenzgauge is that the solutions are more general and we canobserve the quantities the equations predict. Thus
Coulomb Gauge
A= 0 (13)Lorenz Gauge
A +1
c
t = 0 (14)
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
The Lorenz gauge
A +1
c
t = 0 (15)
introduces a symmetry into the problem, which becomes veryuseful when relativity is involved. Using this we obtain thedifferential equations of the form
2A=4
c j (16)
2= 4. (17)
where 2 =2 1c2
2
t2.
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
We now have two inhomogeneous differential equations, tosolve them we will use a Greens function. So that the generalform will be,
(r, t) =
(dr)dtG(R, t t)(r, t) (18)
and
A(r, t) =
(dr
)dt
G(R, t t
)
1
cj(r
, t
), (19)
where R=r r.
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
To make things easier, we will now convert to Fourier spaceusing
(t t) =
d
2
ei(tt), (20)
and
G(R, t t) =
d
2ei(tt
)G(R) (21)
where the transforms obey
2G(R, t t) = 4(R)(t t) (22)
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
Working through the differentials and using the correctboundary conditions at infinity, we arrive at the solution
G(R, t t) =
d2
1R
ei[Rc(tt)]. (23)
Transforming back into cartesian space, this has the form
G(R, t t) = 1
R
R
c (t t). (24)
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
Then using the property of the Dirac delta function thesolutions are of the form
(r, t) =
dr1Rr, t Rc
(25)
and
A(r, t) = dr1
R
j
r, t R
c c
(26)
These make a lot more physical sense and are called theLienard-Wiechert Potentials.
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Gauge Freedom And The Greens Function
If the charge Q changes with time, the infomation that it has
changed can only been appreciated at the observation point,after the retarded time. This is the lag time it takes theinfomation to reach us. The sign in the greens functionsolution, gives us retarded and advanced solutions.
D i i Th Fi ld F Gi Ch A d
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
Given general charge and current densities
(r, tr) =e(tr) (27)
and
A(r, tr) =ev(tr)
c . (28)
we can find expressions for the electric and magnetic fields.
D i i Th Fi ld F Gi Ch A d
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
With the general forms of the potentials
(r, t) =e
t t Rc R dt
(29)
A(r, t) =e
v
c
t t Rc
R
dt (30)
We can now transform them into Fourier space, so thepotentials are easier to manipulate.
D i i Th Fi ld F Gi Ch A d
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
We now have the potentials as they appear in Fourier space:
(r, t) = e
2
ei(tt
R
c)
R ddt (31)
A(r, t) = e
2c
ei(ttR
c)
R vddt (32)
We can now start using these to find E and B via
E= At
(33)
andB=A (34)
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Deriving The Fields From Given Charge And
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8/10/2019 electrodynamics of moving charges
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
For the electric field we therefore get
E(r, t) = e
2
uei(tt
R
c)
R2 ddt
+ e2c
t
u
v
c
R
ei(tt
R
c)ddt.
Using the same ideas but with the curl for B and transformingback to normal space the final expressions are
E(r, t) =e
u
R2
t t Rc
dt
+e
c
t
u v
c
R t t
R
c dt
Deriving The Fields From Given Charge And
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
and
B(r, t) =e
c
v u
R2 t t R
c dt
e
c2
t
v u
R
t t
R
c
dt.
However we cant immediately evaluate the Dirac function,
becauseR is a function oft.
Deriving The Fields From Given Charge And
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
We therefore change the variable to
dt=
1
1
c
R
t
dt. (38)
Therefore
E(r, t) =e
u
R2(t t)
1 1c
Rt
dt
+e
c
t
u v
c
R
(t t)
1 1c
Rt
dt
which becomes after evaluation, and using the substitutionK = 1 1
c
Rt
E(r, t) =
eu
KR2 t=tRc+
t
eu v
c
cKR
t
=t
R
c
. (39)
Deriving The Fields From Given Charge And
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
When this is applied to B we have
B(r, t) = e(v u)cKR2
t=tR
c
+
te(v u)
c2KRt=tR
c
. (40)
However
tF(r, v, t)t
=t
R
c
=
t
F(r, v, t(r, t)) = Ft
t
tt=tRc
(41)
Deriving The Fields From Given Charge And
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On The
Radiation OfMovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
E(r, t) =e u
KR2 +
1
cK
t R
KR2
1
c2K
t v
KR
t=tRc
(42
and
B(r, t) =ev u
cKR2 +
1
c2K
tv u
KR
t=tRc. (43)
Deriving The Fields From Given Charge And
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
UsingR
t = v (44)
R
t =
R v
R (45)
K
t =
v2 R v (u v)2
cR (46)
and after a painstaking task,
Deriving The Fields From Given Charge And
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge Freedom
And The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Deriving The Fields From Given Charge And
Current Densities
We get
E(r, t) =e(u vc)(v R +c2 v2)c2K3R2
v
c3K2Rt=tR
c
(47)
and
B(r, t) =e(v u)(v R +c2 v2)
c3K3R2 +
(v u)
c2K2Rt=tR
c
.
(48)
T A d R l i i
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Tensors And Relativity
We can define a covariant four vector as
xi = (x0, x1, x2, x3) = (ct,x,y,z), (49)
and contravariant as
xi = (x0, x1, x2, x3) = (ct, x, y, z), (50)
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunction
Deriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Tensors And Relativity
The minus signs are due to the metric tensor we have chosen
gik=
1 0 0 00 1 0 00 0 1 00 0 0 1
. (51)
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Tensors And Relativity
To define proper time we need to look at the Lorentz invariantinterval
ds2 =gikdxidxk (52)
which then becomes
ds2 =c2dt2 dx dx. (53)
Factoring out c2dt2 becomes
ds2 =c2dt2
1 1
c2dx2
dt2
(54)
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On TheRadiation Of
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Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Tensors And Relativity
We thus define proper time as the invariant interval
d =ds
c =
dt
. (55)
And with this definition we can define proper velocity
vi = dx0
d
,dx
d= dx
0
dt
dt
d
,dx
dt
dt
d= (c, c) (56)
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Tensors And Relativity
We can thus define proper momentum as
pi =mvi =
m
dx0
d ,m
dx
d
= (mc,mc). (57)
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Poynting And Energy
We see that the Poynting vector is defined by
S= c4
EB= c4
E2u. (58)
We will define E(r, t) and B(r, t) when the acceleration is 0.
E(r, t) =eu vc 1 v2c2
K3R2t=tR
c
(59)
and
B(r, t) =e (v u)1
v2
c2 cK3R2
t=tR
c
. (60)
we see that Poyntings vector is 1R4
dependent, the integralover a surface at infinity, approaches zero. The energy thusremains in a finite space around the charge, and therefore emitsno radiation.
Poynting And Energy
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Poynting And Energy
When the acceleration isnt zero, we thus have the poynting
vector after substituting the values ofK, E2
and u to be
S= e2
43K6R2
K2v2
1
v2
c2
(u v)2 +
2Ku vv v
c
.
(61)
Poynting And Energy
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On TheRadiation Of
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Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
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Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Poynting And Energy
Suppose the particle is briefly at rest at the origin K= 1 andR=r, then the poynting vector is
S=
e2
4c3r2v (u v)
2u=
e2(u v)2
4c3r2 u. (62)
To calculate the energy emitted through a surface, we performthe surface integral
S dA= e
2
4c3a2
(u v)2dA= e2
4c3
(u v)2d. (63)
Poynting And Energy
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Poynting And Energy
Using the sin relationship of the cross product and integratingover the solid angle
d = sin dd. (64)
and thus the total power radiated is
P= S dA= e2
4c3 (u v)2d=2e
2v2
3c3 . (65)
This is the so called Larmor Formula.
Poynting And Energy
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Poynting And Energy
For the general relativistic formula, we use the relativistic fourmomentum defined earlier divided by m2.
P=
2
3
e2
m2c3dpi
d
dpi
d. (66)
Expanding the quantities and making it in terms ofand ,we finally get the general form of the power emitted as
P=23
e2
c6[()2 ( )2]. (67)
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On TheRadiation Of
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Lewis Proctor
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Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Linear Accelerators
Using the equation above and using the fact that, the velocityand the acceleration are in the same direction,we have
P=23
e2c6()2. (68)
To find we have to differentiate p, giving
p=mc(+ ). (69)
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On TheRadiation Of
MovingCharges
Lewis Proctor
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Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Linear Accelerators
We also have=32. (70)
which when substituting into p, gives
p=3mc (71)
and hence
P=
2e2p2
3m2c3 . (72)
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Using the fact that the rate of change of momentum is the
change in energy over displacement, we can calculate the ratiobetween the power radiated and the external power provided tobe
Prad
dE/dt
=2
3
e2
m2c3
dE
dx
dt
dE
dE
dx
=2
3
e2
m2c3
dt
dx
dE
dx
=2
3
e2
m2c3
1
v
dE
dx
.
(73)
The power radiated doesnt become significant unless theexternal power is
dE
dx
mec2
re=
0.551MeVc2
2.8 1015m = 2 1014MeVm1. (74)
Circular Accelerators
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Circular Accelerators
In circular motion the momentum of the particle doesnt
change linearly anymore, it changes radially. Therefore anexpression for the change in momentum in terms of the angularvelocity is needed. We assume
dpd
=
p (75)
Substituting this into P leads to the expression
P=2
3
e2
m2
c322|p|2. (76)
When = cR
the total power lost is therefore
P=2e2c44
3R2 . (77)
Circular Accelerators
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
C cu a cce e ato s
And hence the power radiated per cycle is
Pper cycle=4e234
3R
(78)
To get the correct results we can change units using theformula for eV, giving
Pper cycle=
e
30R E
mc24
(79)
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The Hydrogen Atom
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
ApplicationsLinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
y g
We are going to prove that the planetary model is not feasible,
due to radiation loss. We start with the classical Larmorformula
P=2e2a2
3c3 . (80)
Along with Newtons law for circular motion, with the electronproviding the mass
F=mea, (81)
the force is generated by the Coulomb attraction, therefore
F= e2r2
=mea. (82)
And
a= e2
mr2. (83)
The Hydrogen Atom
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
y g
The power is therefore
P= 2e6
3m2ec3r4
. (84)
Power is the negative rate of change of energy, which comesfrom the electrons kinetic energy and the particles potentialand is equal to
dE
dt =
remec2r
2r2 . (85)
Equating this to the power, and re-arranging, we get thisexpression:
dr
dt =
4r2ec
3r2 . (86)
The Hydrogen Atom
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
y g
Then integrating gives 0
dt=
a0
r
r2dr (87)
Setting our integration variable, we can conserve the r notationwith use of dummy variables, and integrating from t= 0 tot= some unspecified time , and from the Bohr radius towardsthe centre of the atom, where we follow the trajectory of theelectron, we thus have the result
= a304r2ec
= 1.556 1011seconds. (88)
Summary
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Starting from Maxwells equations, we derived expressionsinvolving the potentials alone. We then used anappropiate gauge fix to simplify the equations. From thiswe used the idea of the greens function to give us generalsolutions to these potential equations.
With general charge and current distributions, we acquiredgeneral forms of the potentials and field equations(classical and relativistic), using this we obtained theenergy radiated through an arbitary surface and found a
general result for the radiation of these given densities(also classical and relativistic)
We applied the derived equations to particle acceleratorsand to the hydrogen atom, to check the viability of theplanetary model.
Summary
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On TheRadiation Of
MovingCharges
Lewis Proctor
Theory
Potentials
Gauge FreedomAnd The Greens
FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities
Tensors AndRelativity
Poynting AndEnergy
Applications
LinearAccelerators
CircularAccelerators
The HydrogenAtom
Summary
Outlook
In the future, we can maybe look into the solution of whata relativistically moving charge distribution would look like.Have a look at a few medical applications, due to theadvancement in cancer treatment, radiation could be a bigplayer in the treatment/cure in the future.
For Further Reading I
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On TheRadiation Of
MovingCharges
Lewis Proctor
Appendix
For FurtherReading
J. Schwinger, L.L. Delad, K.K.A. Milton, Y.W. Tsai,Classical Electrodynamics, (1998) .
F. Gronwald, J. NitschTHE PHYSICAL ORIGIN OF GAUGE INVARIANCE INELECTRODYNAMICS AND SOME OF ITSCONSEQUENCES
Otto-von-Guericke-Universitat Magdeburg, 1998.
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