electrodynamics of moving charges

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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2/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom

And The GreensFunction



Poynting AndEnergy

Applications

LinearAccelerators


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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3/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy

Applications

LinearAccelerators


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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4/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy

Applications

LinearAccelerators


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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5/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy

Applications

LinearAccelerators


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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6/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy

ApplicationsLinearAccelerators


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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7/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



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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8/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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9/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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10/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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11/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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12/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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13/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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14/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



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.

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15/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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.

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16/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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17/46



18/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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

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19/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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.

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20/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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)

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21/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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)

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22/46

On The


Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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)

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23/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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,

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24/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials

Gauge Freedom




Poynting AndEnergy



The HydrogenAtom

Summary


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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25/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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)

T A d R l i i
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26/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



The HydrogenAtom

Summary


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)

T A d R l ti it
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27/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials

Gauge FreedomAnd The GreensFunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities


Poynting AndEnergy



The HydrogenAtom

Summary


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)

T A d R l ti it


28/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials



Poynting AndEnergy



The HydrogenAtom

Summary


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)

T A d R l ti it


29/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials



Poynting AndEnergy



The HydrogenAtom

Summary


We can thus define proper momentum as

pi =mvi =

m

dx0

d ,m

dx

d

= (mc,mc). (57)

Poynting And Energy
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30/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials



Poynting AndEnergy



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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31/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials



Poynting AndEnergy



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


32/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials



Poynting AndEnergy



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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33/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials

Gauge FreedomAnd The Greens

FunctionDeriving TheFields FromGiven ChargeAnd CurrentDensities


Poynting AndEnergy



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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34/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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)

Linear Accelerators
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35/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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)

Linear Accelerators
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36/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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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37/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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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38/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



The HydrogenAtom

Summary


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)

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39/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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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40/46

The Hydrogen Atom


41/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy



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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42/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy

Applications

LinearAccelerators


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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43/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy

Applications

LinearAccelerators


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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44/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy

Applications

LinearAccelerators


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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45/46

On TheRadiation Of

MovingCharges

Lewis Proctor

Theory

Potentials




Poynting AndEnergy

Applications

LinearAccelerators


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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46/46

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.
http://find/

electrodynamics of moving charges

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