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UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

LECTURE NOTES 12

   THE LIÉNARD-WIECHERT RETARDED POTENTIALS Vrtr , AND Artr , FOR A MOVING POINT CHARGE

Suppose a point q is moving along a specified trajectory = locus of points of  w tr = retarded position vector (SI units = meters) of the point charge q at retarded time tr .

The retarded time (in free space/) from point charge q to observer is determined by:  ctr r where the time interval ttt r

   and: r rt r  trr   rt w  t 

Observation / field point position of the point charge q at the retarded time tr  at time t (doesn’t move) w tr = retarded position of charge at retarded time tr

  rrrt w  trr   ct ct t

 r = separation distance of point charge q at the retarded position w tr at the retarded time tr  to the observer’s position at the field point Prt, which is at the present time, ttr r c.  r = vector separation distance between the two points, as shown in the figure below:

Position of pt. charge Position of pt. charge q at retarded time, tr q at present time, t

zˆ Trajectory of pt. Source Pt. charge, w(tr)  Stw r   Field Pt. rtw t   r  r  Prt w t  r rt yˆ 

xˆ  n.b. At most one point (one and only one point) on the trajectory w tr of the charged  particle can be “in communication” with the (stationary) observer at field point Prt at the present time t, because it takes a finite/causal amount of time ttt r for EM “news” to   propagate from w tr at the retarded time tr to the observation/field point Prt at position  r at the present time ttr r c.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

In order to make this point conceptually clear, imagine replacing the point charge q moving  along the retarded trajectory w tr with a moving point light source. The stationary observer at  the field point Prt, at the present time t will see the point light source move along the retarded  trajectory w tr ; but it takes a finite time interval for the light (EM “news”) to propagate from  where the light source was at the retarded source point location Sr trr w  t at the retarded  time tr to the observer’s location at the field point Prt, at the present time t.

This situation is precisely what an observer sees when looking at stars, planets, etc. in the night sky!

 Suppose that {somehow} there were e.g. two such source points along the trajectory w tr  “in communication” with the observer at the field point Prt at the present time t with retarded times tt and respectively. rr12

Then: ct t and: ct t thus: ct  t ct  t  ct  t r 1r1 r 2r2 rr 1 2 r1221 r r r    The average velocity of this charged particle in the direction of the observer at r is c !!! {n.b. the velocity component(s) of this particle in other directions are not counted here}. However, we know that nothing can move faster than the c !!!

     Only one retarded point w tr can contribute to the potentials Vrtr , and Artr , at the  field point Prt at any given moment in the present time, t for v < c !

  For v < c, an observer at the field point Prt at a given present time t “sees” the moving charged particle q in only one place.

{Note that a massless particle, such as a photon (which in free space/vacuum does move at the speed of light, c) could/can be “seen” by a stationary observer as being at more than one place at a given {present} time, t !!! Note further that it is also possible that no points along the trajectory of the photon are accessible to an observer….}

A “naïve” / cursory reading of the formula for the retarded scalar potential

  1  rt, Vrt, tot r d  r v 4 o r 1 q might suggest that the retarded scalar potential for a moving point charge is {also} 4o r (as in the static case), except that r = the separation distance is from observer position to the retarded position of the charge q.

However, this would be wrong – for a subtle conceptual reason!

It is true that for a moving point charge q, the denominator factor 1 r can be taken outside of  the integral, but note that {even} for a moving point charge, the integral: rt, d q !!! v tot r

2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede  In order to calculate the total charge of a configuration, one must integrate tot rt, r over the entire charge distribution at one instant of time, but {here} the retardation ttr r c forces  us to evaluate tot rt, r at different times for different parts of the charge configuration!!!

Thus, if the source is moving, we will obtain a distorted picture of the total charge!

  Before integration: r rt r  tr  is a function of rt and r  tr   3 After integration: rt  w r is fixed after integration: tot rt,wrr q  t   r rtw  tr  is a function of r and t because ttr r c.

One might think that this problem would be understandable e.g. for a moving extended charge distribution, but that it would disappear/go away/vanish for point charges. However it doesn’t !!!

In Maxwell’s equations of electrodynamics, formulated in terms of electric charge and current  densities tot and J tot , a point charge = limit of extended charge when the size → zero.  For an extended charge distribution, the retardation effect in  rt, d throws in a factor of: v tot r   11 1 vtr   where:  tr   ˆˆ c 11rrvtrr c  t

ˆ    We define the retardation factor  1 rvtr c, where vtr {more precisely vr tr }  is the velocity of the moving charged particle at the source position rtr at the retarded time tr .

This is a purely geometrical effect, one which is analogous/similar to the Doppler effect. {However, it is not due to special / general relativity (yet)!!}

Consider a long train moving towards a stationary observer. Due to the finite propagation time of EM signals, the train actually appears (a little) longer than it really is! (If c ≈ 10 m/s rather than 3 × 108 m/s, this motional effect would be readily apparent in the everyday world!!)

As shown in the figure below, light emitted from the caboose (end of the train) arriving at the observer at time t must leave the caboose earlier t than light emitted from the front of the rend train t , both arrive simultaneously at the observer at the same present time t. The train is rfront further away from the observer when light from the end of the train is emitted at the earlier time t , compared to the train’s location for the light emitted from the front of the train at the later rend time t . The observer thus sees a distorted picture of the moving train at the present time t. rfront

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

In the time interval tc that the light from the caboose takes to travel the distance L (see figure above) the train moves a distance LLL  . Then since ctc L, then: tLcc  .

But during the same time interval tc , the train moves a distance Lvtc  L  L, or:

L LL  L LL  LL  1 t  but: t thus: t    LL  c vv c c c cv v 1 vc

 Trains moving towards / approaching an observer appear longer, by a factor of 11 vc .

 Conversely, it can similarly be shown that trains moving away / receding from an observer appear shorter by a factor of 11 vc .

  In general, if the train’s velocity vector v makes an angle θ with the observer’s line of sight rˆ (n.b. assuming that the train is far enough away from the observer that the solid angle subtended by the train is such that rays of light emitted from both ends of train are parallel) the extra distance that light from the caboose must cover is Lcos (see figure below). The corresponding time interval is

tLc cos / c. Note that the train also moves a distance LLL  in this same time interval.

Lcos

LLL LLLLcos  LLLcos  tLcos / c but: t   t   i.e.  c c vv c cvv cv

1cos L 1cosvL 11 v Or: L or: L1 or: LLL  with   vc v vcv v cos 1cos  c 1 c

From the above figure, the angle   cos1 rˆvˆ = opening angle between rˆ and vˆ .    v 11 Thus:  cos rˆ where:   . Hence: LLL  . c 1cos  1 rˆ

Again, this retardation effect is due solely to the finite propagation time of the speed of light – it has nothing to do with special / general relativity – e.g. Lorentz contraction and/or time dilation and simultaneity.

4 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

The apparent volume   of the train is related to the actual volume of train   by:

111 vt     where  t  r and 11ˆˆvt c   t ˆˆ r rrrr 11rrvtrr c  t  c  and where rˆ rr r r = unit vector associated with the separation distance between the  position of a {stationary} observer rt at the present time t to a position somewhere on the    ˆ  train rtr at the retarded time tr . Explicitly: rrrrr rt  r  trr  rt  r  t 

 The stationary observer’s position vector rt is constant in time, whereas the retarded position  vector of the moving train rtr changes in time.

   Hence, whenever we carry out integrals of the type  rt, d {or Jrtd,  } v tot r v tot r    where the integrand(s) tot rt, r {or Jrttot , r } are associated with {some kind of} moving charge {current} distribution(s), evaluated at the retarded timetr , the apparent volume   of these  11 1  vt integrals is modified by the factor  where:  t  r , and the ˆˆ r  11rrvtrr c  t c   11 1 retardation factor 11ˆˆvt c   t , i.e. dd  d   d  rrrr ˆˆ  11rrvtrr c  t

The figure shown below graphically depicts this effect, for a snapshot-in-time ttr r c:

True volume at the Apparent volume at present time t the retarded time tr   Observer position vt vt P(r,t) at present r time t (fixed)

 See animated demo of this effect: https://en.wikipedia.org/wiki/Relativistic_Doppler_effect

Note that because the motional correction factor makes no reference to the actual physical size of the “particle”, it is also relevant/important for point charged particles.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

The retarded scalar potential associated with a point electric charge q moving along a  3 retarded trajectory w tr , with: q rt,wrr q  t is:

 33  11111q rt, r qtwwrr qt Vrtr ,  d d  d  44vv  4   v ˆ oorrrr orr1 vtr c

111qqq   4441wˆˆvtc  1 w t   ooorrrr rr r  ˆ   Where: r rr r r, r rt r  trr   rt w  t     vtw  ˆˆ   r And: 1wrrvtcrr  1 w t with:  w tr  , c

 Where: vtw r = velocity vector of charged particle evaluated at the retarded time ttr r c.   The retarded Jrttot , r for a rigid object is related to its retarded charge   density tot rt, r and its retarded velocity vrt, r by the relation:   Jrttot,,,rrr  tot  rtvrt.

The retarded vector potential associated with a point electric charge q moving with retarded velocity       3  vtw r along a retarded trajectory w tr , with: Jqq rt,,,,wrrrrr  rt vrt qvrt  t is:

    3  Jrtqq,,,rrr  rtvrt qv r,w trr   t Art,oo d  d  o d  r 44vvrr 4   v  r    qv r,wtt 3   qvt w oorr  r  d  44v 11wˆˆvt c v t c rrrr r r    qvw  t qvw  t q  o r oor    vtw r 4 1 ˆ w t 44  r r r rr

Thus, we have obtained the so-called Liénard-Wiechert retarded potentials for a point electric   charge q moving with retarded velocity vtw r along a retarded trajectory w tr :

  11qq Vrtr ,   441w ˆ t oorrr r   qvw  t q oor Artr ,   v w  tr 441w ˆ t rrr r

6 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

   vtw  ˆˆ  r Where: 1wrrvtcrr  1 w t with:  w tr  . c

   vtww vt rr1  Vrtr , 2 1 Note that: Artrr,, Vrt  Vrt rr ,w    t using: c  ccc2 c  oo     Vrtr ,  vtw r Or: Artrr,w    t with:  w tr  . c c  Where V is in Volts, A is in Newtons/Ampere (= momentum per Coulomb); they are related to r r  each other by a factor of 1 c and  for the case of a moving point electric charge q.  Recall that the relativistic four-potential is: AVcA  , , hence {here} the retarded relativistic    four-potential for a moving point charge is: AVcAVcrrrr,,w  tVc rr.

Griffith’s Example 10.3:

Find/determine the Liénard-Wiechert retarded potentials associated with point charge q  moving with constant velocity v .

For convenience sake, define tr 0 retarded time the charged particle passes through the origin.

   Then: w tvttvtrrrr  because vtr  v is a constant vector.

The retarded time is: ttr r c or: r ct trr ct with: tttrr

   But: r rrtrtrvtrrr w   but we also have: r ct trr ct

 Solve for the retarded time tr by relating: r rvtcttrr   

222222 Square both sides: rvtcttrr ct 2 ttt rr

2  222  But: rvtrrr rvtrvtr 2 rvtvt rr

222222 Thus: rrvtvtctttt22 rr rr  Solve this quadratic equation for t : cvt22220 ctrvtctr 2  222 r  rr    ab  c

2222222 2 bb2 4 ac ct rv ct  rv  c v r  ct t   t  ** r 2a r cv22

Which sign do we choose? + or  ?? Must make physical sense!

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

Consider the limit as v → 0: ttr r c ← what we want!  And, if v = 0, the point charge q is at rest at the origin ( r  0 ), because it is there at time tr  0 .

  Then: r rr  r  r

Thus, its retarded time should be: ttr r c  ttrcr  when v → 0.

 We must choose the – sign on physical grounds, i.e. we must choose:

 2 ct2222222 rv ct  rv  c v r  ct t  r cv22

   rvt rvt ˆ r r r Now: r ctrr  ct  t and: r   . r ctrr ct t Therefore, the quantity:

   rvtvvrvvv rvt  ˆ rr rr11vc c trr  ct t 1   ct t rr t ct trr c ct t c c c  rv v2 1  222 ct trr t  ct  rv  c v t r cc c

Then, insert the retarded time tr from the expression (**) {above} with the minus sign, i.e.:

 2 ct2 rv ct 2  rv  c 22222 v r  ct t  into the above formula & carry out the algebra: r cv22

1 2  Thus: rr1ˆvc r  ct222222 rv c v r ct with:  1 rˆvc {here}   c  

The general form of the Liénard-Wiechert retarded scalar and vector potentials associated with a  point electric charge q moving with a time-dependent velocity vtw r are:

  111qqq Vrtr ,   4441wˆˆvtc 1 w t ooorrrr rr r    qvww t qv t q ooorr Artr ,    v w  tr 4441wˆˆvtc 1 w t rrrr rr r

   vtw  ˆˆ  r Where: 1wrrvtcrr  1 w t with:  w tr  .  c

8 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

The Liénard-Wiechert retarded scalar and vector potentials associated with a point charge q  moving with constant velocity v are:

  111qq qc Vrtr ,  4441 ˆvc    2222222 ooorr r ct rv c v r ct     qv qv qcv ooo Artr ,  4441 ˆvc 2222222 rr r ct rv c v r ct

  where:  1 rˆvc = retardation factor {here} and: r  rrˆ .    Note again that {here}: Artrr,   Vrtc , where:   vc = constant vector.

The Electromagnetic Fields Associated with a Moving Point Charge

We are now in a position to derive the retarded electric and magnetic fields associated with a moving point charge using the Liénard-Wiechert retarded potentials associated with a moving point charge:

  11qq Vrtr ,   441w ˆ t oorrr r   qvw  t q oor Artr ,   v w  tr 441w ˆ t rrr r

   with: Artrrr,w    t Vrtc , with:  wwtvtcrr  and: c 1  oo {in free space}

 where: r rtw r

ˆ  and: ttr r c = retarded time and:  1wrvtcr = retardation factor.  The equations for the retarded EB and -fields in terms of their retarded potentials are:

  Art,  Ert,, Vrt   r and: B rt,, A  rt rrt rr

Again, the differentiation has various subtleties associated with it because:

   r rt r  trr   rt w  t   both quantities are evaluated at w tr the retarded time tt c and: vtw  t r r rrt

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

     Note that: rrrrtrrrr rwt ctctt  r is a function of both r and t.

    q 11ˆ Now: Vrtr ,  with: rrr 4 1 ˆvt c o r r r

qq11  Vrt,    vt c Thus: r     2 rr r 44oo vtr c   rr rr vtr c   But: r ct tr  r ct  tr  c tr      Then, using Griffiths “Product Rule # 4” AB A B B  A A  B  B A    on the second term r vtr c we obtain:

 (1)    (2)    (3)   (4)   111  1  r vtcrrrrrrrrr vtvt   vtvt  ccc   c

For term (1):

    dv trrrtttrrr dv t dv t rrrrrvtr xyz  vt r  x  r y  r z x y z dtrrr  x dt  y dt  z

dd Again:  since: ttr r c dtr dt

 dv tr tttrrr  rrrrrvtr xyz   at rr  t dt x y z

  dv trr dv t where: atr  = acceleration of charged particle at the retarded time ttr r c. dtr dt

   Second term (2): r rrtrr  rw  t   Thus: vtrrr vt  r ww t rrrr  vt   r vt  t    (2ab ) (2 )

  v trrrr r vxyz t  v t  vt  xxˆˆ  yy zzˆ Term (2a): xyz  vtxvtyvtzxyzrrrrˆˆ  ˆ vt  !!!

10 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

Term (2b):

   vtrr ww t  vxyz  t r v t r  v t r t r xyz  dtwwwrrrtttrrr dt dt vtxyzrrr  vt  vt dtrrr x dt y dt z  tttrrrdw tr vtxyzrrr  vt  vt xyzdtr   dtw r But: vtr  dtr  tttrrr  Thus: vttvtvtvtvtvtvttrrrrr w   xyz    r  rrr   xyz   n.b. vtrrrrrw  t  vt  vt  t is analogous / similar to: rrvtrr a t

   Third term (3): r vtr

    First, work out: vtr Since: A B AByz  AB zy xˆˆ  AB zx  AB xz y  AB xy  AB yx zˆ

Then:

  vtzxzxrrrrvtyyrr vt vt  vt vt vtr   xˆˆ   y    zˆ yz  zx   xy

dvzxzx trrrrttdvy  tr  dv t  tt dv t  dvy  tr  tt dv t rrxˆˆ  rryz   rrˆ dtrr y dt z dt rr  z dt x  dt rr  x dt y

ttrr  tt rr   tt rr  atz r atyxzyxrrrrr xˆˆ  at at y   at at zˆ yz  zx   xy

ttrr  tt rr   tt rr atyzrr  at xˆˆ   at zx rr  at y   at xy rr  at zˆ zy  xz   yx   atrr t

    rrvtrrrrr   at  t  r at  t

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

      Fourth Term (4): vtr r with: r rrtrr  rw  t

     First, work out: r rt wwrr  r   t   0  0  0  0  0  0 Of course!!! Because r is   z y x z y x position vector of field But: r  xˆ   yˆ   zˆ  0    point/observer Pr() y z z x x y   r = constant vector !!!  rxxyyzzˆˆˆ   From the result of term (3) above: vtrrr  at   t we see that: w tvttrrr    

 0      vtrrr  vt   rwwtrr  vt     t rrrr  vt  vt   t 

Collecting all of the above individual results (1) – (4) for the term:

1 rrvtrr c  r r vt c  1     r rrrr vtrr  vt      vt rr  vt   c        (1) (2) (3) (4)

We see that:

1  rrvtrrrrrrrr c  ct at r  t vt  vt  vt   t c    r atrr  t vt r  vt  rr  t   But: ABCBACCAB  

    r atattaat rrrrr rrrr      ta

   And: vv trrr vvt  tvv

Then, some truly amazing/fortuitous cancellations occur:

1  rrvtrrrr c  c t at r  t vtrrrr vt  vt   t c           at t tatvtvtt   tvtvt rrr rrr   rrr  rr  r 1  2 ct rr  vt r  at  rrr  v t  t c  

12 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

Thus: q 1  Vrt,  vt c rr   2 rr  4 o  rr vtr c q 11 ct vt at v2 t t   2 rrr  rrr 4 o  c rr vtr c  q 11vt    r cvt22  att   2 rrrr  4 o  cc rr vtr c 

   rrt rw  t Now, what is t {here}?? ttr  trr  t r r cc c

  But we already found that: r ctr , and noting that: r  rr

11   1   Then: r rr   rr   rr 22r r r      Now: AB A B B A A B B A {Griffiths Product Rule # 4}  r rr   rr   r  r  rrrr   22   r  r  r

11      1      r rrrrrrrrrr 22      22r r r

   But: r vtrr  t {from term (4) above}  rrr  vtrr  t

    rrrrrtrr  r  rw  t  rrrt w r from (2) above:  w tvtt  And: xyz   rr rr  r rrr xyzxˆˆ yzvtt ˆ  r r r xyz   rrvtrr  t

   1  ctr rr  rrr   r   1    rrrvtrr t vt  r  t r r      BAC  CAB rule    1       Thus: ctrrr rr  vt  t vtrrr  t tvtrrr   more cancellations occur!! r  

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

1   Or: ctrrr rr  vt  t r        1  r ˆ r 1 r Now solve fortr : cvttr  rr   r  tr    1    r r r r vtr  r c r vtr  c r   r rr11 rˆˆ r  tr      vt c c vt cc11ˆˆvt c vt c  c rrrrrr r rrrr   ˆˆ 1 r r r ˆ  Thus: tr   , rˆ  r  rrˆ and  1 rvtr c = retardation factor cc1 ˆvt c  r r r

 q 11vt  Vrt, r  c22 vt at t rrr 2 r  r 4 o r cc Then:  q 11vt   r cvt22  atrˆ 2 2 rrr  4o r cc 

    q 1 cvt22 at vt rrr ˆ  r Thus: Vrtr , 2 2 r  4o r  cc

2     q 11vt at vt rrrr ˆ  Or: Vrtr ,1 2   2 r  4  cc c o r 

  ooqv trrr qv t v t Now: Artr ,  nb . .2 Vrtr  , 44ˆ c 1 rrvtr c r 

   Artrr, oo vt 1 vt r qq   tt441 ˆvt c t vt c r r r rr r Then:   o 11vtr   qvtr  4 vt ctt vt c rrrr rr 

  vtrr dvtttrr  dv tr Now: atr where: atr  tdttr t dtr

t What is r {here} ?? t

14 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

Now: ctrr ct  t r 22 22 2  Or: ctrr ctt  r rr  differentiate this expression with respect to t.

22 2 2  ctrr c tt  r r ttt  2 tr r  2cttr  1 2r  tt

     But: rrrt r  trrrr  rt wt   ct ct t and also: r rtw  tr    22 ttrrr  Thus: ctrr212122w cctt    crrr    rt   tr  ttttt

 0      w t tr r r r Because r = vector to field point Pr  crr1  n.b.  0 ttt t is a constant vector – i.e. stationary observer! 

   ttrrwwtdtrr  t r  t r dw tr Thus: ccrr11 r r  r  vtr vtr  tttdtttr   dtr  ttrr tr  rrcvt1 r  Solve for tt t

tt  ttc rr rr r rrcc r vtrr  rr c vt     tt  ttcvtrr r

 tr rrcc111 Thus:     tcvt c ˆˆ rrr r 11rrvtrr c vt c

t 11   r  where:  1 ˆvt c = retardation factor ˆ  r r t 1 rvtr c 

 vtrrr tr at at Then: atr  tt ˆ 1 rvtr c

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

 1 ux ux1  ux 1 ux Using: 1 u2 x xxux 2

Then: 11      2 rrvtr c tt vt c  rr r rr vtr c 11   r vt 2 r r r tct    11rr  vtr 2 vtr  r r tc t t

But: r ctrr  ct  t tt  t tt t 11  r ct t cr  crr c1 where: r  r  ˆ  tt  t  tt  t t  1 rvtr c

  1 ˆvtc1 ˆ  r 11 r r rrvtrr vt Thus: ccc1     t  r

   0    rtw wwtdtvt vt dvt r r r rrtr  rrrtr  1 But:  and: atr tt  ttdttr  tdttr 

Thus:   111  vt rrr   2  vtr  r ttctt vt c rr r r   2 11rrvtrrr v t at 2      r r c  1  1    ˆv tvtat2  2 r rrrr  r c

  Artrr, o 11 vt   qvtr   ttt4  vt c r rr r Then:    11at vt o qvtvtatrrˆ2 2 rrrrr  4r r c

  Artr , 11q  vtr  1    1 ˆ 2 Or: 2 atrrrr rr vt v t at   using o  tc4    c2 o r r   c  o

16 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede   Art, Does this expression = Griffiths result for r , eqn 10.63 bottom of page 437 ?? YES, it does! t

   2 Art, 1 q  v tvt atrr vt r  ˆ rrr 322rratrrr vt  vt tc4   cc o r 

   ˆˆ But: cvtc11 r  1 rrvtrrr c vtr  vt  vt

Thus:    2 at vt Artr , 1 q  v tvtrrr rr 322cvtat1  rrr  tc4  cc o r     2 at vt 1 q vtvtrrr rr 322cv trrrr a  t cv  t 4  ccc o r  1 q    cv t2 a tr c22vt atvt 332rrrr  rrrr   4o  r cc 1 q    cvt atcr c22 vt atvt 332rrrr   r r  rr 4o  r cc  ˆ  But: rrc1  vtccrrr rrr   vtccc  rr  vt    Artr , 1 q r 22   332rrcvtvtatccvtatvtrrr r   r r  rr tc4o  r  c

Or:    Artr , 1 qc r 22   3 rrcvtvtatcrrr r /  cvtatvt r r  rr tc4   o rrcvt  r

 Griffiths Equation 10.63 on page 437

Then {“finally”!} the retarded for a moving point electric charge q:

  Art, Ert,, Vrt   r rrt

2     q 11vt at vt rrrr ˆ  Ertr ,12  2 r 4o    cc c r   q 11vt  1    atr ˆ vt v2 t at 2 rrrr rr    40  rrcc   

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

ˆ With some more algebra, more vector identities, and defining a retarded vector utrr cr vt , the retarded electric field for a moving point charge can equivalently be written as:

 q  r 22 Errrrr rt,  3  c v t utr  ut  at  4   o r utr

      Vrtr ,  vtw r vtr Since: Artrr,w    t with:  w tr  then: Artrr,, Vrt c c c2

The retarded of a moving point charge q can be written as:

    11 Brt,, Arrrrrrr  rt   vt V  rt ,   V rt ,  vt   vt  V  rt ,  cc22  n.b. The relation on the RHS used Griffiths Product Rule # 7: f Af    AAf     We already calculated vtrrr  at   t {term (3) above, page 11 of these lecture notes}  1 rˆˆr and we found that: tr   {see page 14 of these lecture notes}: cc1 ˆvt c  r r

   ˆ at   r r r ˆ Hence: vtrrrr  at   t  at     where:  1 rvtr c ccr

 q   Vrtr ,  {from page 9} and Vrt, is as given above {on page 14}. 4 r o r  Thus: n.b. vtrr vt  0    q 11atr  r Brtrr,, Art   2  4o cc rr  2  vt vt at vt q rrrr1  r ˆ  2  1  r 4  cc c o r 

ˆ After some more algebra, and again using the retarded vector: utrr cr vt  the retarded magnetic field of a moving point electric charge q is:

 1 q  r ˆ 22 Brrrrrrrrt,  3 rrr  cvtvt    atvt    utat  c 4   o r utr

Compare this expression to that for the retarded electric field of a moving point electric charge:

 q  r 22 Errrrr rt,  3  c v t utr  ut  at  4   o r utr

18 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

    The terms in the square brackets have some striking similarities between Br rt, vs. Ertr , . ˆ  Because of the cross product of terms such as r vtr contained in the square brackets of the   ˆ ˆ expression for Br rt, , notice that since: utrr cr vt  or: vtrr cr ut 

ˆˆˆˆˆˆˆ ˆˆ Then: rrrrrrrvtrr c ut  c ut rr ut , i.e. rrvtrr ut .  0   ˆˆ Then, using the ABCBACCAB   rule, and rrvtrr ut  we see that:

 ˆ cvtvt22  atvt  utat rrrrr  rr  rr   ˆ 22 cvtut rr  atut  rr  utat  rr rrr    r utrr  at   ˆ cvtut22    utat  rrrr  r  r

 1 q  r ˆ 22 Thus: Brrrrr rt,  3 rr c v t ut ut  at  c 4   o r utr

 q  r 22 But: Errrrr rt,  3  c v t utr  ut  at  4   o r utr

    B rt,,1 ˆ E rt  We again see that: rrc r  i.e. Br is  Er ; note also that Br is  r where    r rrtrr  rw  t = separation distance vector from retarded source point to field point.   22  2  The first term in Ertr , involving cvtut rr falls off / decreases as r . If both the   1 q ˆ velocity and acceleration are zero, then we obtain the static limit result: Er 2 r 4 o r    The first term in Ertr , is known as the “Generalized” Coulomb field, a.k.a. velocity field which describes the macroscopic, collective behavior of virtual photons!!

     The second term in Ertr , , involving the triple product r ua falls off / decreases as 1 r  i.e. this term dominates at large separation distances!   nd  The second term in Ertr , is responsible for radiation – hence the 2 term is known as the  radiation field. Since it is also proportional to atr it is also known as the acceleration field which describes the macroscopic, collective behavior of real photons!!

   The same terminology obviously also applies to the retarded magnetic field Br rt, .

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede  The EM force exerted on a point test charge qT at the observation/field point r at the present time, t due to the retarded electric and magnetic fields associated with a moving point charge q   is given by the retarded law: Frtrr,,v,, qErtTTT  q  rt Brt r  where   v,T rt = the velocity of the point test charge qT at the observation/field point r at the present time, t.

  Frtrr,,v,, qErtTTT  q  rt Brt r  qq  T r 22  3 cvtutrrr  utat  r  r 4   o r utr  v t  T  ˆ 22 rrcvtutrr  utat  r  r c     ˆ where: r rrtrr  rw  t and: utrr cr vt 

  The Lorentz force Frtr , is the net force acting on a point test charge qT moving with  velocity v,T rt {n.b. which is evaluated at the present time {i.e. the non-retarded} time t }.

    The Lorentz Force Frtr , is due to the retarded electric and magnetic fields Ertr , and    Br rt, associated with the point charge q moving with {its own} retarded velocity vtr and  acceleration atr .    ˆ  n.b. The lower-case quantities r trrrw  t, vtr , utrr cr vt  and atr are all evaluated at the retarded time ttr r c.

20 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

Griffiths Example 10.4:

Calculate the electric and magnetic fields of a point charge q moving with constant velocity.

 Constant velocity  atr  0 (no acceleration).

 q  r 22 ˆ Then: Ertrrr,  3  c vt ut with: utrr cr vt  4   o r utr

  Trajectory: w tvttvtrrrr  for constant velocity.

   The retarded time: ttr r c with: r trrrrrw  trvtt   and: r ctrr  ct  t

ˆ Since: utrr cr vt , then:

ˆ rrrutrr c vt    cvtrrr   cr r trr r vt   cr c w trr r v  t   cr cv trrr tr  c  t  t  v t   cr ctrr v t ctv trr ctr v  t  cr vtr t   ˆ Note also: rrrrrutrrr c vt   c  vt

  It can be shown that, since {here} vtr  v constant velocity vector, then: vtr  vt  v, and thus:

 2222222 (See p. 8 above, i.e. Griffiths r  utr  ct rv c v r ct Example 10.3, p. 433)

Rc 1 v22 sin / c 2 (See Griffiths Problem 10.14, p. 434) Rcvc 122 sin where  

  where R rvt = vector from the present location of the point electric charged particle to the   observation/field point Prt at the present time t moving with constant velocity v .

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 21 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

    The angle   cos1 Rv is the opening angle between R and the constant velocity vector v , as shown in the figure below:  PRt, Observation/Field Point  R Present position  1  of point charge q   cos Rv @ time t.  q v = constant velocity vector

2   qR1  ˆ v Thus: Ert,  3 2 with:    = constant 4 22 2 R c o 1sin       This expression for Ert, shows that Ert, points along the line from the present position of  the point charged particle to the observation/field point PRt, – which is strange, since the “EM news” came from the retarded position of the point charge. We will see/learn that the explanation for this is due to {special} relativity – peek ahead in Physics 436 Lect. Notes 18.5, p. 10-17.  Due to the  22sin  term in the denominator of this expression, the E -field of a fast-moving point charged particle is flattened/compressed into a “pancake”  to the direction of motion,  increasing the E -field strength in the  direction by a factor of  11 2 , whereas in the forward/backward directions { i.e.  and/or anti- to the direction of motion} the strength of the   E -field is reduced by a factor of 1  2 relative to that of the E -field strength when the point electric charge is at rest, as shown in the figure:

2   qR1  ˆ Ert,  3 2 4 22 2 R o 1sin  v with:    = constant c

 Lines of E are compressed into a “pancake”  to the direction of motion as v → c

        rvt rvtttv R v ˆ r r r For Brt, we have: r    since ttr r c or r ctr r rrrc Thus:    0  11Rv   1 1 v    Brt,,ˆ Ert Ert , R Ert ,Ert, r        n.b. R  E ccccrrcc

 2  11vqR 1    ˆ These expressions for E and Or: Brt,,, Ert  Ert 3 2 B were first obtained by cc c4 c22 2 R  o 1sin  in 1888.

22 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede  The lines of B circle around the point charge q as shown in the figure:  When vc (i.e.  1) the Heaviside expressions for EB and reduce to:

  1 q ˆ Ert,  2 R  essentially Coulomb’s law for a point electric charge 4 o R   q B rt, o v Rˆ 2   essentially Biot-Savart law for a point electric charge 4 R

The figure below shows a “snapshot-in-time” at time tsec2  of the classical/macroscopic electric field lines associated with a point electric charge q, initially at rest {at tseco  0 , where the green dot is located}, that undergoes an abrupt, momentary acceleration {i.e. a short impulse lasting ttt11 o  tsec, where the yellow dot is located} in the horizontal direction, to the left in the figure. After the impulse has been applied, the charge continues to move to the left with constant velocity v, at time tsec2  the charge is where the pink dot is located.

The classical/macroscopic electric field lines associated with one “epoch” in time must connect to their counterparts in another “epoch” of time. Here, in this situation, the spatial slopes of the E-field lines are discontinuous due to the abrupt, momentary nature of the acceleration. The spherical shell associated with the discontinuity(ies) in the electric field lines in the “transition” region between the two “epochs” expands at the speed of light, as the EM “news” propagates outward/away from the accelerated charge.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 23 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede

Note that this picture also meshes in nicely (and naturally!) with the microscopic perspective – namely that, when a point charge is accelerated it radiates real photons, which subsequently propagate away from the electric charge at the speed of light. Real photons have a transverse electric field relative to their propagation direction (whereas virtual photons associated with the static/Coulomb field are longitudinally polarized). The spherical shell associated with the discontinuity(ies) in the electric field lines of the “transition” region is precisely where the real photons are located in this “snapshot-in-time” picture, having propagated that far out from the charge after application of the abrupt, momentary impulse-type acceleration of the electric charge.

24 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved.