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Download Lecture Notes 12: Lienard-Wiechert Retarded Potentials for Moving Point Charge, Retarded Electric and Magnetic Fields Associated with Moving Point Charge
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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 Vr r , t AND Ar r , t FOR A MOVING POINT CHARGE Suppose a point electric charge 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/vacuum) from point charge q to observer is determined by: ct r r where the time interval t t tr and: r r t r tr r t w tr Observation / field point at time t (doesn’t move) position of the point charge q at the retarded time tr w tr = retarded position of charge at retarded time tr r r r t w tr ct c t tr 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 P r t , which is at the present time, t tr r c . r = vector separation distance between the two points, as shown in the figure below: Position of pt. charge q at retarded time, tr Position of pt. charge q at present time, t ẑ Trajectory of pt. charge, w(tr) Source Pt. S w tr r r t w tr w tr r t ŷ Field Pt. P r t x̂ 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 P r t at the present time t, because it takes a finite/causal amount of time t t tr for EM “news” to propagate from w tr at the retarded time tr to the observation/field point P r t at position r at the present time t tr r c . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 1 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 P r , t 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 S r tr w tr at the retarded time tr to the observer’s location at the field point P r , t 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 P r t at the present time t with retarded times tr1 and tr2 respectively. Then: r 1 c t tr1 and: r 2 c t tr2 thus: r 1 r 2 c t tr1 c t tr2 c tr2 tr1 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 speed of light c !!! Only one retarded point w tr can contribute to the potentials Vr r , t and Ar r , t at the field point P r t at any given moment in the present time, t for v < c ! For v < c, an observer at the field point P r t 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 tot r , tr 1 Vr r , t d 4 o v r might suggest that the retarded scalar potential for a moving point charge is {also} 1 q 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: tot r , tr d q !!! v 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 r , tr over the entire charge distribution at one instant of time, but {here} the retardation tr t r c forces us to evaluate tot r , tr 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! r r t r tr is a function of r t and r tr Before integration: r w tr is fixed after integration: tot r , tr q 3 w tr r r t w tr is a function of r and t because tr t r c . After integration: 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 1 1 1 1 rˆv tr c 1 rˆ tr v r , tr d throws in a factor of: tot v tr where: tr c We define the retardation factor 1 rˆv tr c , where v tr {more precisely v r tr } is the velocity of the moving charged particle at the source position r tr 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 trend than light emitted from the front of the train trfront , both arrive simultaneously at the observer at the same present time t. The train is further away from the observer when light from the end of the train is emitted at the earlier time trend , compared to the train’s location for the light emitted from the front of the train at the later time trfront . The observer thus sees a distorted picture of the moving train at the present time t. © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 3 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 L L L . Then since ctc L , then: tc L c . But during the same time interval tc , the train moves a distance L vtc L L , or: tc L L L L L 1 L L L L but: tc thus: tc L c v v c v v 1 v c Trains moving towards / approaching an observer appear longer, by a factor of 1 1 v c . Conversely, it can similarly be shown that trains moving away / receding from an observer appear shorter by a factor of 1 1 v c . In general, if the train’s velocity vector v makes an angle θ with the observer’s line of sight r̂ (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 tc L cos / c . Note that the train also moves a distance L L L in this same time interval. L cos tc L cos / c but: tc 1 cos Or: L c v L L L L cos L L L L cos L L tc i.e. v v c v v c v 1 v cos L or: L 1 v c v 1 L or: L v cos v 1 c L 1 v L with c 1 cos From the above figure, the angle cos 1 rˆvˆ = opening angle between r̂ and v̂ . v 1 1 L Thus: cos rˆ where: . Hence: L L. c 1 cos 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: v t 1 1 1 where tr r and 1 rˆv tr c 1 rˆ tr c 1 rˆv tr c 1 rˆ tr and where rˆ r r r r = unit vector associated with the separation distance between the position of a {stationary} observer r t at the present time t to a position somewhere on the train r tr at the retarded time tr . Explicitly: rˆ r r r r r t r tr r t r tr The stationary observer’s position vector r t is constant in time, whereas the retarded position vector of the moving train r tr changes in time. Hence, whenever we carry out integrals of the type tot r , tr d {or J tot r , tr d } v v where the integrand(s) tot r , tr {or J tot r , tr } are associated with {some kind of} moving charge {current} distribution(s), evaluated at the retarded time tr , the apparent volume of these v tr 1 1 1 integrals is modified by the factor where: tr , and the c 1 rˆv tr c 1 rˆ tr 1 1 1 d d retardation factor 1 rˆv tr c 1 rˆ tr , i.e. d d 1 rˆv tr c 1 rˆ tr The figure shown below graphically depicts this effect, for a snapshot-in-time t tr r c : True volume at the present time t Observer position P(r,t) at present time t (fixed) v t Apparent volume at the retarded time tr v tr 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 2005-2015. All Rights Reserved. 5 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 retarded trajectory w tr , with: q r , tr q 3 w tr is: 1 Vr r , t 4 o q r , tr r v d 1 1 4 o r v q 3 w tr r q 3 w tr d d 4 o r v r 1 rˆv tr c 1 1 1 1 q q 1 q 4 o 1 rˆv w tr c r 4 o 1 rˆ w tr r 4 o r Where: rˆ r r r r , r r t r tr r t w tr v w tr 1 rˆv w tr c 1 rˆ w tr with: w tr , And: c Where: v w tr = velocity vector of charged particle evaluated at the retarded time tr t r c . The retarded current density J tot r , tr for a rigid object is related to its retarded charge density tot r , tr and its retarded velocity v r , tr by the relation: J tot r , tr tot r , tr v r , tr . The retarded vector potential associated with a point electric charge q moving with retarded velocity v w tr along a retarded trajectory w tr , with: J q r , tr q r , tr v r , tr qv r , tr 3 w tr is: Ar r , t o 4 J q r , tr v r d o 4 q r , tr v r , tr r v d o 4 qv r , tr 3 w tr v qv r , tr 3 w tr qv w tr o d v r 1 rˆv t c ˆ 4 v t c r r 1 w r r qv w t q qv w tr r o o o v w tr r 4 r 4 1 rˆ w tr r 4 o 4 r d Thus, we have obtained the so-called Liénard-Wiechert retarded potentials for a point electric charge q moving with retarded velocity v w tr along a retarded trajectory w tr : 1 q 1 q Vr r , t 4 o 1 rˆ w tr r 4 o r qv w tr o o q v w Ar r , t tr 4 1 rˆ w tr r 4 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 Where: Fall Semester, 2015 1 rˆv w tr c 1 rˆ w tr Lect. Notes 12 with: w tr Prof. Steven Errede v w tr c . v w tr Vr r , t 1 v w tr 1 Vr r , t Vr r , t w tr Note that: Ar r , t using: c 2 2 c c c c o o v w tr Vr r , t Ar r , t w tr with: w tr . Or: c c Where Vr is in Volts, Ar is in Newtons/Ampere (= momentum per Coulomb); they are related to 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: A V c , A , hence {here} the retarded relativistic four-potential for a moving point charge is: Ar Vr c , Ar Vr c , w tr Vr c . 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 tr v tr tr vtr because v tr v is a constant vector. The retarded time is: tr t r c or: r c t tr ctr with: tr t tr r r r tr r w tr r vtr but we also have: r c t tr ctr But: Solve for the retarded time tr by relating: r r vtr c t tr Square both sides: But: Thus: Solve this quadratic equation for tr : 2 2 r vtr c 2 t tr c 2 t 2 2ttr tr2 2 r vtr r vtr r vtr r 2 2r vtr v 2tr2 r 2 2r vtr v 2tr2 c 2 t 2 2ttr tr2 v 2 tr2 2 c 2t r v tr c 2t 2 r 2 0 c2 a c2t rv b b 2 4ac tr tr 2a b c c t rv c 2 2 c2 v2 2 v 2 r 2 c 2t 2 ** Which sign do we choose? + or ?? Must make physical sense! © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 7 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede Consider the limit as v → 0: tr t 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 r r r r Thus, its retarded time should be: tr t r c tr t r c when v → 0. We must choose the – sign on physical grounds, i.e. we must choose: c t rv c t rv c 2 tr Now: r ctr c t tr 2 2 2 v 2 r 2 c 2t 2 c2 v2 r vtr ˆr r r vtr . r ctr c t tr and: Therefore, the quantity: c c r vtr v r vtr v r v v v r 1 rˆv c ctr 1 tr c t tr 1 c t tr c t tr c c t tr c 1 r v v 2 c t tr tr c 2t r v c 2 v 2 tr c c c Then, insert the retarded time tr from the expression (**) {above} with the minus sign, i.e.: c t rv c t rv c 2 tr 2 2 2 v 2 r 2 c 2t 2 c2 v2 1 Thus: r 1 rˆv c r c c t rv c 2 2 2 into the above formula & carry out the algebra: v 2 r 2 c 2t 2 with: 1 rˆv c {here} 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 v w tr are: 1 q 1 q 1 4 o 1 rˆv w tr c r 4 o 1 rˆ w tr r 4 o qv w tr qv w tr o o o Ar r , t 4 1 rˆv w tr c r 4 1 rˆ w tr r 4 Vr r , t Where: 8 1 rˆv w tr c 1 rˆ w tr with: w tr v w tr c q r q v w tr r . © 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: Vr r , t 1 q 1 q 1 4 o 1 rˆv c r 4 o r 4 o Ar r , t qv o qv o o 4 1 rˆv c r 4 r 4 qc c t rv c 2 2 2 v 2 r 2 c 2t 2 qcv c t rv c 2 2 2 v 2 r 2 c 2t 2 where: 1 rˆv c = retardation factor {here} and: r rrˆ . Note again that {here}: Ar r , t Vr r , t c where: v c = 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: 1 q 1 q Vr r , t 4 o 1 rˆ w tr r 4 o r qv w tr o o q v w Ar r , t tr 4 1 rˆ w tr r 4 r with: Ar r , t w tr Vr r , t c with: w tr v w tr c and: c 1 o o {in free space} where: r r w tr and: tr t r c = retarded time and: 1 rˆv w tr c = retardation factor. The equations for the retarded E and B -fields in terms of their retarded potentials are: Ar r , t Er r , t Vr r , t and: Br r , t Ar r , t t Again, the differentiation has various subtleties associated with it because: r r t r tr r t w tr and: w tr v tr w tr t both quantities are evaluated at the retarded time tr t r c © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 9 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede Note that: r r r r tr r w tr ctr c t tr r is a function of both r and t. Now: Vr r , t 1 1 4 o r 1 rˆv tr c q 1 q 1 r r v t r c 2 4 o r r v tr c 4 o r r v t c r r c t tr r c t tr ctr Thus: Vr r , t But: with: rrˆ r q Then, using Griffiths “Product Rule # 4” A B A B B A A B B A on the second term r v tr c we obtain: (1) (2) (3) (4) 1 1 1 1 r v tr c r v tr v tr r r v tr v tr r c c c c For term (1): dv tr tr dv tr tr dv tr tr r v tr r x x r y y r z z v tr r x dt x r y dt y r z dt z Again: d d dtr dt r r r since: tr t r c dv tr tr t t r y r r z r a tr r tr r v tr r x dt x y z dv tr dv tr where: a tr = acceleration of charged particle at the retarded time tr t r c . dtr dt r r r tr r w tr Thus: v tr r v tr r w tr v tr r v tr w tr Second term (2): (2 a ) Term (2a): 10 v t r v t x v r (2 b ) vz t r xxˆ yyˆ zzˆ y z vx tr xˆ v y tr yˆ vz tr zˆ v tr !!! x r y tr © 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): v tr w tr vx tr v y tr vz tr w tr x y z dw tr tr dw tr tr dw tr tr vx t r v y tr vz t r dtr x dtr y dtr z t t t dw tr vx tr r v y tr r vz tr r x y z dtr dw tr v tr But: dtr t t t Thus: v tr w tr v tr vx tr r v y tr r vz tr r v tr v tr tr x y z n.b. v tr w tr v tr v tr tr is analogous / similar to: r v tr a r tr Third term (3): r v tr First, work out: v tr Since: A B Ay Bz Az By xˆ Az Bx Ax Bz yˆ Ax By Ay Bx zˆ Then: v t v t v t v t v t v t v tr z r y r xˆ x r z r yˆ y r x r zˆ z x y z y x dv t tr dv y tr tr dv y tr tr dvx tr tr dvx tr tr dvz tr tr z r xˆ zˆ yˆ dtr z dtr x dtr y dtr z dtr y dtr x tr t a y tr r xˆ az tr y z t t a y tr r az tr r xˆ z y a t r t r tr t t t az tr r yˆ a y tr r ax tr r zˆ ax tr z x x y tr t ax tr r az tr x z r v tr r a tr tr r a tr tr tr t a y tr r zˆ yˆ ax tr y x © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 11 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Fourth Term (4): v tr r Lect. Notes 12 Prof. Steven Errede with: r r r t r w t First, work out: r r w t r w t r r r r Of course!!! Because r is 0 0 0 0 0 0 position vector of field z y x z y x zˆ 0 xˆ yˆ But: r point/observer P ( r ) y z z x x y r = constant vector !!! r xxˆ yyˆ zzˆ From the result of term (3) above: v tr a tr tr we see that: w tr v tr tr 0 v t r r v t r r w t r v t r w t r v t r v t r t r Collecting all of the above individual results (1) – (4) for the term: 1 r r v t r c r r v t r c 1 r r v tr v tr r r v tr v tr r c (1) (2) (3) (4) We see that: 1 r r v tr c ctr a tr r tr v tr v tr v tr tr c r a t r t r v t r v t r t r But: And: A B C B AC C A B r a tr a r tr tr r a a r tr tr r a v v tr v v tr tr v v Then, some truly amazing/fortuitous cancellations occur: 1 r r v tr c ctr a tr c a tr r t v t v t v t t r r r 12 r r t t r a t v t v t t t v t v t r r r 1 ctr v tr r a tr v 2 tr tr c r r r r r r © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. r UIUC Physics 436 EM Fields & Sources II Thus: Vr r , t Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede 1 v t r c r r 4 o r r v t c 2 r q 1 1 ctr v tr r a tr v 2 tr tr 2 4 o r r v t c c r v tr 1 2 2 1 q c v t a t r r r tr 4 o r r v t c 2 c c r q Now, what is tr {here}?? tr t r c t r r tr c r w tr t c But we already found that: r ctr , and noting that: r r r 1 1 1 Then: r r r r r r r 2 r r 2r Now: A B A B B A A B r r r r r r r r But: B A {Griffiths Product Rule # 4} r r 2r r 2 r r 1 1 1 r r r 2r r 2 r r r r r 2r 2r r v tr tr {from term (4) above} r r r v t t r r r r r r r t r r w t r r r w t r r from (2) above: r And: r r x y z xˆ r y yˆ r z zˆ v tr r tr r x y z x r v tr r tr r w tr v tr r tr 1 ctr r r r r r r 1 r v tr tr r v tr r tr r 1 Thus: ctr r v tr r BAC CAB rule r tr v tr r tr tr r v tr more cancellations occur!! © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 13 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede 1 ctr r r v tr tr Or: r 1 r r 1 r Now solve for tr : c r v tr tr rˆ tr r r r 1 r v t c r v tr rc r r r r r 1 1 rˆ rˆ t r c 1 rˆv tr c c rc 1 rˆv tr c r v tr rc rc r v tr 1 rˆ rˆ r Thus: tr , rˆ c 1 rˆv tr c c r r rrˆ and 1 rˆv tr c = retardation factor 1 v tr 1 2 2 q c v t a t r Vr r , t r r tr 4 o r 2 c c Then: 1 v tr 1 q 2 c 2 v 2 tr r a tr rˆ 2 4 o r c c Thus: Vr r , t q 1 4 o r 2 c 2 v 2 tr r a tr v tr rˆ c c2 2 1 v tr r a tr ˆ v tr r 1 c2 c c Or: 1 q Vr r , t 4 o r 2 Now: qv tr o o qv tr Ar r , t 4 1 rˆv tr c r 4 r v tr n.b. 2 Vr r , t c v tr Ar r , t o v tr 1 o q q t 4 t r 1 rˆv tr c 4 t r r v tr c Then: v tr o 1 1 v tr q t r r v tr c 4 r r v tr c t Now: v tr dv tr tr dv tr tr a tr where: a tr t dtr t t dtr What is tr {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 Now: ctr c t tr r c 2 tr2 c 2 t tr r 2 r r 2 Or: Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede differentiate this expression with respect to t. 2 2 2 c tr c 2 t tr r r t t t r tr 2 2c t tr 1 2r t t But: r r r t r tr r t w tr ctr c t tr and also: r r t w tr Thus: r 2 2 t t c tr 2c c t tr 1 r 2cr 1 r 2r 2r r t w tr t t t t t t cr 1 r t 0 r r w tr 0 n.b. r t t t is a constant vector – i.e. stationary observer! w tr dw tr tr t tr tr r r v t r r Thus: cr 1 rc 1 r t t t dtr t t t t t rc 1 r r v tr r Solve for r t t t rc rc Thus: tr t t r v tr r rc r v tr r t t t Because r = vector to field point P r dw tr v tr dtr tr rc t rc r v tr rc tr rc 1 1 1 t rc r v tr rc 1 rˆv tr c 1 rˆv tr c tr 1 1 where: 1 rˆv tr c = retardation factor t 1 rˆv tr c v tr a tr tr a tr a tr Then: t t 1 rˆv tr c © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 15 UIUC Physics 436 EM Fields & Sources II Using: 1 u x x Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede u 1 x u x 1 u x 1 u 2 2 x x u x Then: 1 1 v t r c r r 2 t r r v tr c r r v t c t r 1 r 1 r v tr 2 r t c t 1 r 1 r v tr v t r r 2 t t r t c r ctr c t tr t tr r But: t t c t tr c t tr 1 1 t t t c r c 1 r where: t 1 rˆv tr c t t t 1 rˆv tr c 1 rˆv tr r v t r r 1 1 c 1 c Thus: c t r 0 dw tr tr v tr v tr dv tr tr 1 r r w tr r w tr a tr and: t dtr t t dtr t t t t But: Thus: r 1 r v tr v t r r t t t c 2 1 r v tr 1 v tr r a tr 2 c r r 1 1 2 t r r v tr c r 1 r 2 1 ˆ 2 r v tr v tr r a tr c Ar r , t o 1 v tr 1 q v tr 4 r t t t r r v tr c Then: 1 a tr v tr ˆ 1 2 v t v t a t r r o q r r r 2 c 4 r r Or: 16 Ar r , t v tr 1 q 1 1 1 2 a tr 2 rˆv tr v tr r a tr using o 2 4 o c r c t r c o © 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 Ar r , t Does this expression = Griffiths result for , eqn 10.63 bottom of page 437 ?? YES, it does! t Ar r , t v 2 tr v tr r a tr v tr q 1 ra tr rˆv tr v tr t c c 4 o 3 r 2 c 2 Thus: v 2 tr v tr r a tr v tr c 1 v tr ra tr c c v 2 tr v tr r a tr v tr q cv tr ra tr cv tr 3 r 2c 2 c c r 2 2 q 2 r r r a t r v tr v t cv t a t c r r r 3 3 2 rc c r 2 2 q r r r a tr v t r c v t a t c c v t r r r 3 3 2 rc c Ar r , t 1 q 3 2 2 4 o r c t c 1 v tr c 1 1 rˆv tr c v tr rˆv tr v tr But: But: 1 4 o 1 4 o 1 4 o rc 1 rˆv tr c rc r r v tr c c rc r v tr Ar r , t r q 1 rc r v tr v tr ra tr c c 2 v 2 tr r a tr v tr 3 3 2 c t 4 o r c Or: Ar r , t qc 1 r rc r v tr v tr ra tr / c c 2 v 2 tr r a tr v tr 3 t c 4 o rc r v t r Griffiths Equation 10.63 on page 437 Then {“finally”!} the retarded electric field for a moving point electric charge q: Ar r , t Er r , t Vr r , t t 1 v t 2 r a t v tr r r rˆ 1 c2 c 4 o r 2 c v tr q 1 1 1 2 2 a tr rˆv tr v tr r a tr c 4 0 r c r Er r , t q 1 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 17 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede With some more algebra, more vector identities, and defining a retarded vector u tr crˆ v tr , the retarded electric field for a moving point charge can equivalently be written as: q r c 2 v 2 tr u tr r u tr a tr Er r , t 3 4 o r u t r v w tr v tr Vr r , t Since: Ar r , t w tr with: w tr then: Ar r , t 2 Vr r , t c c c The retarded magnetic field of a moving point charge q can be written as: 1 1 B r , t Ar r , t 2 v tr Vr r , t 2 Vr r , t v tr v tr Vr r , t c c n.b. The relation on the RHS used Griffiths Product Rule # 7: f A f A A f We already calculated v tr a tr tr {term (3) above, page 11 of these lecture notes} 1 rˆ rˆ {see page 14 of these lecture notes}: and we found that: tr c 1 rˆv tr c c a tr r rˆ Hence: v tr a tr tr a tr where: 1 rˆv tr c c cr Vr r , t q 4 o r {from page 9} and Vr r , t is as given above {on page 14}. n.b. v tr v tr 0 Thus: 1 1 a tr r 4 o c 2 r cr 2 v tr q v tr 1 v tr r a tr ˆ r 1 4 o r 2 c c c After some more algebra, and again using the retarded vector: u tr crˆ v tr Br r , t Ar r , t q the retarded magnetic field of a moving point electric charge q is: 1 q r Br r , t rˆ c 2 v 2 tr v tr r a tr v tr r u tr a tr 3 c 4 o r u t r Compare this expression to that for the retarded electric field of a moving point electric charge: q r c 2 v 2 tr u tr r u tr a tr Er r , t 3 4 o r u t 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 r , t vs. Er r , t . Because of the cross product of terms such as rˆ v tr contained in the square brackets of the expression for Br r , t , notice that since: u tr crˆ v tr or: v tr crˆ u tr rˆ rˆ rˆ u tr rˆ u tr , i.e. rˆ v tr rˆ u tr . Then: rˆ v tr rˆ crˆ u tr c 0 Then, using the A B C B AC C A B rule, and rˆ v tr rˆ u tr we see that: rˆ c 2 v 2 tr v tr r a tr v tr r u tr a tr 2 2 rˆ c v tr u tr r a tr u tr r u tr a tr r u tr a tr rˆ c 2 v 2 tr u tr r u tr a tr Thus: 1 q r ˆ c 2 v 2 tr u tr r u tr a tr r Br r , t 3 c 4 o r u t r But: Er r , t r 2 2 r u tr a tr c v t u t r r 3 4 o r u t r q We again see that: Br r , t 1c rˆ Er r , t i.e. Br is Er ; note also that Br is r where r r r tr r w tr = separation distance vector from retarded source point to field point. The first term in Er r , t involving c 2 v 2 tr u tr falls off / decreases as r 2 . If both the 1 q rˆ velocity and acceleration are zero, then we obtain the static limit result: E r 4 o r 2 The first term in Er r , t 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 Er r , t , involving the triple product r u a falls off / decreases as 1 r i.e. this term dominates at large separation distances! The second term in Er r , t is responsible for radiation – hence the 2nd term is known as the radiation field. Since it is also proportional to a tr 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 r , t . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 19 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 Lorentz force law: Fr r , t qT Er r , t qT vT r , t Br r , t where vT r , t = the velocity of the point test charge qT at the observation/field point r at the present time, t. Fr r , t qT Er r , t qT vT r , t Br r , t qqT r c 2 v 2 t r u t r r u t r a t r 3 4 o r u t r vT t rˆ c 2 v 2 tr u tr r u tr a tr c where: r r r tr r w tr and: u tr crˆ v tr The Lorentz force Fr r , t is the net force acting on a point test charge qT moving with velocity vT r , t {n.b. which is evaluated at the present time {i.e. the non-retarded} time t }. The Lorentz Force Fr r , t is due to the retarded electric and magnetic fields Er r , t and Br r , t associated with the point charge q moving with {its own} retarded velocity v tr and acceleration a tr . n.b. The lower-case quantities r tr r w tr , v tr , u tr crˆ v tr and a tr are all evaluated at the retarded time tr t 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 a tr 0 (no acceleration). r c 2 v 2 tr u tr with: u tr crˆ v tr 3 4 o r u t r Then: Er r , t Trajectory: w tr v tr tr vtr for constant velocity. q The retarded time: tr t r c with: r tr r w tr r v tr tr and: r ctr c t tr Since: u tr crˆ v tr , then: ru tr r crˆ v tr cr rv tr c r r tr rv tr cr cw tr rv tr cr cv tr tr c t tr v tr cr ctr v tr ctv tr ctr v tr c r v tr t Note also: r u tr r crˆ v tr cr r v tr It can be shown that, since {here} v tr v constant velocity vector, then: v tr v t v , and thus: r u tr c t rv c 2 2 2 v 2 r 2 c 2t 2 Rc 1 v 2 sin 2 / c 2 Rc 1 2 sin 2 (See p. 8 above, i.e. Griffiths Example 10.3, p. 433) (See Griffiths Problem 10.14, p. 434) where v c where R r vt = vector from the present location of the point electric charged particle to the observation/field point P r t at the present time t moving with constant velocity v . © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. 21 UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 12 Prof. Steven Errede The angle cos 1 Rv is the opening angle between R and the constant velocity vector v , as shown in the figure below: Present position of point charge q @ time t. R P R, t Observation/Field Point cos 1 R v q v = constant velocity vector 1 2 Rˆ q v , E r t = constant Thus: 3 2 with: 4 o 1 2 sin 2 2 R c This expression for E r , t shows that E r , t points along the line from the present position of the point charged particle to the observation/field point P R, t – 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 2 sin 2 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 1 1 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: E r ,t q 1 2 4 o 1 2 sin 2 3 2 Rˆ 2 R v = constant c with: Lines of E are compressed into a “pancake” to the direction of motion as v → c r vtr r vt t tr v R v r since tr t r c or r ctr For B r , t we have: rˆ r r r r c Thus: 0 1 1 R v 1 1v ˆ B r , t r E r , t E r , t R E r , t E r , t n.b. R E c c r c cc cr Or: 22 1 2 q 1v 1 B r ,t E r ,t E r ,t cc c 4 o c 1 2 sin 2 3 2 Rˆ 2 R These expressions for E and B were first obtained by Oliver Heaviside in 1888. © 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 v c (i.e. 1 ) the Heaviside expressions for E and B reduce to: E r ,t 1 q ˆ R 4 o R 2 essentially Coulomb’s law for a point electric charge q B r , t o 2 v Rˆ essentially Biot-Savart law for a point electric charge 4 R The figure below shows a “snapshot-in-time” at time t2 sec of the classical/macroscopic electric field lines associated with a point electric charge q, initially at rest {at to 0 sec , where the green dot is located}, that undergoes an abrupt, momentary acceleration {i.e. a short impulse lasting t t1 to t1 sec , 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 t2 sec 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 2005-2015. All Rights Reserved. 23 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.