ELECTRODYNAMICS: PHYS 30441
2. Radiation and Retarded Potentials
2.2 Liénard-Wiechert Potentials and Point Charges
Retarded Potentials and the Wave Equation We have arrived at a modified form of the vector and scalar potentials in terms of a charge density and current density source terms evaluated at a retarded time. Before proceeding we are required to verify that these vector and scalar potentials discussed in 2.1, proposed on physical grounds, do indeed satisfy the inhomogeneous wave equations.
Here we will verify that retarded potentials V and A:
⎛⎞ 1 ⌠ ρ−⎜⎟r',t R 1 c V(r,t) = ⎮ ⎝⎠d3 r', 4Rπε ⎮ 0 ⌡ V
Alfred-Marie Emil Johann ⎛⎞ 1 ⌠ Jr',t⎜⎟− R Wiechert, μ c Liénard, A(r,t) = 0 ⎮ ⎝⎠d3 r', German 4Rπ ⎮ French ⌡ Physicist Physicist V (1869-1958) (1861-1928) ∂A where R=− r r ' , B= ∇ xA, E =−∇− V , ∂t Equations were developed in part by Alfred-Marie Liénard in 1898 and satisfy the inhomogeneous wave equation for V: independently by Emil Wiechert in 1900 and continued into the early
2 1900s ⎛⎞2 1 ∂ρ ⎜⎟∇−22V =− ⎝⎠ct ε∂ 0
22 Firstly, consider ∇=∇ where R=r − r ' r R
231(R)2ρ ∇= V d r '∇ ← No θφ , dependence in integration rr∫ 4Rπε0 V
2 22ρ∂ρ11 Now: ∇= +ρ∇ RRRR∂2R R
1123⎛⎞ Recall: ∇=−δR ⎜⎟ (R) 4Rπ ⎝⎠ ------q1 As an aside, remember for a point charge at the origin: V= and ∇=−ρε2 V / 4rπε 0 . q1 q 1 1 1 1 ∇=−δ⇒∇=−δ232(r) 3(r) or for a charge at r': ∇=−δ−23(R), R= r r ' 4rπε00 ε 4rπ 4Rπ
Electrodynamics PHYS 30441, Radiation From Accelerated Charges Part 2, R.M. Jones, University of Manchester -1- ELECTRODYNAMICS: PHYS 30441
Also, ρ=ρ (t − R / c) satifies the 1-D wave equation:
∂ρ2211 ∂ρ ∂ρ 2 ∂ρ 2 −=→=0 ∂∂Rct222 ∂ R2 ct 22 ∂
2 23111⌠ ⎡⎤∂ρ 3 Thus, ∇= V d r ' −4πρδ−(r r ') R ⎮ ⎢⎥22 4Rπε0 ⌡ ⎣⎦ct∂ V
⎡⎤ 2 ⎢⎥ 2311∂ρ−⌠ (r',tR/c)ρ−(r,t R / c) ∇= Vd⎢⎥r'− R 22 ⎮ ct∂ ⎢⎥4Rπε0 ⌡ ε0 ⎢⎥ V ⎣⎦V(r,t− R/c)
22 221∂ρ− (r,t R/c)⎛⎞ 1 ∂ ρ−(r,t R/c) ⇒∇ VV = − or ∇ − Vr,tR/c− =− R 22 ⎜⎟22 () ct∂∂εε0 ⎝⎠ct 0
Thus, the proposed representation in terms of retarded potentials do indeed satisfy the wave equation (it is worth noting that a similar representation for retarded E and B fields in terms of t-R/c do not satisfy the wave equation).
Finally, we note that the advanced potentials with t = t +R/c are entirely consistent with Maxwell’s equations but they violate causality –in nature effect does not precede cause! For completeness we include the advanced potentials:
⎛⎞ 1 ⌠ ρ+⎜⎟r',t R 1 c V(r,t) = ⎮ ⎝⎠d3 r', 4Rπε ⎮ 0 ⌡ V
⎛⎞ 1 ⌠ Jr',t⎜⎟+ R μ c A(r,t) = 0 ⎮ ⎝⎠d3 r'. 4Rπ ⎮ ⌡ V
They do have some theoretical interest as they are a product of the time-reversal invariance of the wave equations.
Exercise:
2 22ρ∂ρ11 Show that ∇= +ρ∇ RRRR∂R 2 R
2 (Hint use spherical coordinates for ∇∇ and ) RR
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Liénard-Wiechert Potentials and Point Charges
We will investigate the scalar and vector potential generated by a moving point charge q.
1 ρ=δ−(r ', t ') q3'⎡⎤ r ' r (t ') , t ' =−−→ t r r ' Integration is difficult as it also depends on r ⎣⎦0 c
⌠ ⎮ ⎛⎞1 3' ρ=τδτ−+−δ−(r',t') q d⎜⎟ t r r' (r ' r0 (τ)) ⎮ c ⎮ ⎝⎠ τ and r' are no ⌡ 1 longer connected contribution only at τ−− =t'=t r r ' c
⌠ ⎡⎤1 δτ−+tR() τ q ⎮ ⎢⎥c ⇒=V(r,t) d τ⎣⎦ ⎮ ' 4πε0 ⎮ rr()−τ0 ⎮ ⌡ R(τ )
Now we use the sifting property of delta functions:
δ−(x x ) 1 δ=[f (x)] 0 , e.g. δ (ax)= δ (x) ∑ d a f(x0 )= 0 f(x) dx
1d1dR ' f(τ=τ−+ ) t R( τ ), f = 1 + (here t0ret= t ) cdττ cd' τ=t0
q1 1 ⇒=V(r,t) ' 4πε0 R(t0 ) ⎡⎤1dR()τ ⎢⎥1+ cdτ ' ⎣⎦⎢⎥τ=t0
Exercise:
1dR()τβ .R dr' Prove =− (where cβ =0 -particle's velocity at time τ ) cdττ R d τ=tret τ=tret
Ans:
' 1dR 1 ' dr0 Consider =∇ r − r . ()r ' 0 o cdτττ=t c d ret τ=tret
' 1dR ()rr.−β0 R.β =− =− cdτ ' R τ=tret rr− 0 τ=tret τ=tret
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Using the result from the exercise
q1 q 1 ⇒=V(r,t) = 44πε πε ⎡ ˆ ⎤ 00()R.R−β R1()−β .R ret ⎣ ⎦ret
1 where the quantities in parenthesis are evaluated at the retarded time: ttR(t=− ). ret c ret
Also, for the vector potential:
μ0 ⌠ 3 J(r ',t') 3' 1 A==⎮ d r' , with J(r',t') cβ (t')ρ (r',t')= qcβ (t')δ [r− r0 (t')] and t'=t- r− r' 4rr'π−⌡ c V and this gives the Liénard-Wiechert vector potential:
μμqc[]ββqc[] A(r,t) ==0 ret 0 ret 44ππ⎡⎤R.R−β ⎡R1()−β .Rˆ ⎤ ⎣⎦ret ⎣⎦ret
∂A For E=−∇ V − and B= ∇ xA we can derive the Lienard-Wiechert fields: ∂t
⎡⎤ ˆ 2 Rxˆˆ R−β x β qd⎢⎥()R1−β() −β ()() E= ⎢⎥ 33+β; =β (τ ), 4dπε ˆˆ2 τ 0 ⎢⎥()1.RR−β c1.RR()−β ⎣⎦ret
⎡⎤2 β−βxRˆ 1 ˆˆ ˆ μ0qc ⎢⎥()()ββ.R( xR) βxR B= 33+ + 4π ⎢⎥2 ˆ 2 ⎢⎥()1.RR−β ˆˆ c1.R()−β R c(1−β .R) R ⎣⎦ret
For slow moving charges these reduce to the results of electro and magneto-statics. To prove this consider ββ<<,1.
1 Also, it is straightforward to show B=×⎡⎤ Rˆ E (as an exercise show this is indeed true) c ⎣⎦ret
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Features of Liénard-Wiechert Potentials
• The Lienard-Wiechert potentials and fields are relativistically correct (Lorentz covariance)
• For point charges, the B-field is always perpendicular to the E-field and the unit vector Rˆ from the retarded position to the observation point
• We have already seen that the E and B fields can be decomposed into velocity and acceleration components: EE=+va E, B=BB v +a
11 with E ,B∝β , and E ,B∝β , vv R 2 aa R
• The energy radiated by a moving charge per unit time per unit solid angle is:
dP ⎧⎫2 dt = lim⎨⎬ P.RRˆ . at large distances R −>∞ () ddΩ ⎩⎭tret
P=ExH is the Poynting vector which repsents the energy flow per unit area ⊥ ar to the direction of energy flow.
The components of the Poynting vector are:
11 PExB,∼∼∝∝ PExB vv v v RR43va v a
1 P∼ E xB∝← the only non-vanishing component aa a a R 2
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Radiation by an Accelerated Charged Particle
Particle Moving with a Constant Velocity ( = cβ)
In this case only the velocity component of the Lienard-Wiechert field contributes:
⎡⎤ˆ 2 q ⎢⎥()R1−β() −β E= 3 4πε ⎢⎥2 0 ⎢⎥()1.RR−β ˆ ⎣⎦ret
2 q ()1−β =−Rˆ β 3 ()r 2 4πε 0 ˆ ()1.RR−β rr
with RRc(tt)rp=+β− ret
or equivalently, Rp=−β− R r c (t tret ); also Rr =− (t tret )c
ˆ ⇒=−RRprβ R r or RR(R prr=−β) (1)
2 q ()1−β ER= 3 p 3 4πε 0 ˆ ()1.RR−β rr
Thus despite the retarded field analysis the field is directed from the actual position of the particle ( R p ) at time t rather than the retarded position!
The objective is now to obtain the denominator in terms of Rp and in terms of the angle between the direction of travel and the field observation point.
22 22 sq(1)⇒=− Rpr R 2R rrrβ .R+ R β (2)
222ˆ 2 sq denominator⇒−β=−β+β Rrrrrr (1 R ) R 2R .R ( .Rr ) (3)
Also, perpendicular components are equal:
22 RRrp×β = ×β
22 2 22 2 R(R.)R(R.rrβ− β = pp β− β) (4)
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22 22 And from (2): Rrp=+β−β R 2R rrr .R R , substitute into (3)
222ˆ 22 2 R(1rrprrrrrr−β= .R) R + 2R.R β−β−β+ R 2R.R (R.) β
222 222 and from (4): (Rrrpp .β− ) R β= (R . β− ) R β
22222ˆ 22222 2 22 ⇒−β=+β−β=+βR(1rrpppp .R) R (R.) R R Rpp(cosθ −= 1) R (1 −β sin θ )
2222ˆ 2 ⇒−β=−βR(1rrp .R) R(1 sinθ ) i.e. this allows the field to be written in terms of the present location of the particle.
2 ˆ G q (1−β )R p E = 2223/2 4R(1sin)πε0p − β θ
Also, since the Lienard-Wiechert equations give: 1 B=× Rˆˆ E (where R is evaluated at the retarted position). c rr In terms of the present position of the particle Rc(tt)Rrr=β −et + p,
ˆ and Rrrr==β−+ R /R c (t t retrpr )/R R /R ; and R r =− c(t t ret )
1 ⇒=β+RR/Rˆ and B=R/RxEβ+ rpr c ()pr
and as Rp lies along the field then the second term is zero:
1 ⇒βB= xE c
This allows the magnetic field to be obtained:
2 ˆ (1−β ) β xR p qq() qcμ0 B=≡2223/2 also note, 4cπε00R(1p −β sin θ ) 4cπε 4 π
Again, the magnetic field does not depend on the position in retarded time but depends on the present position of the particle only!
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Features of Particle Moving with Constant Velocity
• For relativisitic velocities , i.e. β ~ 1: 1 1. At θ = π/2 the E-field is enhanced by a factor of γ= 1−β2
1 2. At θ = 0, π the E-field is suppressed by a factor of 1−β2 = γ2 The consequence of this is the radiation in linear accelerator (applicable for example, to the 2-mile linac in California known as SLAC) is markedly transverse to the direction of motion. For ultra- relavisitic electron beams the radiation is suppressed rapidly by 1/γ2 in the direction of motion and is enhanced by γ transverse to the motion. Thus, the radiation E-field is concentrated in a pancake-like distribution. Indeed, as v→ c the angle of the pancake goes to π/2 and the E-field is then entirely transverse to the direction of motion.
1 1 • The Poynting vector PExB=∝4 and represents an electromagnetic field energy flow μ0 R p attached and convected along with the charge. Also, due to lim R2 P= 0 , there is no radiation { p } Rp →∞ from a uniformly moving charge
Low-velocity beam (β~.1) Relativistic beam (β~.1)
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