Time Harmonic Sources and Retarded Potentials PDF

4 Time harmonic sources and retarded poten- tials The solution of forced wave equation ∂ 2V ρ ∂t 2 ∇ 2 V − µ o ϵo =− ϵo f...

4 Time harmonic sources and retarded poten- tials The solution of forced wave equation ∂ 2V ρ ∂t 2 ∇ 2 V − µ o ϵo =− ϵo for scalar potential V is most conveniently obtained in the frequency domain: Consider a time-harmonic forcing function ρ and a time-harmonic re- sponse V expressed as ρ(r, t) = Re{ρ̃(r)ejωt} and V (r, t) = Re{Ṽ (r)ejωt} in terms of phasors ρ̃(r) and Ṽ (r). ∂ Then, the above wave equation transforms — upon replacing ∂t by jω — into phasor form as ρ̃ ∇2Ṽ + µoϵoω 2Ṽ = −. ϵo For ω = 0 the above equation reduces to Poisson’s equation, which we know has, with an impulse forcing (for 1 C point charge at the origin) 1 1 ρ̃(r) = δ(r), an impulse response solution Ṽ (r) = ≡ 4πϵo|r| 4πϵor 1  where r ≡ |r| denotes the distance of the observing point r from the impulse point located at the origin. – Note that this impulse response ∝ 1/r is symmetric with respect to the origin just like the impulse input δ(r). Note that by substituting the source function δ(r) and response function 4πϵ1o r back We now postulate and subsequently prove that for ω ≥ 0, the impulse re- into the Poisson’s equation we obtain an equality sponse solution of the forced wave equation — i.e., with forcing function ρ̃(r) = δ(r) — is 1 ! " ∇2 = −4πδ(r), |r| e−jk|r| ω which is a useful vector iden- Ṽ (r) = √ c tity. with k ≡ ω µoϵo =. 4πϵo|r| Proof: For ρ̃(r) = δ(r) the source of the forced wave equation (for an arbitrary ω) is symmetric with respect to the origin, implying that the corre- sponding solution Ṽ (r) should also have the same type of symmetry. Then, with no loss of generality, we can claim a solution for the case ρ̃(r) = δ(r) of the form f (r) Ṽ (r) = r where 1 f (r) = 4πϵo for ω = 0, and f (r) is to be determined for an arbitrary ω as follows: 2 – Substituting f (r)/r for Ṽ (r) and δ(r) for ρ(r) in the forced wave equation (see margin), we obtain Forced wave eqn (phasor form): 2 f (r) 2 f (r) δ(r) )+k r r ρ̃ ∇( =− ϵo ∇2Ṽ + k 2Ṽ = − which reduces, for r ̸= 0, to ϵo with 2 f (r) 2 f (r) )+k = 0. ω r r ∇( √ k = ω µ o ϵo =. c – Since (as shown in HW) we have, by using spherical coordinates (reviewed next lecture), f (r) 1 ∂ 2f )= , r r ∂r2 ∇2 ( it follows that we have, for r ̸= 0, 1 ∂ 2f ( 2 + k 2f ) = 0, r ∂r which is in turn satisfied by f (r) = ge∓jkr = ge∓jωr/c with an arbitrary constant g. – Finally, the constraint that f (r) = 1/4πϵo for ω = 0 indicates that 1 g= , 4πϵo 3  and thus The choice −jkr leads to so- e∓jkr called retarded solution of the f (r) = wave equation. The alterna- 4πϵo tive choice +jkr is not used because it leads to an ad- in the solutions f (r)/r of the wave equation with ρ̃(r) = δ(r). vanced solution that depends on future values of the charge distribution not available in This concludes our proof of the postulated solution practice (this causality con- straint is further discussed e−jk|r| ω later in this lecture). Ṽ (r) = √ c with k ≡ ω µoϵo = 4πϵo|r| Note that k is another sym- bol for wavenumber β. In where the sign choice in the exponent favors the physically relevant causal this and higher level courses in EM and signal processing solution as opposed to the acausal alternative (see discussion below). k is favored over β (for a good number of reasons which will become apparent as we learn For the record, by scaling the result above: more). ✬ ✩ ✬ ✩ Likewise, for J˜z (r) = P δ(r), the causal solu- For ρ̃(r) = Qδ(r), the causal solution of the tion of the forced wave equation forced wave equation 2 2 ρ̃ ∇2Ãz + k 2Ãz = −µoJ˜z , ∇ Ṽ + k Ṽ = − , ϵo √ where k ≡ ω µoϵo must be the phasor √ where k ≡ ω µoϵo is the phasor µoP e−jkr Ãz (r) = , Q e−jkr 4π r Ṽ (r) =. ✫ 4πϵo r which describes, with P = I∆z, the vec- ✪ tor potential of the Hertzian dipole defined in Lecture 6. ✫ ✪ 4  We can next argue as follows: e−jk|r| δ(r) → Forced Wave Eqn → 4πϵo|r| and ′ ′ e−jk|r−r | δ(r − r ) → Forced Wave Eqn → 4πϵo|r − r′| imply that ′ ′ ′ 3 ′ d r = Ṽ (r), ρ̃(r′)e−jk|r−r | 3 ′ ρ̃(r )δ(r−r )d r = ρ̃(r) → Forced Wave Eqn → # # 4πϵo|r − r′| giving us, on the right-hand side, the retarded potential solution in the frequency domain. Finally, inverse Fourier transforming the above result back to time do- main, we obtain ′| c ρ(r′ , t − |r−r V (r, t) = d r, ) 3 ′ # 4πϵo|r − r | ′ where we made an explicit use of the time-shift property of the Fourier transform as in |r − r′| ′ ′ c ρ(r′, t − ) ↔ R(r′, ω)e−jω|r−r |/c ≡ ρ̃(r′ )e−jk|r−r |. – Note that V (r, t) is a weighted superposition of the past values of charge density ρ(r, t) (as opposed to future values) because of our 5  use of the causal solution1 (as opposed to acausal solution) of the forced wave equation discussed above. Question: is causality an ad- ditional postulate on top of Maxwell’s equations that needs to It is useful to stress at this point the relationship between a phasor (of a time be invoked to understand radia- tion? harmonic function) and a Fourier transform (of a time domain function) as follows: Answer: no, not really, we need to invoke causality at this stage to pick the relevant root of the solu- A phasor, say, Ṽ (r) is a sample of a Fourier transform function V (r, ω) tion for the forced wave equation at the frequency ω of a time-harmonic function that the phasor repre- simply because we took a short- cut of using a steady-state solu- sents. tion based on Fourier transforms (phasors). Had we solved the Conversely, a Fourier transform V (r, ω) represents a continuous collec- same problem as an initial value problem (using the Laplace trans- tion of phasors Ṽ (r) representing time-harmonic functions of all possi- form), only the retarded potential solution would have figured in our ble ω. answer naturally without having to invoke a separate causality pos- Based on the above correspondence principle we feel free to switch tulate — see J. L. Anderson, between phasor and Fourier transform concepts as convenient. “Why we use retarded potentials”, Am.J. Phys., 60, 465, 1992. 1 This choice is also referred to as Sommerfeld’s radiation condition after Arnold Sommerfeld who also developed an asymptotic formula that retains the causal solution and rejects the acausal one. 6  Having finished the derivation of the retarded potential solution of the forced wave equation for scalar potential, we can re-state our result, and by analogy the result for the retarded vector potential as: ✬ ✬ ✩ ✩ ′ ′ | | c µoJ(r′ , t − |r−r ) 3 ′ A(r, t) = d r, ρ(r′ , t − |r−r V (r, t) = d r, c ) 3 ′ # # 4πϵo|r − r′ | 4π|r − r′| the solution of inhomogeneous wave equation the solution of inhomogeneous wave equation 2 ∂ 2V ρ ∂ 2A ∂t2 ∇ 2 A − µ o ϵo = −µoJ ∂t ∇ V − µ o ϵo 2 = − ✫ ϵo ✫ ✪ ✪ where 1 is the speed of light in free space. µ o ϵo c≡√ These results indicate that retarded potentials z ρ(r′ , t) r − r′ V (r, t) and A(r, t) J(r′ , t) ′ r r are appropriately weighted and delayed sums of ρ(r, t) and J(r, t) y O in convolution-like 3D space integrals. x 7  Next turning our attention to retarded vector potential solutions, we note that the results stated in time and frequency domains are as follows: ✬ ✬ ✩ ✩ Time-domain: Frequency-domain: ′| ′ µoJ(r′ , t − |r−r A(r, t) = d r, c ) 3 ′ Ã(r) = d r, µoJ̃(r′ )e−jk|r−r | 3 ′ # # 4π|r − r′| 4π|r − r′| the solution of the inhomogeneous wave equa- the solution of the inhomogeneous wave equa- tion tion 2 1 ∂ A ω2 c ∂t c ∇2A − 2 2 = −µoJ. ∇2Ã + 2 Ã = −µoJ̃. ✫ ✪✫ ✪ In the next lecture we will learn how to perform vector calculus operations in spherical coordinates and then apply the frequency-domain result obtained above to the calculation of radiation from short current elements. 8