ELECTROMAGNETIC WAVES

Transcription

1 ELECTROMAGNETIC WAVES Now we will study eletromagneti waves in vauum or inside a medium, a dieletri. (A metalli system an also be represented as a dieletri but is more ompliated due to damping or attenuation for long wave-lengths. We will see that the dieletri funtion in this ase is omplex valued. We start with the more simple ase of non onduting media. With the simplifiations that the dieletri funtion and magneti permeability are onstants with no spatial dependene we get in regions of no free arriers or urrents ( D = 0 E = 0 E = 0 B = 0 1 B E + = 0 εµ E B = 0 Using the relation ( C = ( C C leads to εµ E E = 0 εµ B B = 0 The same wave equation for B and E fields.

2 Bo E. Sernelius 9: The solution to these equations are i kr t Er (, t = E e ( 0 i kr t Br (, t = B e ( 0 where the amplitudes E 0 and B 0 are onstant vetors. These represent plane, monohromati, eletromagneti waves propagating in the diretion ˆk. The amplitudes depend on one another both as regards the diretion and size. (It is assumed that one takes the real value of the fields to represent the real fields. To see this we return to what was said in the first leture about Fourier transforms: Fourier transforming differential equations has the following substitutional effets: i t ik ik Let us now use this in the wave equations: εµ E εµ E = 0 k E + E = 0 or k = εµ = n propagation onstant or wave number n = εµ index of refration The same result is of ourse obtained if used on the B field. Let us apply the proedure on Maxwell's equations

3 Bo E. Sernelius 9: E = 0 ik E = 0 B = 0 ik B = 0 1 B E + = 0 ik E i B = 0 εµ E εµ B = 0 ik B + i E = 0 The first shows that both E and B are transverse. The third that also the fields are orthogonal to eah other and that k, E, and B form a right handed set of vetors and that B0 E0 = n εµ This is also onsistent with the last of the equations. For a travelling wave the eletri and magneti fields osillate in phase. For a standing wave, whih is the superposition of two waves with the same frequeny but with k vetors in opposite diretion, the phase relations are suh that the eletri and magneti fields reah their maxima 90 degrees out of phase both in time and in spae.

4 Bo E. Sernelius 9:4 The Poynting vetor S = E H = E B 4π 4πµ = E0 B0 os ( k r t kˆ 4πµ = = where εµ E0 os ( k r t kˆ 4πµ E0 os ( k r t 4πη η µ ε is the wave impedane of the medium. S = E0 kˆ 8πη

5 Bo E. Sernelius 9:5 Convolution integrals The onstitutive relations are for homogeneous and isotropi systems expressed as onvolution integrals: Dr (, t = d r' dt' ε ( r r', t t' Er ( ', t' Br (, t = d r' dt' µ ( r r', t t' Hr ( ', t' Similarly we have: ind J ( r, t = d r' dt' σ ( r r', t t' E( r', t' Pr (, t = d r' dt' α ( r r', t t' Er ( ', t' Mr (, t = d r' dt' χ ( r r', t t' Hr ( ', t' where the first equation is Ohm s law and the other two give the polarization and magnetization of the system in the linear response regime in terms of the dieletri and magneti polarizabilities, α and χ, respetively. The polarization and magnetization are related to the fields aording to: D = E + 4πP B = H + 4πM The onvolution integrals have the nie property that their Fourier transforms will be just the produt of the two fators in the integrand. Thus, we have ( = ( ( ( = ( ( Dq, ε q, Eq, Bq, µ q, Hq, and ind J q, σ q, E q, ( = ( ( ( = ( ( ( = ( ( Pq, α q, Eq, Mq, χ q, Hq,

6 Bo E. Sernelius 9:6 These last two equations mean that ( = + ( ( = + ( ε q, 1 4πα q, µ q, 1 4πχ q, Another useful fat is that the Fourier transform of the inverse to a orrelation funtion is just unity divided by the Fourier transform of the funtion itself: 1 ( = ( ε q, 1 ε q, while ε 1 ( r, t 1 ε ( r, t

7 Bo E. Sernelius 9:7 COMPLEX REPRESENTATION We have seen that it is onvenient to represent osillatory funtions of time and spae by omplex exponentials, with the "real" physial funtion understood to be the real part of the omplex funtion. It is very useful when we have many funtions all with the same spae-time fator expi(k.r-t but possibly with different phases. As we have seen the differentiation and integration are redued to multipliation and division. We put the phase in the omplex amplitudes, and the ommon spae-time fator [expi(k.r-t] an usually be anelled out or ignored. This works sine most of the operations are linear, like addition, differentiation, integration and these ommutes with taking the real part. So we an perform all operations in the omplex plane and take real part at the end. There is however an important exeption: multipliation of two fields like in the alulation of the Poynting vetor does not ommute with taking the real part. The produt of real parts of two omplex numbers is not in general equal to the real part of the produt. In general the result is very involved, but if we are ontent with the timeaverage of the produt we an by using a trik get away with a simple rule. Let us have two funtions it iα it Ft ( = Fe 0 = F0e e it iβ it Gt ( = Ge 0 = G0e e Now, Re[ Ft ( ] Re[ Gt ( ] = F0 G0 ( ostosα + sintsinα ostosβ + sintsin β ( [ ( ] = F0 G0 1 os ± α β * * = Re 1 F0 G0 Re 1 F0 G0 [ ] = [ ]

8 Bo E. Sernelius 9:8 Thus in short hand notation the time-average produt theorem beomes F G 1 * F G = 1 * 0 0 F0 G0 Disussion: Sine we have (α-β in the expression for the time average. the spatial fator expi(k.r in ommon for F and G is suppressed. Applying this to Poynting vetor gives * S = E H * = E H π 8π where i kr E = E ( t 0e i kr H = H ( t 0e where E 0 and H 0 are both omplex valued vetors. This an also be applied to the time-averaged energy density for a plane wave in a linear medium: ( 1 * * = E0 D0 + H0 B0 16π 1 1 = ( = 0 16π ε E µ H 8π ε E We see that S = E 0 kˆ = kˆ = kˆ = v kˆ 8πη ηε n