Chapter 16 changes for u and e

Chapter 16 changes in the presence of and PhL 2.5.03 Section 16.2 on deriving the Grand Finale formula. You might think you could just replace B with H and E with D and have the stuff cleared out of Maxwell's equations, but that is not the case. Looking at page 178, the problem is that the Faraday's law will then say that curl D = -1/c t H * . So we need another plan. Go back to the starting point 16.31 and add as shown based on 7.1 page 202 which shows the four Maxwell's in this case, notice there is a single insertion point. Go ahead and do the time derivatives and get 16.32. We are going to end up rescaling the B field, so the div equations won't change. We care about the curl equations. These are: xE = ikB k = = * = k' = 1/ xB = -ikE = -ik-2 E = -1 Now define the usual modified k (call it k') as shown above right. Then we get xE = ik'B xE = ik'(B) xE = ik'B' xB = -ik'-1E x(B) = -ik'E xB' = -ik'E Therefore, we have shown that we can rewrite 16.32, our starting point, by making two replacements: 1) replace k with k' k' = k / = 1/ 2) replace B with B' B' = B Once we have done this, all four equations look exactly the same. This is the main point!!! The triplet equation groups page 543. These are as shown, except make the two replacements shown above. The multipole fields page 545. These are as shown, except make the two replacements shown above. Notice that all Bessel functions will now have (k'r) as argument. Grand Finale 16.47 page 546. Put in k' for k everywhere, and the first equation is for B'. These are just the same changes made in the steps above. Section 16.3 equations. Equations like 16.48 and 16.51 are just limits with constants not shown, so we are not bothered by not seeing 1/ in 16.51. The swap rule 16.52 of course should have our B' in place of B. I would put a B' in 16.53 and therefore also in 16.56. Same in 16.57 along with replacing k with k' there. Now in 16.58 we have to make a different change! The true formula for u has ED and BH. So we would then write this as EE + (1/) BB. If we now rescale to our B = B'/, we get EE+ (1/) -2 B' B' = [ EE + B'B']. In the radiation limit being done here, we have |E| = |B'| from modified 16.56. Then we can write the energy u = 2 (B')2. Therefore, 16.60 will have an extra out front. Then in 16.61 we really should have H there and H = B/. But then put in B = B'/, and then 16.61 will have 1/() out front. Using 16.57 with its actual k' and B' then gives 16.62 with both B' inside, and with (1/) out front. Thus, the result 16.65 will have this 1/ out front. The ratio in 16.66 will then say that M/U(modified) = M/U (shown) * (1/) * (1/). So the ratio is then no longer as shown and I do not know how to interpret the modified result. Section 16.4 on distribution. The true time averaged power will be ExH = (1/) E x B = (1/) E x B' = E x B'. Thus, we should have this square root factor out in front of 16.70. Step 1: Let's go through the general multipole expansion and add the presence of uniform and . Start with 7.1 page 202 which shows the four Maxwell's in this case, notice there is a single insertion point. So, I have marked up how the two 3-equation sets on page 543 are altered. One alteration is that you have to think of k as /v instead of /c, but only in places that got their k from the wave equation! This means inside any Bessel function like j(kr), the multiplying k in there must be the k = /v version of k. However, the external factors of k are unchanged, namely, the 1/k that you see in the multipole fields on 545 except you have to add one factor as shown for E in the electric mode case only. Looking then at the general expansion 16.47, we have to add same factor that we added into 16.42. Note that this is NOT just a rescaling of the aE coefficient because the two expansions are coupled. On page 548 I added new factors as shown. Next, how is the huge 16.139 going to be altered? Section 16.5 on moments. Looking at 16.77, Jackson is starting with the full equations including H, and he has left the polarization charge grouped in with . So I have no modifications to make to this entire section. Section 16.7 on center-fed antenna. Everything goes exactly as stated. However, when we get to the radiated power formula 16.121, we should add the same factor out front as on page 550 described above, and in all other power formulas in this section. Section 16.8 on Plane Wave Expansions. First consider 16.130. As usual, k = k' in the exponent. The second equation there should be modified according to 7.13 on page 204 which says B0 = -1 E0. Thus the second equation is correct if we just replace B by B'. So we just make the usual 2 changes in 16.130. As for 16.131, it is just a case of the Grand Finale 16.47, and we know to think of B' on the left of the second, and all k are k'. Luckily for us, all the solution for the coefficients is unchanged and we end up with 16.139 with just our usual 2 changes. Section 16.9 on the Conducting Sphere. As usual, replace B with B' everywhere and k with k'. We do this in the assumed form of the outgoing waves as well as in 16.139. The boundary condition can be taken as nB' = 0 so nothing changes throughout. The final results 16.151 and 16.152 are the same except interpret k and B as k' and B'. Now we come to the power formulas. There should be a factor out front in 16.155 for the same reason as discussed above (really it is ExH = (1/) ExB = (1/) ExB' ).