This is an exercise problem, BUT my question is not the exercise, but the fundamental assumption of the problem. That's the reason I don't post it in the homework section as it has nothing to do with solving the problem. I have issue with the given assumption all together.
This is a case of Normal Incidence where the incident wave travels in +z direction in medium 1. The incident wave hitting the medium 2 boundary on xy plane where z=0. Incident E is in x direction and H in y direction.
This is the original question word by word:
In writing Eq.9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave.....along the x direction. Prove that this must be so.
Hint: Let the polarization vectors of the transmitted and reflected waves be:
##\hat n_T=\hat x\cos\theta_T+\hat y \sin\theta_T\;\hbox { and } \;\hat n_R=\hat x\cos\theta_R+\hat y \sin\theta_R##
and prove from the boundary conditions that ##\theta_T=\theta_R##.
##\hbox{(9.76)}\;\Rightarrow\;\tilde E_R(z)=\hat x E_{0R}e^{-jk_1z}\;\hbox { and }\;\tilde B_R(z)=-\hat y \frac 1 {v_1}E_{0R}e^{-jk_1z}##
##\hbox{(9.77)}\;\Rightarrow\;\tilde E_T(z)=\hat x E_{0T}e^{-jk_2z}\;\hbox { and }\;\tilde B_T(z)=\hat y \frac 1 {v_2}E_{0T}e^{-jk_2z}##
This is from "Introduction to Electrodynamics" 3rd edition by Dave Griffiths, the gold book for undergrad physics. The pages of relevant are page 384 to 386.
My first issue is: By assuming the polarity of ##\tilde E_T## is ##\hat n_T=\hat x\cos\theta_T+\hat y \sin\theta_T## and ##\tilde E_R## is ##\hat n_R=\hat x\cos\theta_R+\hat y \sin\theta_R##, the author already assuming the ##\tilde E_R## and ##\tilde E_T## are tangential to the boundary. I accept that ##\tilde E_{0T}## has the same polarity as ##\tilde E_{0I}## as this is proven by the boundary condition that tangential E is continuous across the boundary. So this in natural the polarity follows.
BUT I have found nothing in over 5 books explaining why ##\tilde E_{0R}## is the same polarity. In fact, there is no proof I can find that ##\tilde E_{0R}## is tangential to the boundary. By assuming that ##\tilde E_R## is ##\hat n_R=\hat x\cos\theta_R+\hat y \sin\theta_R##, the author already make an important assumption that is not proven yet here.
I might still have more to follow later.
Thanks
Alan
This is a case of Normal Incidence where the incident wave travels in +z direction in medium 1. The incident wave hitting the medium 2 boundary on xy plane where z=0. Incident E is in x direction and H in y direction.
This is the original question word by word:
In writing Eq.9.76 and 9.77, I tacitly assumed that the reflected and transmitted waves have the same polarization as the incident wave.....along the x direction. Prove that this must be so.
Hint: Let the polarization vectors of the transmitted and reflected waves be:
##\hat n_T=\hat x\cos\theta_T+\hat y \sin\theta_T\;\hbox { and } \;\hat n_R=\hat x\cos\theta_R+\hat y \sin\theta_R##
and prove from the boundary conditions that ##\theta_T=\theta_R##.
##\hbox{(9.76)}\;\Rightarrow\;\tilde E_R(z)=\hat x E_{0R}e^{-jk_1z}\;\hbox { and }\;\tilde B_R(z)=-\hat y \frac 1 {v_1}E_{0R}e^{-jk_1z}##
##\hbox{(9.77)}\;\Rightarrow\;\tilde E_T(z)=\hat x E_{0T}e^{-jk_2z}\;\hbox { and }\;\tilde B_T(z)=\hat y \frac 1 {v_2}E_{0T}e^{-jk_2z}##
This is from "Introduction to Electrodynamics" 3rd edition by Dave Griffiths, the gold book for undergrad physics. The pages of relevant are page 384 to 386.
My first issue is: By assuming the polarity of ##\tilde E_T## is ##\hat n_T=\hat x\cos\theta_T+\hat y \sin\theta_T## and ##\tilde E_R## is ##\hat n_R=\hat x\cos\theta_R+\hat y \sin\theta_R##, the author already assuming the ##\tilde E_R## and ##\tilde E_T## are tangential to the boundary. I accept that ##\tilde E_{0T}## has the same polarity as ##\tilde E_{0I}## as this is proven by the boundary condition that tangential E is continuous across the boundary. So this in natural the polarity follows.
BUT I have found nothing in over 5 books explaining why ##\tilde E_{0R}## is the same polarity. In fact, there is no proof I can find that ##\tilde E_{0R}## is tangential to the boundary. By assuming that ##\tilde E_R## is ##\hat n_R=\hat x\cos\theta_R+\hat y \sin\theta_R##, the author already make an important assumption that is not proven yet here.
I might still have more to follow later.
Thanks
Alan