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2thumbsGuy
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Homework Statement
Suppose that an electromagnetic wave is moving in the +x-direction, from one medium to another; call them medium 1 and medium 2. The boundary conditions on the electric and magnetic fields are then
E1= E2, [itex]\frac{1}{μ1}[/itex]B1=[itex]\frac{1}{μ2}[/itex]B2
These equations relate the components of electric and magnetic fields just to the left and just to the right of the interface between the two media, parallel to the surface. (There are no components perpendicular to the surface, so no boundary conditions in this direction.)
As described in class, there will be an incident wave EI, a reflected wave ER, and a transmitted wave ET. Apply the boundary conditions given above and show that the reflected and transmitted waves are related to the incident wave.
Homework Equations
ER=[itex]\frac{(1-β)}{(1+β)}[/itex]EI, Et=[itex]\frac{2}{(1+β)}[/itex] EI
where
β=[itex]\frac{μ_1}{v_1}[/itex] [itex]\frac{μ_2}{v_2}[/itex]=√[itex]\frac{μ_1}{ɛ_2}[/itex][itex]\frac{μ_2}{ɛ_1}[/itex]=[itex]\frac{μ_1}{μ_2}[/itex][itex]\frac{n_1}{n_2}[/itex]
HINT: Recall that B=[itex]\frac{1}{c}[/itex] E,n=[itex]\frac{c}{v}[/itex],c=([itex]\frac{1}{√ɛμ}[/itex]), and that the speed of the EM wave is different in different media.
The Attempt at a Solution
ER=((α-β)/(α+β)) EI and ET=(2/(α+β)) EI where α ≡ (cosθT)/(cosθI ) and β≡μ1/μ2 n1/n2.
If the E and B fields are parallel to the incident medium and do not have any components vertical to the medium, this means the E and B fields are parallel to the surface (perpendicular to the normal).
α is a function of the angle from the EM wave to the normal, and since the angle between the angle of the incident wave to the normal is 0, α=(cosθ_T)/(cosθ_I )=cos0/cos0=1
In addition, β, which is a constant specific to how the mediums 1 & 2 affect how light travels in them, equals μ1/μ2 n1/n2 (among other things) So, more completely:
ER=((cosθT)/(cosθI )-β)/((cosθT)/(cosθI )+β) EI, ET=2/((cosθT)/(cosθI )+β) EI
Then, at the boundary, if:
EI=ER+ET=((cosθT)/(cosθI)-β)/((cosθT)/(cosθI)+β) EI)+2/((cosθT)/(cosθI)+β) EI)
Then: EI=(((cosθT)/(cosθI)-β)/((cosθT)/(cosθI)+β)+2/((cosθT)/(cosθI)+β)) EI=(((cosθT)/(cosθI)-β+2)/((cosθT)/(cosθI )+β))EI
Since (cosθT)/(cosθI)=1
ER+ET=((1-β+2)/(1+β)) EI⇒(ER+ET)/EI = (1-β+2)/(1+β)=1
This means that (ER+ET)/EI is proportional, making them equal at the boundary of the mediums.
Please tell me if I'm missing some critical point, all these subtleties are lost on me.