- #1

Hi!

I have a problem understanding what happens when a plane electromagnetic wave hits a surface consisting of two medias of different optical density.

My old school litterature tells me two formulas which I really don't understand where they come from.

The same teacher starts by telling me these facts:

1) E1t=E2t

2) H1t=H2t

3) D1n=D2n

4) B1n=B2n

where n stands for normal to the surface and t stands for tangential to the surface.

In my drawing I have tried to use 1 and 3 to wind up with my teacher's curious formulas like

[tex]\frac{E_{rn}}{E_{1n}}=-\frac{sin(\theta_1-\theta_2)}{sin(\theta_1+\theta_2)}...1.1[/tex]

and

[tex]\frac{E_{rt}}{E_{1t}}=\frac{tan(\theta_1-\theta_2)}{tan(\theta_1+\theta_2)}...1.2[/tex]

but I get a totally different answer like for instance for the tangential part

[tex]E_1sin(\theta_1)=E_2sin(\theta_2)+E_rsin(\theta_r)...1.3[/tex]

which may be rewritten as

[tex]\frac{E_{rt}}{E_{1t}}=\frac{E_rsin(\theta_r)}{E_1sin(\theta_1)}=\frac{E_1sin(\theta_1)-E_2sin(\theta_2)}{E_1sin(\theta_1)}...1.4[/tex]

or

[tex]\frac{E_{rt}}{E_{1t}}=\frac{E_rsin(\theta_r)}{E_1sin(\theta_1)}=1-\frac{E_2sin(\theta_2)}{E_1sin(\theta_1)}...1.5[/tex]

which isn't even close to my teacher's formula.

So what am I doing wrong?

Best regards, Roger

PS

I like complex numbers and by using them I kind of think that my failure of understanding might not be regarding the physics but regarding the math, my thoughts goes like this:

[tex]K=\frac{E_{rt}}{E_1}=1-\frac{E_2sin(\theta_2)}{E_{1t}sin(\theta_1}...2.3[/tex]

Here I have tried to comply with my teacher's formula by testing this

[tex]K\approx 1-\frac{e^{j\theta_2}}{e^{j\theta_1}}=1-e^{j(\theta_2-\theta_1)}...2.4[/tex]

Here I am guessing wildly and state that there can be no single real 1 in a complex number, which makes

[tex]K\approx e^{j(\theta_2-\theta_1)/2}(e^{-j(\theta_2-\theta_1)/2}-e^{j(\theta_2-\theta_1)/2})...2.5[/tex]

or

[tex]K\approx -2je^{j(\theta_2-\theta_1)/2}\frac{(e^{j(\theta_2-\theta_1)/2}-e^{-j(\theta_2-\theta_1)/2}}{2j}...2.6[/tex]

which equals

[tex]K\approx -2je^{j(\theta_2-\theta_1)/2}sin((\theta_2-\theta_1)/2)...2.7[/tex]

where we have a constant (complex) amplitude which only varies in phase and has been added 90 degrees in phase due to j, in any case the substraction of the angles is here correct and I am starting to wonder if the transponate also should be considered and in that case perhaps it "stays" in the nomenator of 2.4 which would yield

[tex]K\approx -2je^{j(\theta_2-\theta_1)/2}\frac{sin((\theta_2-\theta_1)/2)}{sin((\theta_2+\theta_1)/2)}...2.8[/tex]

still, I only get half the angles and it should also read tangens.

This is the best I can do about this, I have given up so please help.

I have a problem understanding what happens when a plane electromagnetic wave hits a surface consisting of two medias of different optical density.

My old school litterature tells me two formulas which I really don't understand where they come from.

The same teacher starts by telling me these facts:

1) E1t=E2t

2) H1t=H2t

3) D1n=D2n

4) B1n=B2n

where n stands for normal to the surface and t stands for tangential to the surface.

In my drawing I have tried to use 1 and 3 to wind up with my teacher's curious formulas like

[tex]\frac{E_{rn}}{E_{1n}}=-\frac{sin(\theta_1-\theta_2)}{sin(\theta_1+\theta_2)}...1.1[/tex]

and

[tex]\frac{E_{rt}}{E_{1t}}=\frac{tan(\theta_1-\theta_2)}{tan(\theta_1+\theta_2)}...1.2[/tex]

but I get a totally different answer like for instance for the tangential part

[tex]E_1sin(\theta_1)=E_2sin(\theta_2)+E_rsin(\theta_r)...1.3[/tex]

which may be rewritten as

[tex]\frac{E_{rt}}{E_{1t}}=\frac{E_rsin(\theta_r)}{E_1sin(\theta_1)}=\frac{E_1sin(\theta_1)-E_2sin(\theta_2)}{E_1sin(\theta_1)}...1.4[/tex]

or

[tex]\frac{E_{rt}}{E_{1t}}=\frac{E_rsin(\theta_r)}{E_1sin(\theta_1)}=1-\frac{E_2sin(\theta_2)}{E_1sin(\theta_1)}...1.5[/tex]

which isn't even close to my teacher's formula.

So what am I doing wrong?

Best regards, Roger

PS

I like complex numbers and by using them I kind of think that my failure of understanding might not be regarding the physics but regarding the math, my thoughts goes like this:

[tex]K=\frac{E_{rt}}{E_1}=1-\frac{E_2sin(\theta_2)}{E_{1t}sin(\theta_1}...2.3[/tex]

Here I have tried to comply with my teacher's formula by testing this

[tex]K\approx 1-\frac{e^{j\theta_2}}{e^{j\theta_1}}=1-e^{j(\theta_2-\theta_1)}...2.4[/tex]

Here I am guessing wildly and state that there can be no single real 1 in a complex number, which makes

[tex]K\approx e^{j(\theta_2-\theta_1)/2}(e^{-j(\theta_2-\theta_1)/2}-e^{j(\theta_2-\theta_1)/2})...2.5[/tex]

or

[tex]K\approx -2je^{j(\theta_2-\theta_1)/2}\frac{(e^{j(\theta_2-\theta_1)/2}-e^{-j(\theta_2-\theta_1)/2}}{2j}...2.6[/tex]

which equals

[tex]K\approx -2je^{j(\theta_2-\theta_1)/2}sin((\theta_2-\theta_1)/2)...2.7[/tex]

where we have a constant (complex) amplitude which only varies in phase and has been added 90 degrees in phase due to j, in any case the substraction of the angles is here correct and I am starting to wonder if the transponate also should be considered and in that case perhaps it "stays" in the nomenator of 2.4 which would yield

[tex]K\approx -2je^{j(\theta_2-\theta_1)/2}\frac{sin((\theta_2-\theta_1)/2)}{sin((\theta_2+\theta_1)/2)}...2.8[/tex]

still, I only get half the angles and it should also read tangens.

This is the best I can do about this, I have given up so please help.

#### Attachments

Last edited by a moderator: