Question

I am stuck on a problem involving malus' law for N polarizers each rotated a small angle theta from the last. the initial light is polarized vertically and after the last polarizer, the polarization is rotated 90 deg. from original. the irradiance at that point is .9*initial irradiance. What is N? I cannot separate N completely in my solution due to exponential*trig function

Answer 1

Take ln of you equation and that should do the trick.

Remember \[\ln (a ^{n})=n\;\ln a\]

Answer 2

Thank you. The equations I am starting with are \[\Theta=\Pi/(2N)\]\[I=I_0*(\cos^2(\Theta))^N\] and \[I=.9*I_0\] I end up with \[e^N(1+\cos(2*\Theta))=1.8\] or \[e^N(\cos^2(\Theta))=.9\] If I take the ln, I will have \[N+2\ln(\cos(\Theta))=\ln(1.8)\] I am unsure what to do about the cosine portion. Even if I use exponential identities I have sums of exponentials and taking the ln wouldn't be helpful???

Answer 3

I don't know if this is a better direction but another equation I get through modification is
[e^(N+i2*Theta)+e^(N-i2*Theta)+2]/2

Answer 4

Assume N is big enough (hence θ small enough) so that you can write \[\cos \theta \approx 1- \frac{\theta ^{2}}{2}\] then \[\ln (1+\epsilon)\approx \epsilon\]I cannot think of any expression for ln (cos θ) right now, but there might be one allowing you to solve the problem exactly.

Answer 5

Using spreadsheet, it seems N=23 gives out 0.898 without approximations. So it is legitimate to assume θ small and use the approximations I suggested earlier.

Answer 6

Thank you again. This seems to be close to the answer the book gives of 24. I am reviewing some of my math methods notes and book to find a relationship. I am now looking at using complex notation for \[e ^{z}=e ^{x+iy}\]

Answer 7

N=24 yields 0,902, so both answers are correct.

Answer 8

Thank you very much. I have been unable to find any alternate methods that woould be easier.