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Two identical imperfect polarizers are placed in the way of a natural beam of light. When the polarizers' planes are parallel, the system transmits eta = 10.0 times more than in the case of crossed planes. Find the dergee of polarization of light produced (a) by each polarizer separately, (b) by the whole system when the planes of the polarizers are parallel. |
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Answer» Solution :Let us represent the natural light as a sum of two mutually perpendicualr componets, both with intensity `I_(0)`. Suppose that each polarizer TRANSMITS a fraction `alpha_(1)` of the light with oscillation plane parallel to the principle direction of the polarizer and a fraction `alpha_(2)` with oscillation plane perpendicular to the principle direction of the polaizer. Then the intensity of light transmitted through the two polarizers is equal to `I_(||) = alpha_(1)^(2)I_(0) + alpha_(2)^(2)I_(0)` when their principle direction are parallel and `I_(_|_) = alpha_(1)alpha_(2)I_(0) + alpha_(2)alpha_(1)I_(0) = 2alpha_(1)alpha_(2)I_(0)` when they are crossed. But `(I_(_|_))/(I_(||)) = (2alpha_(1)alpha_(2))/(alpha_(1)^(2) + alpha_(2)^(2)) = (1)/(eta)` so `(alpha_(1) - alpha_(2))/(alpha_(1)+alpha_(2)) = sqrt((eta - 1)/(eta+ 1))` (b) Now the degee of polarization PRODUCED by either polarizer when used SINGLY is `P_(0) = (I_(max) - I_(min))/(I_(max) + I_(min)) = (alpha_(1) - alpha_(2))/(alpha_(1) + alpha_(2))` (ASSUMING, of course, `alpha_(1) gt alpha_(2)`) Thus `P_(0) = sqrt((eta - 1)/(eta + 1)) = sqrt((9)/(1)) = 0.905` (b) When both polarizer are used with theri principle directions parallel, the transmitted light, when analysed, has maximum intensity, `I_(max) = alpha_(1)^(2)I_(0)` and minimum intensity, `I_(min) = alpha_(2)^(2)I_(0)` so `P = (alpha_(1)^(2) - alpha_(2)^(2))/(alpha_(1)^(2) + alpha_(2)^(2)) = (alpha_(1) - alpha_(2))/(alpha_(1) + alpha_(2)).((alpha_(1) + alpha_(2))^(2))/(alpha_(1)^(2) + alpha_(2)^(2))` `= sqrt((eta - 1)/(eta + 1)) (1+ (2 alpha_(1) alpha_(2))/( alpha_(1)^(2) +alpha_(2)^(2)))` `= sqrt((eta - 1)/(eta + 1)) (1+(1)/(eta)) = (sqrt(eta^(2) - 1))/(eta) = sqrt(1-(1)/(eta^(2))) = 0.995`. |
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