An unpolarized beam of light is incident on a stack of ideal polarizing filters.

Question

An unpolarized beam of light is incident on a stack of ideal polarizing filters. The axis of the first filter is perpendicular to the axis of the last filter in the stack. Find the fraction by which the transmitted beam's intensity is reduced in the following three cases.
(a) Three filters are in the stack, each with its transmission axis at 48.4° relative to the preceding filter.


(b) Four filters are in the stack, each with its transmission axis at 35.8° relative to the preceding filter.


(c) Seven filters are in the stack, each with its axis at 13.3° relative to the proceeding filter.


(d) Comment on comparing the answers to parts (a), (b), and (c).

Explanation / Answer

a) initial intensity (unpolarized light) I after 1st filter, intensity becomes I/2 after 2nd filter, intensity becomes I/2 cos^2(48.4) after 3rd filter, intensity becomes I/2cos^2(48.4)cos^2(48.4) = I/2cos^4(48.4) intensity reduced I - I/2 cos^4(48.4) = I[1- cos^4(48.4)/2] the required fraction = 1 - cos^4(48.4)/2 = 0.9028 b) initial intensity (unpolarized light) I after 1st filter, intensity becomes I/2 after 2nd filter, intensity becomes I/2 cos^2(35.8) after 3rd filter, intensity becomes I/2cos^2(35.8)cos^2(35.8) = I/2cos^4(35.8) after 4th filter, intensity becomes I/2cos^2(35.8)cos^2(35.8)cos^2(35.8) = I/2cos^6(35.8) intensity reduced I - I/2 cos^6(35.8) = I[1- cos^6(35.8)/2] the required fraction = 1 - cos^6(35.8)/2 = 0.8577 c) Let's assume n filters in stack, eachwith its axis at 90.0o/(n-1) relative to the precedingfilter initial intensity (unpolarized light) I after 1st filter, intensity becomes I/2 after 2nd filter, intensity becomes I/2 cos^2(90.0o/(n-1)) after 3rd filter, intensity becomes I/2 cos^4(90.0o/(n-1)) after 4th filter, intensity becomes I/2 cos^6(90.0o/(n-1)) ... after nth filter, intensity becomes I/2cos^2(n-1)(90.0o/(n-1)) intensity reduced I - I/2 cos^2(n-1)(90.0o/(n-1)) = I[1 -cos^2(n-1)(90.0o/(n-1))/2] the required fraction = 1 - cos^2(n-1)(90.0o/(n-1))/2 now n = 7, the fraction = 1 - cos^12(13.3o)/2 = 0.6392 d) if n gets more and more, the angle gets less and less, thefraction gets less and less. The limit value is 1/2 (becausecos(very small) ˜ 1)

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