# Four polarizers are placed in succession with their axesvertical, at 30 degrees to the vertical, and at 60 degrees to thevertical, and at 90degrees to

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Four polarizers are placed in succession with their axesvertical, at 30 degrees to the vertical, and at 60 degrees to thevertical, and at 90degrees to the vertical.
A. calculate what fraction of the incident unpolarizedlight is transmitted by the four polarizers.
B. Can the transmitted light be decreased by removingone of the polarizers? If so, which one?
C. Can the transmitted light intensity be extingusishedby removing one of the polarizers? If so, which one(s)?

2021-03-12

A) The intensity of the light passing through thesuccessive polarizers is
$$I_1=(\frac{1}{2})I_o$$
$$\displaystyle{I}_{{2}}={I}_{{1}}{{\cos}^{{2}}\theta_{{2}}}$$
$$\displaystyle={\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}\theta_{{2}}}$$
$$\displaystyle{I}_{{3}}={I}_{{2}}{{\cos}^{{2}}\theta_{{3}}}$$
$$\displaystyle{\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}\theta_{{2}}}{{\cos}^{{2}}\theta_{{3}}}$$
$$\displaystyle{I}_{{4}}={I}_{{3}}{{\cos{\theta}}_{{4}}}$$
$$\displaystyle={\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}\theta_{{2}}}{{\cos}^{{2}}\theta_{{3}}}{{\cos}^{{2}}\theta_{{4}}}$$
$$\displaystyle{\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}{30}}{{\cos}^{{2}}{30}}{{\cos}^{{2}}{30}}$$
$$\displaystyle={0.21}{I}_{{o}}$$
B) If we remove the second polarizer we get
$$\displaystyle{I}_{{1}}={\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}$$
$$\displaystyle{I}_{{3}}={I}_{{1}}{{\cos}^{{2}}\theta_{{3}}}'$$
$$\displaystyle={\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}\theta_{{3}}}'$$
$$\displaystyle{I}-{4}={I}_{{3}}{{\cos}^{{2}}\theta_{{4}}}'$$
$$\displaystyle{\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}\theta_{{3}}}'{{\cos}^{{2}}\theta_{{4}}}'$$
$$\displaystyle{\left({\frac{{{1}}}{{{2}}}}\right)}{I}_{{o}}{{\cos}^{{2}}{60}}{{\cos}^{{2}}{30}}$$
$$\displaystyle={0.094}{I}_{{o}}$$
Thus we can decrease the intensity byremoving the second or third polarizer.
C) If we remove the second and thirdpolarizers we will have two polarizers with their axesperpendicular to each other and so no light willbe transmitted.