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Two parallel plane-polarized beams of light of equal intesity whose oscillation planes N_(1) and N_(2) from a small angle varphi between them (Fig.) fall on a Nicol prism. To equalize the intensities of the beams emerging behind the prism, its principle direction N must be aligned along the bisecting line A or B. find the value of the angle varphi at which the rotation of the Nicol prism through a small angle delta varphi lt lt varphi from the position A results in the fractional change of intensities of the beams DeltaI//I by the value eta = 100 times exceeding that resulting due to rotation through the same angle from the position B. |
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Answer» Solution :If the principle direction `N` of the Nicol is along `A` or `B`, the intensity o flight transmitted is the same whether the LIGHT incident is one with oscillation plane `N_(1)` or one with `N_(2)`. If `N` MAKES an ANGLE `delta varphi` with `A` as shown then the fractional difference in intensity transmitted (when the light incident is `N_(1)` or `N_(2)`) is `((DeltaI)/(I))_(A) = (cos^(2)(90^(@) - (varphi)/(2)-delta varphi)-cos^(2)(90^(@) + (varphi)/(2)-delta varphi))/(cos^(2)(90^(@) -(varphi)/(2)))` `=(sin^(2)((varphi)/(2)+deltavarphi) - sin^(2)((varphi)/(2) - delta varphi))/(sin^(2)((varphi)/(2)))` `=(2sin((varphi)/(2)).2 cos((varphi)/(2))delta varphi)/(sin^(2)((varphi)/(2))) = 4cot((varphi)/(2))delta varphi` If `N` makes an angle `delta varphi(LT lt varphi)` with `B` then `((DeltaI)/(I))_(B) = (cos^(2)(varphi//2 - delta varphi) - cos^(2) (varphi//2 + delta varphi))/(cos^(@)varphi//2) = (2cos((varphi)/(2)).2sinvarphi//2 delta phi)/(cos^(@)varphi//2) = 4tanvarphi//2 delta varphi` Thus `eta = ((Delta I)/(I))_(A)//((DeltaI)/(I))_(B) = cot^(2)varphi//2` or `varphi = 2tan^(-1)(1)/(sqrt(eta))` This given `varphi = 11.4^(@)` for `eta = 100`. 
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