Two polarizing sheets have their transmission axes crossed so that no light is transmitted. A third sheet is inserted so that its transmission axis makes an angle $θ$ with the transmission axis of the first sheet. If the middle polarizing sheet is rotating at an angular speed $ω$ about an axis parallel with the light beam, find an expression for the final intensity transmitted (after passing through all three sheets) as a function of time. (Original intensity was $I_{0}$ ) (unpolarized)

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$I = \frac{I_{0}}{8} \sin^{2} 2 ω t$

$I = \frac{I_{0}}{4} \sin^{2} 2 ω t$

$I = \frac{I_{0}}{4} \cos^{2} 2 ω t$

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Detailed Solution

Intensity after passing through $1^{s t}$ polaride $= \frac{I_{0}}{2}$
Let $2^{n d}$ polaride is at angle $θ$ from first polaride & $3^{r d}$ polaride is at angle 90- $θ$ from $2^{n d}$ one then intensity after passing through $2^{n d}$ polaride = $\frac{I_{0}}{2} \cos^{2} θ$
Intensity after passing through $3^{r d}$ polaride $= \frac{I_{0}}{2} \cos^{2} θ \cos^{2} (90 - θ)$

$I_{f i n a l} = \frac{I_{0}}{2} \cos^{2} θ \sin^{2} θ = \frac{I_{0}}{8} \sin^{2} 2 θ = \frac{I_{0}}{8} \sin^{2} 2 ω t$

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