Question

An unpolarized beam of light is incident on a group of four polarizing sheets, which are arranged in such a way that the characteristic direction of each polarizing sheet makes an angle of ${30}^{\circ }$ with that of the preceding sheet. The fraction of incident unpolarized light transmitted is

• A. $\frac{27}{128}$
• B. $\frac{128}{27}$
• C. $\frac{37}{128}$
• D. $\frac{128}{37}$

Answer 1

The correct option is A $\frac{27}{128}$

Let $I_0$ be the intensity of unpolarized light, then intensity of light transmitted From $1st$ polarizing sheet $=\frac{I_0}{2}$
From $2nd$ polarizing sheet,
$I^{\prime }=\frac{I_0}{2}(\cos {30}^{\circ }{)}^2=\frac{I_0}{2}{\left(\frac{\sqrt{3}}{2}\right)}^2=\frac{I_0}{2}\left(\frac{3}{4}\right)$
From $3rd$ polarizing sheet,
$I^{\prime \prime }=I^{\prime }(\cos {30}^{\circ }{)}^2=\frac{I_0}{2}\left(\frac{3}{4}\right){\left(\frac{\sqrt{3}}{2}\right)}^3=\frac{I_0}{2}{\left(\frac{3}{4}\right)}^2$
From $4th$ polarizing sheet,
$I^{\prime \prime \prime }=I^{\prime \prime }(\cos {30}^{\circ }{)}^2=\frac{I_0}{2}{\left(\frac{3}{4}\right)}^2{\left(\frac{\sqrt{3}}{2}\right)}^2=I_0\times \frac{27}{128}$
$\therefore \frac{I^{\prime \prime \prime }}{I_0}=\frac{27}{128}$.