Natural light falls at the Brewster angle on the surfcae of glass. Using the Fresnel equations, find
(a) the reflection coefficient,
(b) the degree of polarization of refracted light.

Explanation

From Fresnel's equations
`{:(I'_(bot)=I_(bot)(sin^(2)(theta_(1)-theta_(2)))/(sin^(2)(theta_(1)+theta_(2)))),(I'_(||)=0):}}` at Brewste's angle
`I'_(||) = 0`
`I'_(bot) = I_(bot) sin^(2)(theta_(1) - theta_(2))`
`= (1)/(2)I (sin theta_(1) cos theta_(2) - cos theta_(1) sintheta_(2))^(2)`
Now `tan theta_(1) = n, sin theta_(1) = (n)/(sqrt(n^(2) + 1))`
`cos theta_(1) = (1)/(sqrt(n^(2) + 1)), sintheta_(2) = cos theta_(1)`
`cos theta_(2) = sin theta_(1)`
`I'_(bot) = (1)/(2)I ((n^(2) - 1)/(n^(2) + 1))^(2)`
Thus reflection coefficient `= rho = (I'_(bot))/(I)`
`= (1)/(2) ((n^(2) - 1)/(n^(2) + 1))^(2) 0.074`
on putting `n= 1.5`
(b) For the refracted light
`I''_(bot) = I_(bot) - I'_(bot) = (1)/(2)I {1-((n^(2) - 1)/(n^(2) + 1))^(2)}`
`= (1)/(2)I (4n^(2))/((n^(2) + 1)^(2))`
`I'_(||) = (1)/(2)I`
at the Brewster's angle.
Thus the degree of polarization of the refraced light is
`P = (I''_(||) - I''_(bot))/(I''_(||) + I''_(bot)) = ((n^(2) + 1)^(2)-4n^(2))/((n^(2) + 1)^(2) + 4n^(2))`
`= ((n^(2) - 1)^(2))/(2(n^(2)+1)^(2) -(n^(2) -1)^(2)) = (rho)/(1-rho)`
On putting `rho = 0.074` we get `P = 0.080`.