Relative intensity of polarized and unpolarized light in incident beam

[nanobanano]

1. A beam of light is a mixture of polarized light and unpolarized light. When it is sent through a Polaroid sheet, we find that the transmitted intensity can be varied by a factor of five depending on the orientation of the Polaroid. Find the relative intensities of these two components of the incident beam.
2. Homework Equations :
unpolarized light: I' = 1/2*Io'
polarized light: I = Iocos^2(Φ)
3. The Attempt at a Solution :
The intensity of transmitted unpolarized light will always be 1/2 the incident beam's intensity since it does not depend on the orientation of the Polaroid. The greatest the intensity of transmitted light from a polarized beam is equal to the intensity of the incident beam I = Io (when the axis of transmission is parallel to the electric field component). Since the two transmitted beams can vary by a factor of five, I've set 5I' = I and then substituted the equations for each (with the polarized intensity being equal to the incident beam intensity) resulting in Io' = 10*Io.

 

[haruspex]

resulting in Io' = 10*Io.

If I understand your notation, I_o is the incoming polarised intensity, I'_o the incoming unpolarised intensity, etc.
Incoming total intensity = I_o+I'_o
Minimum outgoing total intensity = I'_o/2
Maximum outgoing total intensity = I_o+I'_o/2
With your result, that gives a ratio of 5:6, not 1:5.

 

[Delta2]

Isn't the maximum equal to 5 times the minimum? so $I_0+I'_0/2=5(I'_0/2)$ from which I get $I_0=2I'_0$ i.e ration 2:1 polarized to unpolarized.