What is the ratio of the intensity of light emerging from the third polarizer, I3, to the intensity of light incident on the first polarizer, I0?

Question

A beam of initially unpolarized light passes through a sequence of three ideal polarizers. The angle between the axes of the first and second polarizers is 16.9°, and the angle between the axes of the second and third polarizers is 52.9°.

 

What is the ratio of the intensity of light emerging from the third polarizer, I3, to the intensity of light incident on the first polarizer, I0?

 

I honestly am not sure how to approach this question or what equations to use. Any help would be appreciated & writing out some steps would really help!

Answer 1

Lincoln is right on!  Perhaps to add just a little context, to orient us:

 

Malus' law is the one relating, for the case of (linearly) polarized light falling on a (linear) polarizer,  the output intensity I_out from the polarizer to the input intensity I_in, as:

 

I_out = I_incos²(θ)   [NOTE:  cos²(θ) means (cos(θ))²]

 

where θ is the angle between the plane of the polarized light and the axis of the polarizer.

 

Also, as Lincoln mentioned, if unpolarized light falls on a polarizer, its intensity is simply cut in half (for reasons we do not have to go into here).

 

So, for this case, we have three polarizers, and can thus talk about three intervals of intensity of light:

 

I_o -- entering the first polarizer

I₁ -- leaving the first polarizer and entering the second

I₂ -- leaving the second polarizer and entering the third

I₃ -- leaving the third polarizer

 

We can relate these intensities by the rules above.

 

The light entering the first polarizer with intensity I_o is unpolarized.  That means, by the "1/2 rule," I₁ = I_o/2.  Also, I₁ is now polarized in the same direction as the first polarizer.

 

Thus, when the light with intensity I₁ comes to the second polarizer, the angle between the light's polarization plane and the axis of the second polarizer is just the angle between the axes of the two polarizers, as listed in the problem.  So we can relate the intensity of the light coming out of the second polarizer, I₂, to the intensity of the light coming into the second polarizer, I₁, by Malus' law:

 

I₂ = I₁cos²(16.9^o) = (I_o/2)cos²(16.9^o)

 

You can calculate a value for cos²(16.9^o)

 

We can relate I₃ to I₂ the same way:

 

I₃ = I₂cos²(52.9^o) = (I_o/2)cos²(16.9^o)cos²(52.9^o)

 

By evaluating the cosine-squared terms, you will get I₃ equals some decimal (less than 1) times I_o.  Then it is one step to writing the ratio of I₃/I_o.

 

I hope this helps!  If you want to talk more about any of the details, just let me know.

Answer 2

HINT: You should apply Malus's Law to the polarizers, keeping in mind that the intensity decreases by a factor of 2 when the unpolarized light passes through the first polarizer. To obtain the intensity after passing through the second and third polarizers, apply Malus's Law each time.