# Homework Help: Intensity of polarized light that has passed through two polarizing sheets

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1. Nov 8, 2014

### Kant Destroyer

1. The problem statement, all variables and given/known data
A beam of polarized light of intensity 43.0 W/m2 is sent through a system of two polarizing
sheets. Relative to the polarization direction of that incident light, the polarizing directions of the sheets
are at angles θ for the first sheet and 90 degrees for the second sheet. If the intensity of the final, transmitted light is 8.6 W/m2, what is the value of θ?

2. Relevant equations
I = I0cos2(θ)

3. The attempt at a solution
I recognized that to solve the problem I had to work backwards to solve for θ.

I used the equation I = I0cos2(θ) and found that Ifinal = I1cos2(90-θ), where I1 is the intensity of the light after passing through the first polarizing sheet. Now I'm unsure where to head with this because I have two unknown variables, I1 and (90-θ). I am also lacking another equation to use a system of equations as far as I can tell.

2. Nov 8, 2014

### haruspex

What fraction of light makes it through the first polarising sheet?

3. Nov 8, 2014

### Kant Destroyer

The intensity of the light that is between the first and second polarizing sheets can be seen in my original post as I1.

4. Nov 8, 2014

### haruspex

Sure, but how does it relate mathematically to l0?

5. Nov 8, 2014

### Kant Destroyer

The relationship between I0 and I1 is: I1 = I0cos2(θ). Are you suggesting that I substitute this in for I1 in my equation and then solve for θ? I have tried this and I am not sure how to solve for theta in this instance.

6. Nov 8, 2014

### haruspex

How is theta involved? It hasn't reached the second filter yet.

7. Nov 8, 2014

### Kant Destroyer

As described in the question, theta is the angle of orientation of the first sheet relative to the polarization direction of the light entering the sheet. Per the relevant equation, this means that theta is used to find the intensity of the light after it has traveled through the first polarizing sheet.

8. Nov 8, 2014

### haruspex

You have I0 initially, I1 between the two sheets, I2 finally.
The last two are related by $I_2 = I_1 \cos^2(\theta)$.
I'm asking how I1 relates to I0. Remember that the light between the two filters doesn't 'know' it's about to go through a second filter, so the angle theta can have nothing to do with this relationship. If it helps, mentally throw the second filter away and answer my question.

9. Nov 8, 2014

### Kant Destroyer

I think you misunderstand my original post because I failed to use another variable for the angle between the two polarizing sheets and instead used (90-θ). Theta is meant to represent the angle of orientation of the first polarizing sheet, and per the relevant equation it is used to find I1.

10. Nov 8, 2014

### haruspex

Sorry, dealing with too many threads at once. let me start again.
You have Ifinal = I1cos2(90-θ), I1 = I0cos2(θ) and the value of Ifinal / I0 . So your difficulty is in simplifying the trig, yes?
What trig formulae do you know which involve expressions like sin(θ)cos(θ)?

11. Nov 8, 2014

### Kant Destroyer

The only trig formulas I remember are sin2+cos2 = 1 and cos2(θ) = 1/2 + 1/2cos(2θ).

12. Nov 8, 2014

### haruspex

You need the one for expanding $\sin(2\theta)$, and one for $\cos(90-\theta)$. You don't know those?

13. Nov 8, 2014

### Kant Destroyer

I'm not even sure how sin(2θ) plays into the mix in this problem seeing as they are both cos2 functions, but no I do not know those. If I remember correctly the cos(90-θ) is actually just sin(θ) but I'm not sure about that.

14. Nov 8, 2014

### haruspex

Right. So making that substitution, write out the equation for Ifinal as a function of I0.

15. Nov 8, 2014

### Kant Destroyer

Ifinal/cos2(θ)sin2(θ) = I0

EDIT: Misread your response. Ifinal = I0cos2(θ)sin2(θ)

16. Nov 8, 2014

### haruspex

Right. You ought to know that $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

17. Nov 8, 2014

### Kant Destroyer

so cos2(θ)sin2(θ) = 1/4sin2(2θ) ?

18. Nov 8, 2014

### haruspex

If you mean (1/4)sin2(2θ) , yes. Now you can use the two trig equations you posted earlier to get rid of the remaining quadratic.

19. Nov 8, 2014

### Kant Destroyer

Thank you very much for taking the time to help me.

20. Nov 8, 2014

### haruspex

OK. Sorry about my earlier misdirection.