I am working on a question about polarization of light.
Unpolarized light passes through 9 polarizers(the polarizers are in sequence) and the angle between the first and the last polarizer is 90degrees.

What should be the angle between the axis of the polarizers to maximize the light transmitted?

And how do i find this maximum intensity?

My thought: If the axis of the first and the last is 90degrees, then rest of the 7 polarizers between 1st and last should have their axis parallel and at 45degrees, but i'm not sure because I can't prove it. I have only answered questions that says 2 or 3 polarizers but not 9 of them.
Thanks.

If we assume that all the 7 polarizers can be aligned parallel then proving that θ = 45 is not too difficult.
Let us assume that there is only one polarizer inclined at θ1 to the vertical between the first and ninth polarizer.
( since all others are //) .
Let Ao be the initial amplitude , then from the law of Malus (I think)
A1 = Ao cos θ1 which is the amplitude after the first polarizer . Similarly
A2 = A1 cos θ2 . But θ2 =( 90 - θ1). So
A2 = A1 cos θ2 = Ao cos θ1 cos( 90 – θ1) = Ao cos θ1 sin θ1
= (1 /2 ) 2 sin θ1 cos θ1 = (1 /2 ) sin (2 θ1) . Thus
A2 = (1 /2 ) sin (2 θ1) which will have a maximum value when sin 2 θ1 = 1
or 2 θ1 = 90 or θ1 = 45 deg.
The intensity I is proportional to the square of the amplitude. Thus,
I 2 = Io (cos 45)^2 = ( ½) Io.

I 2 = ( ½) Io.

However this solution is no fun as 7 polarizers are aligned. Why don’t you try finding out the intensity when each succeeding polarizer is increasingly aligned by 45 deg, which means that the axis is rotated by 8 x 45 = 360 deg?.
I think that works out to I 9 = ( 1/16 ) Io.

I think I understand what you're talking about but i stilll can't do it. I came up with this idea but it's not working yet.
- we know the angle between axis of the first and the last is 90 degrees and the initial light is unpolarized.

- After the1st polarizer, the intensity I_1 = ½Io,
- After 2nd polarizer, intensity I_2 = I_1*cos(θ)^2;
- After 3rd polarizer, intensity is I_3 = I_2*cos(θ)^2;
- After 4th polarizer, intensity is I_4 = I_3*cos(θ)^2;
And I think after the last polarizer, final intensity will be cos(90-θ)^2.
So if we had 9polarizers then the equation becomes the one below:
(½Io)(7*cos^2(θ))(cos^2(90-θ)).. I plotted this function and it gave me the 45 degrees as the angle at maximum peak but it says the intensity t that point is 0.875Io(87.5% of the initial intensity) which is not right because 50% of it was already gone after the first polarizer.
I don't know what to do now. Out of ideas.
Please help . . . .
Thanks

What should be the angle between the axis of the polarizers to maximize the light transmitted?

The question is kind of ambiguous.

Can we align the 7 polarizers as we first solved it ? In this case using 7 polarizers where one would have sufficed is meaningless.

Can we align each succeding polarizer at 45 deg to the previous as in our second attempt? In this case the total angle > 90 deg

Should the angle between each polarizer be the same? If so then
8 x theta = 90 and we cant work out max intensity as the intesity gets fixed with theta.