Three polarizers are placed in succession with their axes of polarization as follows: P1 vertical, P2 at angle theta to the vertical, and P3 at 90 degrees to the vertical. Unpolarized light of intensity I.0 is incident on the first polarizer. The final intensity after the passing through the 3 polarizers is measured to be (3/32)I.0
Determine the angle theta between the first polarizer P1 and second polarizer P2.
When unpolarized light passes through a polarizer, the intensity is reduced by a factor of ½. ... When polarized light of intensity I0 is incident on a polarizer, the transmitted intensity is given by I = I0cos2θ, where θ is the angle between the polarization direction of the incident light and the axis of the filter.
because each polarizer allows 1/2 cos^2Theta where theta is some angle from the vertical. So if Theta for P1 is zero, then
Itrans=1/2 Io
Then, the light getting thru P2 is.5Iocos^2Theta, and finally, the light getting thru the final P3 is .5 Io (cos^2Theta *cos^2(90-Theta)
If Io is 1.0, then
.5(cosTheta*sin(90-Theta)=3/32
cosTheta * (sin90sinTheta-cos90sinTheta)=6/32
but cos90=0 so
sinTheta*cosTheta=1/2 sin 2*theta=6/32
sin Theta= sqrt (12/32)=.612
Theta=37 deg
To determine the angle theta between the first polarizer P1 and the second polarizer P2, we can use Malus' Law, which relates the intensity of polarized light transmitted through a polarizer to the angle between the polarizer's axis of polarization and the direction of the incident light.
According to Malus' Law, the intensity (I_t) transmitted through a polarizer is given by the equation:
I_t = I_0 * cos^2(theta)
where I_0 is the initial intensity of the unpolarized light and theta is the angle between the polarizer's axis of polarization and the direction of the incident light.
In this problem, we are given that the initial intensity of the unpolarized light is I_0, and the final intensity after passing through three polarizers is (3/32)I_0. We can use Malus' Law to relate the final intensity (I_f) to the initial intensity (I_0) and the angle between the second polarizer (P2) and the incident light.
The first polarizer (P1) is vertically aligned, so the angle between its axis of polarization and the incident light is 0 degrees. The second polarizer (P2) is at an angle theta to the vertical, and the final polarizer (P3) is at 90 degrees to the vertical.
Using Malus' Law for the second polarizer (P2) and the final polarizer (P3), we can express the final intensity (I_f) as follows:
I_f = I_0 * cos^2(theta) * cos^2(90 - theta)
Since the final intensity is given as (3/32)I_0, we can set up the equation:
(3/32)I_0 = I_0 * cos^2(theta) * cos^2(90 - theta)
Simplifying the equation, we get:
(3/32) = cos^2(theta) * sin^2(theta)
To solve this equation, we need to find the angle (theta) that satisfies this relationship.
One approach is to plot a graph of the equation (3/32) = cos^2(theta) * sin^2(theta) and find the points where the equation is satisfied. However, a more straightforward approach is to use trigonometric identities to simplify the equation further.
Using the identity sin^2(theta) = 1 - cos^2(theta), we can rewrite the equation as:
(3/32) = cos^2(theta) * (1 - cos^2(theta))
Multiplying out the terms, we get:
(3/32) = cos^2(theta) - cos^4(theta)
Rearranging the equation, we have:
cos^4(theta) - cos^2(theta) + (3/32) = 0
This is a quadratic equation in terms of cos^2(theta). We can solve this quadratic equation using the quadratic formula:
cos^2(theta) = [ -b ± sqrt(b^2 - 4ac) ] / 2a
where a = 1, b = -1, and c = 3/32.
Substituting these values into the quadratic formula, we get:
cos^2(theta) = [ -(-1) ± sqrt((-1)^2 - 4(1)(3/32)) ] / (2 * 1)
cos^2(theta) = [ 1 ± sqrt(1 - 12/32) ] / 2
cos^2(theta) = [ 1 ± sqrt(20/32) ] / 2
cos^2(theta) = [ 1 ± sqrt(5/8) ] / 2
Since cos^2(theta) must be a real number between 0 and 1, we can take the positive square root and solve for cos^2(theta):
cos^2(theta) = [ 1 + sqrt(5/8) ] / 2
Now, we can solve for theta by taking the arccosine (inverse cosine) of cos^2(theta):
theta = arccos(sqrt([ 1 + sqrt(5/8) ] / 2))
Using a calculator or mathematical software, we can find the value of this expression, which will give us the angle theta between the first polarizer P1 and the second polarizer P2.