We want to rotate the direction of polarization of a beam of polarized light through 90 degrees by sending the beam through one or more polarizing sheets.(a) What is the mini-mum number of sheets required? (b) What is the minimum number of sheets required if the transmitted intensity is to be more than 60% of the original intensity?
(a)If we place one sheet with its polarization direction 90⁰ to the polarized light then the transmitted intensity will be zero (since I=I₀•cos²α = I₀•cos²90⁰=0).
If we use 2 sheets, the 1st sheet rotates by angle α (0< α<90⁰) and the 2nd sheet rotate by the angle (90⁰-α)
I=I₀•cos²α•cos²(90⁰-α)= I₀•cos²α•sin²α.
Thus, we need two sheets.
(b) I=I₀•cos²ⁿ(90⁰/n)
n=2 I₂=I₀•cos⁴(90⁰/2)=0.25•I₀
n=3 I₃=I₀•cos⁶(90⁰/3)=0.422•I₀
n=4 I₄=I₀•cos⁸(90⁰/4)=0.53•I₀
n=5 I₅=I₀•cos¹⁰(90⁰/5)=0.6054•I₀
I ո ≥0.6•I₀
Thus, the minimum value n=5 sheets.
Since when can polarizing sheets rotate a linearly polarized light beam?
I don't really know. I did ln(1/100)=-4.605 and ln(cos(90)^2)=-1.605 and 4.605/1.605=2.869. The answer for part (a) should be 2 but 2.869 is closer to 3 so am I doing anything wrong?
I still don't understand part (a). I know that cos^2(90)=0 but how did you get to the answer of 2 sheets?
2 because....that's the minimum no. of angles of that can ad up to 90
To rotate the direction of polarization of a beam of polarized light through 90 degrees, we can use polarizing sheets. Each polarizing sheet acts as a polarizer, allowing only light with a specific direction of polarization to pass through.
(a) To determine the minimum number of sheets required, we need to understand how a single polarizing sheet works. When unpolarized light passes through a polarizing sheet, only the component of the light that matches the axis of polarization of the sheet can pass through, while the other component is blocked.
So, in order to rotate the direction of polarization by 90 degrees, we essentially need to pass the light through two polarizing sheets that have perpendicular axes of polarization. The first sheet will block the component of the initial polarization, and the second sheet will allow only the component perpendicular to the initial polarization to pass through.
Therefore, the minimum number of sheets required to rotate the direction of polarization by 90 degrees is two.
(b) If we want the transmitted intensity to be more than 60% of the original intensity, we need to consider Malus's Law, which states that the intensity of the transmitted light through a polarizer is given by:
I_t = I_0 * cos^2(theta)
Where I_t is the transmitted intensity, I_0 is the initial intensity, and theta is the angle between the polarization direction of the incident light and the axis of polarization of the polarizer.
To determine the minimum number of sheets required for a transmitted intensity of more than 60% of the original intensity, we can equate the transmitted intensity to 60% of the original intensity:
0.6 * I_0 = I_0 * cos^2(theta)
Canceling out I_0 from both sides, we get:
0.6 = cos^2(theta)
Taking the square root of both sides, we have:
sqrt(0.6) = cos(theta)
By calculating the inverse cosine (cos^-1) of sqrt(0.6), we can find the angle theta that satisfies this equation.
Now, since we want to rotate the polarization direction by 90 degrees, the angle between the incident light's polarization direction and the polarization axis of the first sheet should be 90 degrees. This means that the angle theta should be 0 degrees for the first sheet.
For subsequent sheets, we need to add an angle of 90 degrees each time. Therefore, to achieve a transmitted intensity of more than 60% of the original intensity, we would need at least ceil(90 degrees / theta) sheets, where ceil() denotes the ceiling function (rounding up to the nearest integer).
Note that the actual number of sheets required may be slightly higher due to losses in intensity during transmission through each sheet.
By following this approach, you can determine the minimum number of sheets required to rotate the direction of polarization and achieve the desired transmitted intensity.