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) To rotate the direction of polarization of a beam of polarized light through 90 degrees, we need to use at least two polarizing sheets. Each sheet will rotate the polarization by 90 degrees, resulting in a total rotation of 180 degrees, which is equivalent to rotating the direction of polarization by 90 degrees.
(b) If we want the transmitted intensity to be more than 60% of the original intensity, we can achieve this by using at least three polarizing sheets. Each sheet will reduce the intensity by approximately 50%, so using three sheets will result in an overall transmission of 50% * 50% * 50% = 12.5% of the original intensity, which is lower than the desired 60% threshold. Therefore, we need to use at least four sheets to achieve a transmitted intensity of more than 60% of the original intensity.
To rotate the direction of polarization of a beam of polarized light through 90 degrees, we can use the property of polarizing filters that only allow light waves vibrating in a specific direction to pass through.
(a) To rotate the direction of polarization by 90 degrees, we need at least two polarizing sheets. The first sheet will polarize the light in a specific direction, and the second sheet will further rotate the polarization by 90 degrees.
(b) If we want the transmitted intensity to be more than 60% of the original intensity, we need to consider the transmission properties of polarizing sheets. The intensity of transmitted light through a polarizing sheet is given by Malus's law: I_transmitted = I_0 * cos^2(theta), where I_0 is the original intensity and theta is the angle between the polarization direction of the sheet and the original polarization direction.
To achieve a transmitted intensity of more than 60% (0.6 * I_0), we need to find the minimum number of polarizing sheets that will reduce the intensity by less than 40% (0.4 * I_0).
Using Malus's law, we can calculate the angle theta required to achieve this reduction in intensity:
0.4 * I_0 = I_0 * cos^2(theta)
0.4 = cos^2(theta)
cos(theta) = sqrt(0.4)
theta = arccos(sqrt(0.4))
For each polarizing sheet, the transmitted intensity will be reduced by cos^2(theta). To achieve a reduction of less than 40% for n sheets, we need:
cos^2(theta)^n > 0.4
cos(theta)^2n > 0.4
cos(theta^n) > sqrt(0.4)
Solving this equation for n will give us the minimum number of sheets required.