How Do You Solve a Polarization Problem with Multiple Half-Wave Plates?

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To solve the problem of determining the intensity ratio after two polarizers with a stack of half-wave plates in between, one must understand how half-wave plates affect polarization. Each half-wave plate rotates the polarization direction of light passing through it by twice the angle of its fast axis relative to the vertical. With ten half-wave plates, each rotated by pi/20 radians, the cumulative effect results in a gradual rotation of the light's polarization. The final intensity can be calculated using Malus's Law, which states that the intensity after passing through a polarizer is proportional to the square of the cosine of the angle between the light's polarization direction and the polarizer's axis. Thus, the problem can be approached by calculating the cumulative rotation and applying Malus's Law to find the intensity ratio.
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One problem in my homework assignment that I am having trouble with is:

Take two ideal polarizers (the first with its axis vertical and the second, horizontal) and insert between them a stack of 10 half-wave plates, the first with its fast axis rotated pi/20 rad. from the vertical, and each subsequent one rotated pi/20 rad. from the previous one. Determine the ratio of intensity after the second polarizer to that after the first polarizer.

Does anyone know how to solve this problem?
 
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What is the effect of a half-wave plate, at some angle, where the angle is with respect to the axes of the polarizer and half-wave plate?
 
Thread 'Have I solved this structural engineering equation correctly?'

Thread 'Have I solved this structural engineering equation correctly?'

Hi all, I have a structural engineering book from 1979. I am trying to follow it as best as I can. I have come to a formula that calculates the rotations in radians at the rigid joint that requires an iterative procedure. This equation comes in the form of: $$ x_i = \frac {Q_ih_i + Q_{i+1}h_{i+1}}{4K} + \frac {C}{K}x_{i-1} + \frac {C}{K}x_{i+1} $$ Where: ## Q ## is the horizontal storey shear ## h ## is the storey height ## K = (6G_i + C_i + C_{i+1}) ## ## G = \frac {I_g}{h} ## ## C...
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