Two sheets of polaroid are arranged with their polarisation axes at angles of 0 and

90 degrees relative to the horizontal. Randomly polarized light is shone so that it
passes through both filters. What fraction of the light is transmitted?
A further, third sheet of polaroid is now introduced between them with its polarisation
axis at an angle θ to the horizontal. What fraction of the original light intensity now
passes through this arrangement?
What is the maximum possible transmitted fraction?

The answer to part 1 is zero.

Sheet Polaroid has inherent losses of about 50%, even for the favored plane of polarization. Therefore I have no further comment about part 2 of your question. Their answer will be wrong. Some light will get through because the polarizer in the middle repolarizes the light.

To determine the fraction of light transmitted through the arrangement of polaroids, we need to understand how polaroid filters work.

Polaroid filters are designed to only allow light that is polarized in a specific direction to pass through while blocking light polarized in other directions. The axis of polarization refers to the direction of the filter's molecules, which act like miniature slits that only allow light waves oscillating in a specific direction to pass through.

In the first scenario, where the two polaroids have their polarizing axes at angles of 0 and 90 degrees relative to the horizontal, the randomly polarized light will only have half of its intensity transmitted. This is because the first filter will allow half of the light to pass through (polarized along its axis of 0 degrees), and the second filter will block this orientation of light (as its axis is at 90 degrees). Therefore, only half of the original light intensity is transmitted.

Now, let's consider the second scenario where a third polaroid is introduced with its polarization axis at an angle θ to the horizontal. To determine the fraction of light transmitted, we need to analyze the relative angles between the polarizing axes.

The intensity of light transmitted through two polaroids can be calculated using Malus's Law. According to this law, the transmitted intensity is given by the equation:

I_transmitted = I_0 * cos^2(θ)

Where I_transmitted is the transmitted intensity, I_0 is the initial intensity, and θ is the angle between the polarizing axes of the two filters.

In this case, since the first two polaroids have their axes at 0 and 90 degrees, the angle between them is 90 degrees. Therefore, the transmitted intensity is given by:

I_transmitted = I_0 * cos^2(90 - θ)
= I_0 * sin^2(θ)

So, the fraction of the original light intensity that passes through this arrangement is sin^2(θ).

Finally, to determine the maximum possible transmitted fraction, we need to find the value of θ that maximizes sin^2(θ). Since sin^2(θ) is a periodic function with a maximum value of 1, the maximum transmitted fraction occurs when θ is 45 degrees.

Therefore, the maximum possible transmitted fraction is 1.

To recap:
1. In the first scenario, with polaroids at angles of 0 and 90 degrees, half of the light intensity is transmitted.
2. In the second scenario, with the introduction of a third polaroid, the transmitted fraction is given by sin^2(θ).
3. The maximum possible transmitted fraction is 1 and occurs when the angle θ is 45 degrees.