Given:
- Intensity of unpolarised light, I = 32 W/m^2
- Intensity of emerging light, I' = 3 W/m^2

To Find:
- Angle between the transmission axis of the first two polarizers

Explanation:
1. When unpolarised light passes through a polarizer, it becomes polarized light with its electric field oscillating in a specific direction.
2. The intensity of polarized light passing through a polarizer can be given by the Malus' Law:
I' = I * cos^2(θ)
Where I is the intensity of incident unpolarised light and θ is the angle between the transmission axis of the polarizer and the direction of the electric field of the incident light.
3. In this case, the unpolarised light passes through three polarizers, and the transmission axis of the last polarizer is crossed with the first.
4. When the transmission axes of two polarizers are crossed, the intensity of the emerging light becomes zero, as no light can pass through.
5. Therefore, in this case, the intensity of the emerging light should be zero, but it is given as 3 W/m^2.
6. This implies that the angle between the transmission axis of the first two polarizers is such that the intensity of light passing through them is reduced by a factor of 3/32.
7. Using the Malus' Law, we can write the equation for the intensity of light passing through the first two polarizers:
I1 = I * cos^2(θ1)
I2 = I1 * cos^2(θ2)
Where θ1 is the angle between the transmission axis of the first polarizer and the direction of the electric field of the incident light, and θ2 is the angle between the transmission axis of the second polarizer and the direction of the electric field of the light after passing through the first polarizer.
8. According to the given information, I2 = 3 W/m^2. Therefore:
3 = (32 * cos^2(θ1)) * cos^2(θ2)
3/32 = cos^2(θ1) * cos^2(θ2)
Taking the square root on both sides:
√(3/32) = cos(θ1) * cos(θ2)
9. Since the angles between the transmission axes of the first two polarizers are complementary, we can write:
θ1 + θ2 = 90°
cos(θ1) = sin(θ2)
10. Substituting the value of cos(θ1) from step 8 into the above equation:
√(3/32) = sin(θ2) * cos(θ2)
(√3/√32) = sin(θ2) * cos(θ2)
(√3/√32) = (1/2) * sin(2θ2)
(√3/√32) = (1/2) * 2sin(θ2) * cos(θ2)
(√3/√32) = (√2/4) * sin(2θ2)
(√3/√32) =