When linearly polarized light passes through a polarizer, the intensity of the light that emerges depends on the angle between the polarization axis of the polarizer and the polarization direction of the incident light. The relationship between the intensity of the incident light and the intensity of the transmitted light is given by Malus's law.
Malus's law states that the intensity (I) of the transmitted light through a polarizer is proportional to the square of the cosine of the angle (θ) between the polarization axis of the polarizer and the polarization direction of the incident light.
In this case, we have two polarizers with polarization axes at 10 degrees and 60 degrees, respectively. Let's calculate the fraction of irradiance that will emerge.
Step 1: Calculate the angle between the first polarizer and the incident light: θ₁ = 40° - 10° = 30°
Step 2: Calculate the fraction of intensity that emerges from the first polarizer: I₁ = cos²(θ₁)
Step 3: Calculate the angle between the second polarizer and the transmitted light from the first polarizer: θ₂ = 60° - 40° = 20°
Step 4: Calculate the fraction of intensity that emerges from the second polarizer: I₂ = cos²(θ₂)
Step 5: Calculate the total fraction of intensity that emerges: I_total = I₁ * I₂
Now let's calculate the values:
θ₁ = 30° I₁ = cos²(30°) = 0.75
θ₂ = 20° I₂ = cos²(20°) = 0.92
I_total = 0.75 * 0.92 = 0.69
Therefore, approximately 69% (0.69 fraction) of the irradiance will emerge after passing through the two polarizers.