To achieve the desired reduction of light intensity through the three Polaroids, we need to understand how polarized light behaves when passing through these filters. When two Polaroids are crossed, they block each other's light transmission. However, by strategically placing a third Polaroid in between, we can manipulate the intensity of the light that passes through.
Understanding Polarization and Malus's Law
Polarization refers to the orientation of light waves. When unpolarized light passes through a Polaroid, it becomes polarized in the direction of the Polaroid's axis. Malus's Law states that when polarized light passes through a second Polaroid, the intensity of the transmitted light (I) can be calculated using the formula:
I = I₀ * cos²(θ)
Here, I₀ is the intensity of the incoming polarized light, and θ is the angle between the light's polarization direction and the axis of the Polaroid.
Setting Up the Problem
In this scenario, we have:
- Polaroid A (initially polarizing unpolarized light)
- Polaroid B (crossed with A)
- Polaroid C (to be placed between A and B)
We want the intensity of light transmitted by Polaroid B to be 1/8th of the intensity of the unpolarized light incident on A. If we denote the initial intensity of unpolarized light as I₀, then the target intensity through Polaroid B is:
I_B = I₀ / 8
Calculating the Required Angles
First, when unpolarized light passes through Polaroid A, the intensity becomes:
I_A = I₀ / 2
Now, we need to consider the placement of Polaroid C. Let’s denote the angle between the axis of Polaroid A and Polaroid C as θ₁, and the angle between Polaroid C and Polaroid B as θ₂.
According to Malus's Law, the intensity after Polaroid C (I_C) can be calculated as:
I_C = I_A * cos²(θ₁)
Substituting I_A:
I_C = (I₀ / 2) * cos²(θ₁)
Next, the intensity after Polaroid B (I_B) becomes:
I_B = I_C * cos²(θ₂)
Substituting I_C:
I_B = [(I₀ / 2) * cos²(θ₁)] * cos²(θ₂)
Setting Up the Equation
We want I_B to equal I₀ / 8, so we set up the equation:
[(I₀ / 2) * cos²(θ₁)] * cos²(θ₂) = I₀ / 8
Dividing both sides by I₀:
(1/2) * cos²(θ₁) * cos²(θ₂) = 1/8
Multiplying both sides by 2 gives:
cos²(θ₁) * cos²(θ₂) = 1/4
Choosing the Angles
A common approach is to set θ₁ and θ₂ to be equal, which simplifies our calculations. Let’s assume:
θ₁ = θ₂ = θ
Then, the equation becomes:
cos²(θ) * cos²(θ) = 1/4
Which simplifies to:
cos⁴(θ) = 1/4
Taking the square root of both sides gives:
cos²(θ) = 1/2
Thus, we find:
cos(θ) = 1/√2
This corresponds to an angle of:
θ = 45°
Final Arrangement
To achieve the desired intensity reduction, place Polaroid C at an angle of 45° relative to the axis of Polaroid A. This configuration will allow the light intensity transmitted through Polaroid B to be reduced to 1/8th of the original intensity of the unpolarized light incident on Polaroid A.
In summary, the placement of Polaroid C at 45° between A and B effectively manipulates the polarization and intensity of the light, demonstrating the fascinating principles of optics and polarization.
