Two polarizers are oriented at 55° to each other and plane-polarized light is incident on them. If only 25% of the light gets through both of them, what was the initial polarization direction of the incident light?

Verified step by step guidance

1. Start by recalling Malus's Law, which states that the intensity of light passing through a polarizer is given by: \( I = I_0 \cos^2(\theta) \), where \( I_0 \) is the initial intensity, \( \theta \) is the angle between the light's polarization direction and the polarizer's axis, and \( I \) is the transmitted intensity.
2. Since there are two polarizers, the light first passes through the first polarizer, reducing its intensity to \( I_1 = I_0 \cos^2(\theta_1) \), where \( \theta_1 \) is the angle between the initial polarization direction of the light and the first polarizer's axis.
3. The light then passes through the second polarizer, which is oriented at an angle of 55° relative to the first polarizer. The intensity after the second polarizer is given by \( I_2 = I_1 \cos^2(55°) \). Substituting \( I_1 \), we get \( I_2 = I_0 \cos^2(\theta_1) \cos^2(55°) \).
4. We are told that only 25% of the initial light intensity gets through both polarizers, so \( I_2 = 0.25 I_0 \). Substituting this into the equation, we have \( 0.25 I_0 = I_0 \cos^2(\theta_1) \cos^2(55°) \). Cancel \( I_0 \) from both sides to get \( 0.25 = \cos^2(\theta_1) \cos^2(55°) \).
5. Solve for \( \cos^2(\theta_1) \) by dividing both sides by \( \cos^2(55°) \): \( \cos^2(\theta_1) = \frac{0.25}{\cos^2(55°)} \). Finally, take the square root to find \( \cos(\theta_1) \), and use the inverse cosine function to determine \( \theta_1 \), which represents the initial polarization direction of the incident light relative to the first polarizer.

Key Concepts

Malus's Law

Malus's Law states that when polarized light passes through a polarizer, the intensity of the transmitted light is proportional to the cosine square of the angle between the light's polarization direction and the polarizer's axis. Mathematically, it is expressed as I = I0 * cos²(θ), where I0 is the initial intensity, I is the transmitted intensity, and θ is the angle between the light's polarization direction and the polarizer's axis.

Polarization of Light

Polarization of light refers to the orientation of the light waves' oscillations in a particular direction. Natural light consists of waves vibrating in multiple planes, while polarized light has waves oscillating predominantly in one direction. This property is crucial in understanding how light interacts with polarizers, as only the component of light aligned with the polarizer's axis can pass through.

Intensity Ratio in Polarizers

When light passes through multiple polarizers, the intensity of the transmitted light can be calculated using the angles between the light's polarization direction and each polarizer. For two polarizers at an angle θ to each other, the transmitted intensity can be found using the formula I = I0 * cos²(θ1) * cos²(θ2), where θ1 is the angle between the incident light and the first polarizer, and θ2 is the angle between the first and second polarizer. This relationship helps determine the initial polarization direction based on the final intensity.