Problem 39 - Physics

Question

Suppose that the transmission axis of the first analyzer is rotated \(27^{\circ}\) relative to the transmission axis of the polarizer, and that the transmission axis of each additional analyzer is rotated \(27^{\circ}\) relative to the transmission axis of the previous one. What is the minimum number of analyzers needed for the light reaching the photocell to have an intensity that is reduced by at least a factor of 100 relative to that striking the first analyzer?

Step-by-Step Solution

Answer

At least 10 analyzers are needed.

Step 1: Understanding the Problem

The problem involves determining how many analyzers are required to reduce the light intensity by a factor of 100. We have a polarizer and multiple analyzers with each rotated at \(27^{\circ}\) relative to the previous.

Step 2: Identify Key Formula

The intensity of light passing through the analyzers can be calculated with Malus's Law: \( I = I_0 \cos^2 \theta \), where \( I_0 \) is the initial intensity and \( \theta \) is the angle between the transmission axes.

Step 3: Calculate Initial Reduction per Analyzer

Each analyzer will reduce the intensity according to Malus's Law. First, calculate the reduction factor for one analyzer: \( \cos^2(27^{\circ}) \).

Step 4: Calculate Cumulative Intensity

For \( n \) analyzers, the intensity after passing through all analyzers is given by: \[ I_n = I_0 (\cos^2(27^{\circ}))^n \]

Step 5: Set Up the Inequality for Required Reduction

Since the light needs to be reduced by a factor of 100, we must have: \[ I_n \leq \frac{I_0}{100} \] Hence, \[ (\cos^2(27^{\circ}))^n \leq \frac{1}{100} \]

Step 6: Solve for the Number of Analyzers

Determine \( n \) by solving the inequality: \[ n \geq \frac{\log(1/100)}{\log(\cos^2(27^{\circ}))} \]After substituting \( \cos(27^{\circ}) \approx 0.891 \), we calculate: \[ n \geq \frac{\log(0.01)}{\log(0.891^2)} \] Calculate \( n \) to find the minimum number of analyzers.

Step 7: Calculate and Round Up

Performing the calculation, we find: \( n \approx \frac{-2}{-0.212} \approx 9.43 \). Since \( n \) must be a whole number, round up to obtain \( n = 10 \).

Key Concepts

Polarization

Polarization is a key concept in understanding how light interacts with different materials and devices like polarizers and analyzers. In essence, polarization refers to the orientation of the oscillations of light waves in relation to the direction of the wave's travel. Normally, light waves oscillate in multiple planes, but a polarizer filters the light so that only waves oscillating in a specific plane are transmitted.
By rotating a polarizer, you change the plane in which the light waves are allowed to oscillate. This is important when multiple polarizing devices are used in tandem, such as in this exercise with analyzers, since their orientation affects the transmission and intensity of light. This leads us to the idea of intensity reduction and how light changes as it passes through each device.

Intensity Reduction

Intensity reduction is a crucial outcome of passing light through polarizing devices like analyzers. When light goes through a polarizer or analyzer, its intensity—essentially the brightness or power of the light—is diminished. This reduction occurs because only a portion of the light's oscillating wave is aligned with the polarizer's axis and is thus transmitted.
Utilizing Malus's Law, we can calculate the intensity of light after it passes through an analyzer. Malus's Law states: \( I = I_0 \cos^2 \theta \), where \( I_0 \) is the initial intensity and \( \theta \) is the angle between the light's initial polarization direction and the analyzer's axis. As the angle increases, fewer oscillations align with the analyzer, and thus, the intensity decreases, which is what the exercise aims to quantify.

Light Transmission

Light transmission refers to the passage of light through an object or series of objects. In our exercise, this involves light moving through a polarizer first and then through a series of analyzers. Each time the light passes through an analyzer, its intensity is altered based on the transmission axis of the analyzer.
One interesting aspect of this is the cumulative effect of multiple analyzers, as each successive device reduces the intensity further based on its alignment. Thus, the light's overall transmission intensity after passing through all the analyzers is a compounded effect of each analyzer's orientation, typically calculated using Malus's Law for each step.

Analyzers

Analyzers are optical devices used to examine the polarization of light. They are frequently used in conjunction with polarizers to understand and control light properties. In the context of this exercise, analyzers are used to progressively reduce light intensity by a specific amount.
Each analyzer rotates the transmitted light further, filtering it based on its own orientation angle relative to the previous device. The problem statement involves figuring out how many analyzers are needed to reduce the light intensity hitting a photocell to a specific level. Each analyzer contributes to the compounded reduction due to their unique rotation relative to the previous one. Using the mathematics of Malus's Law repeatedly allows us to predict the number of analyzers needed for the required reduction.