Question

To visually examine sunspots through a telescope, astronomers have to reduce the intensity of the sunlight to avoid harming their retinas. They accomplish this intensity reduction by mounting two linear polarizers on the telescope. The second polarizer has a polarizing angle of \(110.6^{\circ}\) relative to the horizontal. If the astronomers want to reduce the intensity of the sunlight by a factor of \(0.7645,\) what polarizing angle should the first polarizer have with the horizontal? Assume that this angle is smaller than that of the second polarizer.

Short answer

Answer: The polarizing angle of the first polarizer with the horizontal is approximately 81 degrees.

Step-by-step solution

Find the angle between the polarizers

As per Malus' Law, we know that \(I = I_{0}cos^2(\theta)\). Given the desired intensity reduction factor (\(0.7645\)), we can write: \(I = 0.7645 I_{0}\) and \(0.7645 = cos^2(\theta)\) Now, we have to find the angle \(\theta\) between the first and second polarizer.

Solve for the angle between the polarizers

Now let's solve for \(\theta\): \(0.7645 = cos^2(\theta)\), so \(cos(\theta) = \pm\sqrt{0.7645}\) Since angles between \(0\) and \(180^{\circ}\) have cosine values between \(1\) and \(-1\), we just take the positive square root value: \(cos(\theta) ≈ 0.874\) To find the angle \(\theta\), we can use the inverse cosine function: \(\theta ≈ cos^{-1}(0.874) ≈ 29.6^{\circ}\)

Find the polarizing angle of the first polarizer

We are given the polarizing angle of the second polarizer as \(110.6^{\circ}\), and we have found the angle between the first polarizer and the second polarizer to be \(29.6^{\circ}\). To find the polarizing angle of the first polarizer, we can subtract the angle between the polarizers from the second polarizer's angle: \(\text{Polarizing angle of first polarizer }= 110.6^{\circ} - 29.6^{\circ}\)

Calculate the polarizing angle of the first polarizer

Now let's perform the subtraction: \(\text{Polarizing angle of first polarizer }= 110.6^{\circ} - 29.6^{\circ} ≈ 81^{\circ}\) So, the polarizing angle of the first polarizer with the horizontal should be approximately \(81^{\circ}\).

Key concepts

Malus' Law

Malus' Law plays a critical role in understanding light intensity and polarization. It states that the intensity (\(I\)) of polarized light after passing through a polarizing filter is directly proportional to the square of the cosine of the angle (\( \theta \-\theta\theta\theta\theta ) between the light's initial polarization direction and the axis of the filter.

Let's say we have an incident light beam with an initial intensity of \(I_0\). After passing through the polarizer oriented at angle \(theta\) to the light's polarization direction, the intensity gets reduced to \(I = I_0 cos^2(theta)\).

If a second polarizer is introduced into the path of the polarized light coming from the first polarizer, and it's set at an angle \(theta\) relative to the first, we can apply Malus' Law again to determine the resulting intensity. This principle is used in various applications like photography, LCD displays, and, as in our example, safely observing sunspots through a telescope.

Polarization of Light

The phenomenon of polarization of light refers to the direction in which the electric field of the light wave oscillates. Unpolarized sunlight has waves vibrating in all possible directions perpendicular to the direction of propagation. When this light passes through a polarizing filter, it emerges with waves oscillating in only one direction - it becomes polarized.

Types of Polarizers

There are several types of polarizers, but the most common are linear polarizers which allow light with a particular orientation of the electric field to pass through while blocking the other orientations. They create linearly polarized light, which has a single plane of oscillation.

When discussing sunspot observation, astronomers use linear polarizers to control the light intensity reaching the telescope. By rotating the polarizers at specific angles to each other, they can selectively reduce the brightness and prevent eye damage without losing the required detail in their observations.

Intensity Reduction

Intensity reduction is the process of decreasing the brightness of light, which is particularly important when observing the Sun to prevent damage to sensitive instruments or the human eye. Polarizers achieve intensity reduction by selectively filtering out light that does not align with their polarization axes.

By adjusting the angle between two polarizing filters, astronomers can achieve a specific reduction factor. In the textbook exercise, the goal was to reach an intensity reduction factor of \(0.7645\).

Understanding how intensity reduction works can also reveal why polarized sunglasses are effective in reducing glare. They allow for comfortable vision by blocking scattered, horizontally polarized light and only letting through light polarized in other directions. This same principle ensures astronomers can make detailed observations of the sun without risking their eyesight.