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Chapter 16: Problem 13

What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity \(1.00 \mathrm{~kW} / \mathrm{m}^{2}\) to reduce the intensity to \(10.0 \mathrm{~W} / \mathrm{m}^{2}\) ?

Short Answer

Expert verified
The axis of the polarizing filter needs to make an angle of \( \theta = \cos^{-1}(0.1) \) degrees with the direction of polarized light.

Step by step solution

01

Recall Malus's Law

Malus's Law states that the intensity of polarized light after passing through a polarizing filter is given by the equation: \( I = I_0 \cos^2(\theta) \), where \( I_0 \) is the initial intensity, \( I \) is the transmitted intensity, and \( \theta \) is the angle between the light's initial polarization direction and the axis of the polarizing filter.
02

Calculate the Angle

Using Malus's Law, we set up the equation \( 10 = 1000 \cos^2(\theta) \) to find the angle \( \theta \). Dividing both sides by 1000, we get \( \cos^2(\theta) = \frac{10}{1000} = 0.01 \). Taking the square root of both sides gives us \( \cos(\theta) = \sqrt{0.01} = 0.1 \).
03

Determine the Angle \( \theta \)

To find the angle \( \theta \), we use the inverse cosine function: \( \theta = \cos^{-1}(0.1) \). Compute this using a calculator set to degree mode to find the value of \( \theta \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polarized Light
When discussing light waves, we often imagine them moving like waves on the surface of a pond鈥攚ith undulations spreading out in all directions. However, unlike waves on a pond, light waves can vibrate in multiple planes. Polarized light is light that has been filtered so that all of its waves vibrate in the same plane. Imagine shaking a rope up and down rapidly; this is similar to how polarized light moves, but instead of a rope, imagine the light waves themselves.

Why is this important? Polarized light is used in a variety of applications, from reducing glare on water and road surfaces in sunglasses to enhancing colors and contrast in photography. Most light sources, like the sun or a light bulb, emit unpolarized light that vibrates in all planes. Polarizers can be used to convert this light into polarized light, with significant applications in scientific experimentation and technology.
Polarizing Filter
A polarizing filter is a key component when manipulating light, and its purpose is to transform unpolarized light into polarized light or to dim light that is already polarized. It works by blocking certain planes of vibration while allowing others to pass through. You can think of it as slats in a fence that only allow waves oriented in a certain direction to go through.

When a beam of unpolarized light encounters a polarizing filter, only the portion of light waves vibrating in the direction parallel to the filter's axis will pass through. Any light waves that are vibrating at 90 degrees to the axis of the filter are completely blocked. This process is an excellent demonstration of wave-particle duality, as light behaves both as a wave (that can be filtered) and as a particle (that has intensity).
Light Intensity
The light intensity, often referred to as illuminance or light level, is a measure of the brightness of light as it is perceived by the human eye. It is quantified in watts per square meter (\( W/m^2 \) in the International System of Units. In the context of polarized light, intensity controls how much energy comes through a polarizing filter and can determine the brightness of the image.

According to Malus's Law, the intensity of polarized light after it passes through a polarizing filter changes depending on the angle between the direction of the incoming light and the axis of the filter. The transmitted intensity is highest when the light is aligned with the filter's axis and decreases as the angle increases. This is crucial in various technologies, such as liquid crystal displays (LCDs) and photography, where controlling the intensity of light is necessary for optimal results.

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Most popular questions from this chapter

What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?

Waves on a swimming pool propagate at \(0.750 \mathrm{m} / \mathrm{s}\). You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.0 s. How far away is the other end of the pool?

An aircraft maintenance technician walks past a tall hangar door that acts like a single slit for sound entering the hangar. Outside the door, on a line perpendicular to the opening in the door, a jet engine makes a \(600-\mathrm{Hz}\) sound. At what angle with the door will the technician observe the first minimum in sound intensity if the vertical opening is \(0.800 \mathrm{~m}\) wide and the speed of sound is \(340 \mathrm{~m} / \mathrm{s} ?\)

Show that if you have three polarizing filters, with the second at an angle of \(45^{\circ}\) to the first and the third at an angle of \(90.0^{\circ}\) to the first, the intensity of light passed by the first will be reduced to \(25.0 \%\) of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.)

Figure shows two 7.50-cm-long glass slides illuminated by pure \(589-\mathrm{nm}\) wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are \(1.00 \mathrm{~mm}\) apart?
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