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Chapter 24: Problem 11

The human eye is most sensitive to light having a frequency of about \(5.5 \times 10^{14} \mathrm{~Hz}\), which is in the yellow-green region of the electromagnetic spectrum. How many wavelengths of this light can fit across the width of your thumb, a distance of about \(2.0 \mathrm{~cm} ?\)

Short Answer

Expert verified
About 36,697 wavelengths of the light can fit across the width of your thumb.

Step by step solution

01

Determine the Speed of Light

The speed of light in a vacuum is approximately constant at \( c = 3.00 \times 10^{8} \text{ m/s} \). This will be used to find the wavelength of the light because wavelength (\( \lambda \)) and frequency (\( f \)) are related through the speed of light: \( c = \lambda \times f \).
02

Calculate the Wavelength

Using the formula \( \lambda = \frac{c}{f} \), substitute the known values: \( \lambda = \frac{3.00 \times 10^{8} \text{ m/s}}{5.5 \times 10^{14} \text{ Hz}} \). Solving this gives:\[ \lambda = 5.45 \times 10^{-7} \text{ m} \].
03

Convert Wavelength to Centimeters

Since the wavelength is in meters and the width of the thumb is in centimeters, convert the wavelength from meters to centimeters. 1 meter is equivalent to 100 centimeters, thus:\[ \lambda = 5.45 \times 10^{-7} \text{ m} \times 100 \text{ cm/m} = 5.45 \times 10^{-5} \text{ cm}. \]
04

Calculate the Number of Wavelengths

To find out how many wavelengths fit across the thumb, divide the thumb width by the wavelength:\[ \text{Number of wavelengths} = \frac{2.0 \text{ cm}}{5.45 \times 10^{-5} \text{ cm}} \].This results in approximately 36697 wavelengths.

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Key Concepts

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

Speed of Light
The speed of light is one of the fundamental constants in physics, denoted by the symbol \(c\). It is the speed at which light travels in a vacuum. On average, light moves at \(3.00 \times 10^{8}\) meters per second. This speed is crucial because it serves as a vital link between the wavelength and frequency of light.
  • The speed of light is not just important for physics but forms the basis for technologies like GPS and telecommunications.
  • Knowing the speed of light helps in calculating different properties of light, such as wavelength, when frequency is known or vice versa.
Understanding the speed of light helps to predict how light behaves in different environments, like when passing through water or glass.
Frequency of Light
Frequency, denoted by \(f\), is a measure of how many times a wave repeats itself over a second. The SI unit for frequency is the Hertz (Hz). For light, frequency determines its color. In the visible spectrum, it is the frequency that makes light appear as different colors to the human eye.
  • Light of different frequencies will show up as different colors; for instance, lower frequencies are red, while higher ones are violet.
  • In this exercise, the frequency given is \(5.5 \times 10^{14}\) Hz, which places the light in the yellow-green region.
Frequency is a pivotal concept when dealing with the electromagnetic spectrum, as it helps us identify where a specific type of light falls.
Converting Units
Converting units is a critical skill, especially in science, where measurements may need to be in a specific form for a calculation. In our example, the wavelength calculated was initially in meters. Yet, we required it in centimeters to compare with the thumb's width.
  • The conversion between meters and centimeters is straightforward: 1 meter = 100 centimeters.
  • Converting units ensures that all values in your calculations make sense within a context, such as comparing distances.
Systematic unit conversion keeps our operations accurate and understandable, and it鈥檚 a foundational skill in not just physics, but all of science.
Electromagnetic Spectrum
The electromagnetic spectrum encompasses all types of electromagnetic radiation, from radio waves to gamma rays. Visible light, which is what we see, is just a small part of this spectrum.
  • Colors we see, like the yellow-green in this exercise, result from different wavelengths in the visible range.
  • The electromagnetic spectrum is critical for many technologies, including medical imaging and communication.
Knowing the place of a particular frequency or wavelength within the electromagnetic spectrum allows scientists and engineers to use these properties for a wide range of applications, from understanding astronomical phenomena to designing lasers.

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Most popular questions from this chapter

A lidar (laser radar) gun is an alternative to the standard radar gun that uses the Doppler effect to catch speeders. A lidar gun uses an in frared laser and emits a precisely timed series of pulses of infrared electromagnetic waves. The time for each pulse to travel to the speeding vehicle and return to the gun is measured. In one situation a lidar gun in a stationary police car observes a difference of \(1.27 \times 10^{-7} \mathrm{~s}\) in round-trip travel times for two pulses that are emitted \(0.450 \mathrm{~s}\) apart. Assuming that the speeding vehicle is approaching the police car essentially head-on, determine the speed of the vehicle.

A communications satellite is in a synchronous orbit that is \(3.6 \times 10^{7} \mathrm{~m}\) directly above the equator. The satellite is located midway between Quito, Equador, and Bel茅m, Brazil, two cities almost on the equator that are separated by a distance of \(3.5 \times 10^{6} \mathrm{~m}\). Find the time it takes for a telephone call to go by way of satellite between these cities. Ignore the curvature of the earth.

A source is radiating light waves uniformly in all directions. At a certain distance \(r\) from the source a person measures the average intensity of the waves. (a) Does the average intensity increase, decrease, or remain the same, as \(r\) increases? (b) If the magnitude of the electric field is determined from the average intensity, is the electric field the rms value or the peak value? In both cases, justify your answers. A light bulb emits light uniformly in all directions. The average emitted power is \(150.0 \mathrm{~W}\). At a distance of \(5.00 \mathrm{~m}\) from the bulb, determine (a) the average intensity of the light, (b) the rms value of the electric field, and (c) the peak value of the electric field.

The drawing shows light incident on a polarizer whose transmission axis is parallel to the \(z\) axis. The polarizer is rotated clockwise through an angle \(\alpha\) between 0 and \(90^{\circ}\). While the polarizer is being rotated, does the intensity of the transmitted light increase, decrease, or remain the same if the incident light is (a) unpolarized, (b) polarized parallel to the \(z\) axis, and (c) polarized parallel to the \(y\) axis? Provide a reason for each of your answers.

A distant galaxy emits light that has a wavelength of \(434.1 \mathrm{nm}\). On earth, the wavelength of this light is measured to be \(438.6 \mathrm{nm}\). (a) Decide whether this galaxy is approaching or receding from the earth. Give your reasoning. (b) Find the speed of the galaxy relative to the earth.
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