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Chapter 38: Problem 1

A photon of green light has a wavelength of 520 nm. Find the photon's frequency, magnitude of momentum, and energy. Express the energy in both joules and electron volts.

Short Answer

Expert verified
Frequency: \(5.77 \times 10^{14}\) Hz, Energy: \(3.82 \times 10^{-19}\) J or 2.39 eV, Momentum: \(1.27 \times 10^{-27}\) kg鈰卪/s.

Step by step solution

01

Identify Given Values

We are given that the wavelength of green light \( \lambda = 520 \text{ nm} \). Remember to convert this to meters for calculations: \( \lambda = 520 \times 10^{-9} \text{ m} \).
02

Use the Speed of Light Formula to Find Frequency

The formula to relate speed of light \( c \), wavelength \( \lambda \), and frequency \( f \) is \( c = f \cdot \lambda \). Here, the speed of light \( c = 3.00 \times 10^8 \text{ m/s} \). Solve for frequency \( f \):\[ f = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \text{ m/s}}{520 \times 10^{-9} \text{ m}} = 5.77 \times 10^{14} \text{ Hz} \]
03

Calculate the Energy of the Photon in Joules

Use Planck's equation \( E = h \cdot f \), where Planck's constant \( h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \). Insert the frequency from Step 2:\[ E = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \times 5.77 \times 10^{14} \text{ Hz} = 3.82 \times 10^{-19} \text{ J} \]
04

Convert the Energy from Joules to Electron Volts

1 electron volt \( (\text{eV}) \) is equivalent to \( 1.60 \times 10^{-19} \text{ J} \). Convert the energy calculated in Step 3:\[ E = \frac{3.82 \times 10^{-19} \text{ J}}{1.60 \times 10^{-19} \text{ J/eV}} = 2.39 \text{ eV} \]
05

Calculate the Photon鈥檚 Momentum

Use the momentum formula \( p = \frac{h}{\lambda} \), where \( h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \). Insert the values:\[ p = \frac{6.63 \times 10^{-34} \text{ J} \cdot \text{s}}{520 \times 10^{-9} \text{ m}} = 1.27 \times 10^{-27} \text{ kg} \cdot \text{m/s} \]

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Key Concepts

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

Wavelength
In physics, wavelength is the distance between consecutive crests or troughs of a wave. For light or any electromagnetic wave moving through a vacuum, the speed is constant, and that's the speed of light, denoted as \( c \). For visible light such as green light, which this exercise deals with, the wavelength is typically measured in nanometers (nm), which is a billionth of a meter.
  • Green light has a wavelength of about 520 nm, which is 520 x 10^-9 meters when converted to standard scientific units.

  • Wavelength is inversely proportional to frequency, meaning the longer the wavelength, the lower the frequency.

A key formula involving wavelength, frequency \( f \), and the speed of light \( c \) is \( c = f \cdot \lambda \). By rearranging this formula, you can find the frequency if the wavelength is known, as was done in the original solution.
Frequency
Frequency \( f \) is a measure of how often the waves of light (or any electromagnetic radiation) pass a point in space per second. It is measured in Hertz (Hz), with one Hertz equivalent to one cycle per second.
  • For the photon of green light with a wavelength of 520 nm, the frequency was calculated using the formula \( f = \frac{c}{\lambda} \).

  • The speed of light \( c \) is approximately \( 3.00 \times 10^8 \text{ m/s} \), allowing us to calculate the frequency as \( 5.77 \times 10^{14} \text{ Hz} \).

It's important to understand that frequency is part of what defines the energy of a photon. According to Planck's equation \( E = h \cdot f \), where \( h \) is Planck's constant (\( 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \)), the energy is directly proportional to the frequency.
Momentum
Momentum is a concept typically used in contexts involving motion, mass, and speed. But for photons, which are massless particles of light, momentum is defined using a different approach. The momentum \( p \) of a photon can be calculated using the formula \( p = \frac{h}{\lambda} \), where \( h \) is Planck's constant and \( \lambda \) is the wavelength of the photon.
  • This equation shows that even though photons have no mass, they possess momentum because they have energy.

  • Thus, the greener light photons, with a wavelength of 520 nm (\( 520 \times 10^{-9} \text{ m} \)), has a momentum of \( 1.27 \times 10^{-27} \text{ kg} \cdot \text{m/s} \).

Considering the small magnitude of a photon's momentum, it illustrates why light, despite being powerful, doesn't exert any perceivable push when it hits objects in our day-to-day experiences.

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Most popular questions from this chapter

A horizontal beam of laser light of wavelength 585 nm passes through a narrow slit that has width 0.0620 mm. The intensity of the light is measured on a vertical screen that is 2.00 m from the slit. (a) What is the minimum uncertainty in the vertical component of the momentum of each photon in the beam after the photon has passed through the slit? (b) Use the result of part (a) to estimate the width of the central diffraction maximum that is observed on the screen.

An x ray with a wavelength of 0.100 nm collides with an electron that is initially at rest. The x ray's final wavelength is 0.110 nm. What is the final kinetic energy of the electron?

A photon of wavelength 4.50 pm scatters from a free electron that is initially at rest. (a) For \(\phi = 90.0^\circ\), what is the kinetic energy of the electron immediately after the collision with the photon? What is the ratio of this kinetic energy to the rest energy of the electron? (b) What is the speed of the electron immediately after the collision? (c) What is the magnitude of the momentum of the electron immediately after the collision? What is the ratio of this momentum value to the nonrelativistic expression \(mv\)?

A photon with wavelength 0.1100 nm collides with a free electron that is initially at rest. After the collision the wavelength is 0.1132 nm. (a) What is the kinetic energy of the electron after the collision? What is its speed? (b) If the electron is suddenly stopped (for example, in a solid target), all of its kinetic energy is used to create a photon. What is the wavelength of this photon?

A 75-W light source consumes 75 W of electrical power. Assume all this energy goes into emitted light of wavelength 600 nm. (a) Calculate the frequency of the emitted light. (b) How many photons per second does the source emit? (c) Are the answers to parts (a) and (b) the same? Is the frequency of the light the same thing as the number of photons emitted per second? Explain.
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