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Chapter 38: Problem 2

The human eye is most sensitive to green light of wavelength 505 nm. Experiments have found that when people are kept in a dark room until their eyes adapt to the darkness, a \(single\) photon of green light will trigger receptor cells in the rods of the retina. (a) What is the frequency of this photon? (b) How much energy (in joules and electron volts) does it deliver to the receptor cells? (c) To appreciate what a small amount of energy this is, calculate how fast a typical bacterium of mass 9.5 \(\times\) 10\(^{-12}\) g would move if it had that much energy.

Short Answer

Expert verified
(a) 5.94 脳 10鹿鈦� Hz; (b) 3.94 脳 10鈦宦光伖 J, 2.46 eV; (c) 2.87 脳 10鈦宦� m/s.

Step by step solution

01

Convert Wavelength to Frequency

The frequency of a photon can be calculated using the speed of light equation: \( c = \lambda u \), where \( c \) is the speed of light \( (3 \times 10^8 \text{ m/s}) \), \( \lambda \) is the wavelength \( (505 \text{ nm}) \), and \( u \) is the frequency. First, convert \( \lambda \) from nm to meters: \( 505 \text{ nm} = 505 \times 10^{-9} \text{ m} \). Then rearrange the equation for frequency: \( u = \frac{c}{\lambda} \). Substitute the given values: \( u = \frac{3 \times 10^8}{505 \times 10^{-9}} = 5.94 \times 10^{14} \text{ Hz} \).
02

Calculate Energy in Joules

The energy of a photon is given by the equation \( E = h u \), where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \) and \( u \) is the frequency. Substitute the frequency calculated in Step 1: \( E = 6.626 \times 10^{-34} \times 5.94 \times 10^{14} = 3.94 \times 10^{-19} \text{ J} \).
03

Convert Energy to Electron Volts

To convert energy from joules to electron volts (eV), use the conversion factor \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \). Thus, \( E = \frac{3.94 \times 10^{-19}}{1.602 \times 10^{-19}} = 2.46 \text{ eV} \).
04

Calculate Bacterium's Speed

The kinetic energy of the bacterium can be equated to the photon energy: \( \frac{1}{2}mv^2 = E \). Convert the mass of the bacterium from grams to kilograms: \( 9.5 \times 10^{-12} \text{ g} = 9.5 \times 10^{-15} \text{ kg} \). Rearrange the equation for velocity \( v \): \( v = \sqrt{\frac{2E}{m}} \). Substitute the known values: \( v = \sqrt{\frac{2 \times 3.94 \times 10^{-19}}{9.5 \times 10^{-15}}} = 2.87 \times 10^{-2} \text{ m/s} \).

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Key Concepts

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

Photon Energy
Photon energy is a crucial concept in quantum mechanics, particularly when studying light and electromagnetic radiation. A photon is a particle of light, often described as a "packet" of energy. The energy of a single photon is directly related to its frequency, which is part of what makes wavelengths and colors of light unique.

The formula used to calculate the energy of a photon is:
  • \(E = h \cdot u\)
  • \(E\) represents the energy of the photon
  • \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34} \text{ J s}\)
  • \(u\) is the frequency of the photon
This equation highlights that even a minuscule amount of energy can be significant at a microscopic level, such as triggering receptor cells in the eye.

Moreover, understanding photon energy is vital in many fields including photonics and quantum computing, as photons are often used for transmitting information and performing computations.
Frequency Calculation
Frequency calculation is an essential step in determining the characteristics of light, which is important in both physics and engineering.

To find the frequency of light, the speed of light \(c\) is considered constant (at about \(3 \times 10^8 \text{ m/s}\)), and is related to a photon's wavelength \(\lambda\) by the equation:
  • \(c = \lambda \cdot u\)
  • \(u = \frac{c}{\lambda}\)
Here, \(u\) represents the frequency. For this calculation, make sure the wavelength is in meters, converting from nanometers by using \(1 \text{ nm} = 10^{-9} \text{ m}\).

Once the wavelength is converted, the frequency can be determined, which allows for further calculations such as finding the photon's energy.
Planck's Constant
Planck's constant is a fundamental figure in quantum physics. It serves as a bridge between the macroscopic and quantum worlds, helping to measure action in the quantum domain. Represented by the letter \(h\), Planck's constant has a value of approximately \(6.626 \times 10^{-34} \text{ J s}\).

This constant plays a critical role in calculating photon energy, where it signifies the proportionality factor between the frequency \(u\) of a photon and its energy \(E\), as seen in the equation \(E = h \cdot u\).

Planck's constant not only aids in understanding the quantization of light but also underpins the concept of quantized energy levels in atoms, providing the basis for much of modern quantum mechanics.

In essence, Planck's constant allows us to understand phenomena at a microscopic scale, which would otherwise be invisible or inexplicable with classical physics alone.

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Most popular questions from this chapter

A 2.50-W beam of light of wavelength 124 nm falls on a metal surface. You observe that the maximum kinetic energy of the ejected electrons is 4.16 eV. Assume that each photon in the beam ejects a photoelectron. (a) What is the work function (in electron volts) of this metal? (b) How many photoelectrons are ejected each second from this metal? (c) If the power of the light beam, but not its wavelength, were reduced by half, what would be the answer to part (b)? (d) If the wavelength of the beam, but not its power, were reduced by half, what would be the answer to part (b)?

What would the minimum work function for a metal have to be for visible light (380-750 nm) to eject photoelectrons?

An incident x-ray photon of wavelength 0.0900 nm is scattered in the backward direction from a free electron that is initially at rest. (a) What is the magnitude of the momentum of the scattered photon? (b) What is the kinetic energy of the electron after the photon is scattered?

(a) What is the minimum potential difference between the filament and the target of an x-ray tube if the tube is to produce x rays with a wavelength of 0.150 nm? (b) What is the shortest wavelength produced in an x-ray tube operated at 30.0 kV?

X rays with initial wavelength 0.0665 nm undergo Compton scattering. What is the longest wavelength found in the scattered x rays? At which scattering angle is this wavelength observed
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