/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 114 The retina of a human eye can de... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 114

The retina of a human eye can detect light when radiant energy incident on it is at least \(4.0 \times 10^{-17} \mathrm{~J}\). For light of 600 -nm wavelength, how many photons does this correspond to?

Short Answer

Expert verified
The radiant energy corresponds to approximately 121 photons.

Step by step solution

01

Calculate the frequency

The frequency (f) of the light can be calculated using the speed of light (c), and the wavelength (位). The relationship is given as \(f = c / \lambda\). Here, c is \(3.0 \times 10^8 m/s\), and \(\lambda\) is 600 nm or \(600 \times 10^{-9} m\). So, \(f = (3.0 \times 10^8 m/s) / (600 \times 10^{-9} m) = 5.0 \times 10^{14} Hz\).
02

Calculate the energy of a single photon

The energy of a photon (E) can be calculated using Planck鈥檚 constant (h) and the frequency (f). The relationship is given as \(E = h \times f\). Here h is \(6.63 \times 10^{-34} Js\) and \(f = 5.0 \times 10^{14} Hz\). So, \(E = 6.63 \times 10^{-34} Js \times 5.0 \times 10^{14} Hz = 3.315 \times 10^{-19} J\).
03

Compute the number of photons

The number of photons (N) can be calculated by dividing the radiant energy (E_r) by the energy of a single photon (E). The relationship is given as \(N = E_r / E\). Here, \(E_r = 4.0 \times 10^{-17} J\), and \(E = 3.315 \times 10^{-19} J\). Therefore, \(N = 4.0 \times 10^{-17} J / 3.315 \times 10^{-19} J = 120.66\). As the number of photons must be an integer, it can be rounded to 121.

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Key Concepts

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

Radiant Energy
Radiant energy refers to the energy carried by light or electromagnetic waves. In the context of photons, which are the smallest units of light, radiant energy is the energy carried by a single photon or a collection of photons. When we consider light interacting with the human eye, as in the exercise, the question becomes how many photons with a certain amount of radiant energy are needed to be detectable.

For instance, the minimum amount of energy required to trigger a response in the retina, as mentioned in the exercise, is a good example of a threshold radiant energy. It's essential to understand that the energy carried by a photon is directly proportional to its frequency and inversely proportional to its wavelength.
Planck's Constant
Planck's constant, denoted as 'h', is a fundamental physical constant important in the field of quantum mechanics. It relates the energy of a photon to its frequency as per the equation \( E = h \times f \) where \( E \) is the photon energy and \( f \) is the frequency of the light.

This constant is a bridge between the macroscopic and quantum worlds, giving us insight into how energy quantization works at the smallest scales. In the given exercise, Planck's constant is pivotal for calculating the energy of an individual photon, which is a stepping stone to understanding how many such photons is required to reach the minimum radiant energy detectable by the human eye.
Frequency of Light
The frequency of light is the number of wave cycles that pass a given point per second. It's usually measured in Hertz (Hz). High-frequency light has more energy per photon than low-frequency light.

In the context of photon energy calculation, knowing the frequency allows us to determine the energy for a single photon using Planck's constant. For instance, in the exercise, we found the frequency of 600-nm wavelength light by using the relationship between speed, wavelength, and frequency. This is fundamental as the frequency is a key component in calculating the photon's energy.
Speed of Light
The speed of light in a vacuum, commonly symbolized as 'c', is a universal constant and is approximately equal to \(3.0 \times 10^8 m/s\). This speed is critical in physics and is used for various calculations, including finding the frequency of light when its wavelength is known.

In the given exercise, we use the speed of light to connect the wavelength of a photon to its frequency. This information is essential because it allows us to calculate the radiant energy of a photon, which, in turn, is used to determine how many such photons would be needed to reach a certain energy threshold like the one detected by the human eye.
Photon Number Computation
Photon number computation involves determining the number of photons present in a given amount of radiant energy. It's a critical concept when dealing with individual particle interactions with matter, like photons hitting a retinal cell.

In practical applications, such as the exercise provided, we calculate the number of photons by dividing the total radiant energy by the energy of a single photon. This calculation can be seen as counting how many energy packets (photons) of a specific size (energy) we need to match or exceed a particular energy quota. Understanding this helps to crystallize the relationship between the energy of light and its particle nature as a collection of photons.

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Most popular questions from this chapter

What is the de Broglie wavelength, in centimeters, of a 12.4-g hummingbird flying at \(1.20 \times 10^{2} \mathrm{mph} ?\) \((1 \mathrm{mile}=1.61 \mathrm{~km})\)

Define the following terms and give an example of each: transition metals, lanthanides, actinides.

In 1996 physicists created an anti-atom of hydrogen. In such an atom, which is the antimatter equivalent of an ordinary atom, the electrical charges of all the component particles are reversed. Thus, the nucleus of an anti-atom is made of an anti-proton, which has the same mass as a proton but bears a negative charge, while the electron is replaced by an anti-electron (also called positron) with the same mass as an electron, but bearing a positive charge. Would you expect the energy levels, emission spectra, and atomic orbitals of an antihydrogen atom to be different from those of a hydrogen atom? What would happen if an anti-atom of hydrogen collided with a hydrogen atom?

For each of the following pairs of hydrogen orbitals, indicate which is higher in energy: (a) \(1 s, 2 s ;\) (b) \(2 p\) \(3 p ;\) (c) \(3 d_{x y}, 3 d_{y c}\) (d) \(3 s, 3 d ;\) (e) \(4 f, 5 s\)

The UV light that is responsible for tanning the skin falls in the 320 - to 400 -nm region. Calculate the total energy (in joules) absorbed by a person exposed to this radiation for \(2.0 \mathrm{~h}\), given that there are \(2.0 \times 10^{16}\) photons hitting Earth's surface per square centimeter per second over a 80-nm (320 nm to \(400 \mathrm{nm}\) ) range and that the exposed body area is \(0.45 \mathrm{~m}^{2}\). Assume that only half of the radiation is absorbed and the other half is reflected by the body. (Hint: Use an average wavelength of \(360 \mathrm{nm}\) in calculating the energy of a photon.
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