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Chapter 3: Problem 23

Calculate the energy and momentum of a photon of wavelength \(500 \mathrm{~nm}\).

Short Answer

Expert verified
The energy of the photon is \(3.98 x 10^{-19} Joules\) and the momentum of the photon is \(1.33 x 10^{-27} kg.m/s\).

Step by step solution

01

Calculating the Energy

Using the formula for energy of a photon, \(E = \frac{hc}{\lambda}\), where \(h = 6.626 x 10^{-34} Js\) is the Planck constant, \(c = 3.00 x 10^8 m/s\) is the speed of light, and \(\lambda = 500nm = 500 x 10^{-9}m\) is the wavelength. Substituting the given values, energy of the photon can be calculated.
02

Result of Energy

After calculation, \(E = \frac{(6.626 x10^{-34} Js)(3.00 x 10^8 m/s)}{500 x 10^{-9} m} = 3.98 x 10^{-19} Joules\)
03

Calculating the Momentum

Using the formula for momentum of a photon, \(p = \frac{E}{c}\), where \(E\) is the energy calculated in the previous step. Substituting \(E = 3.98 x 10^{-19} Joules\) and \(c = 3.00 x 10^8 m/s\), momentum of the photon can be calculated.
04

Result of Momentum

After calculation, \(p = \frac{3.98 x 10^{-19} Joules}{3.00 x 10^8 m/s} = 1.33 x 10^{-27} kg.m/s\)

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Key Concepts

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

Planck's constant
Understanding Planck's constant (\(h\)) is essential when delving into the quantum mechanics world, especially when working with photon energy. Named after Max Planck, who introduced quantum theory, this constant is a fundamental physical constant denoting the quantization of energy. Its value is approximately \(6.626 x 10^{-34} Js\) (joule-seconds).

Planck's constant serves as a bridge between the energy of light quanta (photons) and their frequency. The relation \( E = hu \) connects energy (E) and frequency (\(u\)), demonstrating that energy is proportional to frequency, with Planck's constant being the proportionality factor. In the context of our exercise, we use Planck's constant along with the speed of light to calculate the energy of a photon of a given wavelength.
Speed of light
The speed of light in a vacuum, denoted as \(c\), is another crucial constant in physics and cosmology. This speed is approximately \(3.00 x 10^8 m/s\) and is considered the maximum speed at which all energy, matter, and information in the universe can travel.

It is not only a speed limit but also an essential component in equations linking space and time. For our photon calculations, the speed of light is used in tandem with Planck's constant to determine the energy of the photon. Additionally, when calculating the momentum of a photon, the constant \(c\) plays a vital role as it forms part of the momentum formula for particles that have no rest mass, like photons.
Wavelength of photon
The wavelength of a photon corresponds to the distance between successive peaks of the electromagnetic wave associated with the photon. It is denoted by the symbol \(\lambda\) and typically measured in meters (m), although other units like nanometers (nm) are often used for light waves.

Photon wavelength is inversely proportional to its frequency and energy. That is, shorter wavelengths correspond to higher frequency and energy, and vice versa. In our example, a wavelength of \(500 nm\) is converted to meters (\(500nm = 500 x 10^{-9}m\)) to calculate the photon's energy using Planck's constant and the speed of light.
Momentum of a photon
Despite having no mass, photons have momentum, a somewhat counterintuitive concept resulting from the wave-particle duality of light. A photon's momentum (\(p\)) can be calculated using the previously determined energy (\(E\)) and the speed of light (\(c\)), through the formula \(p = \frac{E}{c}\).

The momentum of a photon is directly proportional to its energy and inversely proportional to the speed of light. This relationship illustrates that even massless particles like photons can exert pressure, known as radiation pressure, which is due to their momentum. The exercise shows that a photon with a wavelength of \(500 nm\) carries a specific momentum, a critical factor in applications like photon pressure on solar sails in space engineering.

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Most popular questions from this chapter

Light of wavelength \(500 \mathrm{~nm}\) is incident on a metallic surface. If the stopping potential for the photoelectric effect is \(0.45 \mathrm{~V}\), find (a) the maximum energy of the emitted electrons, (b) the work function, and (c) the cutoff wavelength.

An electron initially at rest recoils from a head-on collision with a photon. Show that the kinetic energy acquired by the electron is \(2 h f \alpha /(1+2 \alpha)\), where \(\alpha\) is the ratio of the photon's initial energy to the rest energy of the electron.

The work function for potassium is \(2.24 \mathrm{eV}\). If potassium metal is illuminated with light of wavelength \(350 \mathrm{~nm}\), find (a) the maximum kinetic energy of the photoelectrons and (b) the cutoff wavelength.

Planck's fundamental constant, \(h\). Planck ultimately realized the great and fundamental importance of \(h\), which, much more than a curve-fitting parameter, is actually the measure of all quantum phenomena. In fact, Planck suggested using the universal constants \(h, c\) (the velocity of light), and \(G\) (Newton's gravitational constant) to construct "natural" or universal units of length, time, and mass. He reasoned that the current units of length, time, and mass were based on the accidental size, motion, and mass of our particular planet, but that truly universal units should be based on the quantum theory, the speed of light in a vacuum, and the law of gravitation - all of which hold anywhere in the universe and at all times. Show that the expressions \(\left(\frac{h G}{c^{3}}\right)^{1 / 2},\left(\frac{h G}{c^{5}}\right)^{1 / 2}\), and \(\left(\frac{h c}{G}\right)^{1 / 2}\) have dimensions of length, time, and mass and find their numerical values. These quantities are called, respectively, the Planck length, the Planck time, and the Planck mass. Would you care to speculate on the physical meaning of these quantities?

X-radiation from a molybdenum target \((0.626 \AA)\) is incident on a crystal with adjacent atomic planes spaced \(4.00 \times 10^{-10} \mathrm{~m}\) apart. Find the three smallest angles at which intensity maxima occur in the diffracted beam.
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