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Chapter 25: Problem 15

A photon and electron have same de-Broglie wavelength. Give that \(v\) is the speed of electron and \(c\) is the velocity of light. \(E_{e}, E_{p}\) are the kinetic energy of electron and photon respectively. \(p_{e}, p_{h}\) are the momentum of electron and photon respectively. Then which of the following relation is correct? (a) \(\frac{E_{c}}{E_{p}}=\frac{v}{2 c}\) (b) \(\frac{E_{c}}{E_{p}}=\frac{2 c}{v}\) (c) \(\frac{p_{e}}{p_{h}}=\frac{c}{2 v}\) (d) \(\frac{p_{c}}{p_{h}}=\frac{2 c}{v}\)

Short Answer

Expert verified
The correct relation is \( \frac{E_e}{E_p} = \frac{v}{2c} \) (option a).

Step by step solution

01

Understanding de-Broglie Wavelength

The de-Broglie wavelength is given by \( \lambda = \frac{h}{p} \), where \( h \) is the Planck constant and \( p \) is the momentum. Since the photon and electron have the same de-Broglie wavelength, it follows that their momenta are equal, \( p_e = p_h \).
02

Expressing Momentum for Photon and Electron

The momentum of a photon \( p_h \) can be expressed as \( E_p/c \), where \( E_p \) is the energy of the photon and \( c \) is the speed of light. For an electron, the momentum \( p_e \) is given by \( m v \), where \( m \) is the mass of the electron and \( v \) is its velocity. Equating these, we have \( m v = \frac{E_p}{c} \).
03

Expressing Energies and Finding Relation

The kinetic energy of the electron is given by \( E_e = \frac{1}{2} m v^2 \). Since \( m v = \frac{E_p}{c} \), substituting the mass \( m \) from this in the kinetic energy equation, we get \( E_e = \frac{1}{2} \left(\frac{E_p}{c}\right) v \). Therefore, \( E_e = \frac{E_p v}{2c} \).
04

Comparing Options with Derived Expression

The derived expression \( E_e = \frac{E_p v}{2c} \) can be rearranged to \( \frac{E_e}{E_p} = \frac{v}{2c} \), which matches option (a). Therefore, the correct relation is given by option (a).

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Key Concepts

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

de-Broglie wavelength
The de-Broglie wavelength is a concept introduced by French physicist Louis de Broglie, which likens particles to waves. This wave-particle duality means that not only light (photons) but also particles like electrons have wavelengths associated with them. This idea is encapsulated in the formula: \[ \lambda = \frac{h}{p} \]where \( \lambda \) is the de-Broglie wavelength, \( h \) is Planck鈥檚 constant, and \( p \) represents the momentum of the particle. The beauty of this equation lies in its universality:
  • Both light particles (photons) and matter particles (electrons) can be described as having wave-like properties.
  • This wave property is evident if the de-Broglie wavelength is comparable to the dimensions of the system being observed, such as an electron moving through a crystalline solid.
  • The same de-Broglie wavelength indicates that two particles, say a photon and an electron, have the same momentum quantity.
Understanding this concept is pivotal in modern physics, especially in fields like quantum mechanics and quantum chemistry.
Momentum
Momentum, a fundamental concept in physics, denotes the quantity of motion an object possesses. It is determined by both the mass of the object and its velocity, mathematically defined as:\[ p = mv \]where \( p \) stands for momentum, \( m \) is the mass, and \( v \) is the velocity. In the context of photons, which have no rest mass, the momentum is derived differently:
  • The photon鈥檚 momentum \( p_h \) is obtained from its energy \( E_p \) using the equation \( p_h = \frac{E_p}{c} \), where \( c \) is the speed of light.
  • For electrons, the momentum \( p_e \) is calculated with \( p_e = mv \), leveraging their rest mass.
Thus, even massless particles like photons possess momentum due to their energy, which reinforces the versatility of momentum concepts across different scales.
Kinetic Energy
Kinetic energy is the energy of motion, showing how changing the speed or mass of an object affects its energetic properties. The kinetic energy \( E \) of a moving particle like an electron is expressed by:\[ E_e = \frac{1}{2} mv^2 \]where \( E_e \) is the kinetic energy of the electron, \( m \) is its mass, and \( v \) is its velocity. This formula emphasizes:
  • The direct proportion between kinetic energy and the square of velocity, highlighting how energy rises sharply with speed.
  • The correlation between kinetic energy and mass, supporting why heftier particles carry more energy at the same speed.
  • Photons, despite having no mass, still carry kinetic energy through their inherent energy property, \( E_p \), derived from their frequency \( f \) via \( E_p = hf \).
Understanding these expressions helps demystify energy transformations in physics, crucial for comprehending particle behaviors.
Photon
Photons are elementary particles, discrete packets of electromagnetic energy that display both wave-like and particle-like properties. Key characteristics of photons include:
  • They are massless, allowing them to travel at the speed of light in a vacuum \( c \approx 3 \times 10^8 \) m/s.
  • They carry energy \( E_p \) that is proportional to their frequency \( u \) through \( E_p = hu \), with \( h \) being Planck's constant.
  • Even though they lack mass, they possess momentum \( p_h \), expressed as \( p_h = \frac{E_p}{c} \).
Photons play critical roles in numerous phenomena, such as quantum tunneling and the photoelectric effect. They are integral to understanding electromagnetic radiation, forming the foundation for technologies like lasers and the transmission of quantum information.
Electron
Electrons are subatomic particles with a negative charge and a fundamental role in atoms. They orbit the atomic nucleus and are essential to chemical bonding and electricity. Significant aspects of electrons include:
  • They possess a rest mass, approximately \( 9.11 \times 10^{-31} \) kg, which is significantly smaller than other subatomic particles.
  • Their behavior is influenced by wave-particle duality, described by de-Broglie wavelength, especially evident in small-scale phenomena.
  • Electrons exhibit properties of both particles and waves, serving as key subjects of quantum mechanics where they show properties like spin and quantized energy levels.
Understanding electrons is crucial for a myriad of scientific applications and technologies, including semiconductors, superconductors, and all forms of electronic devices.

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Most popular questions from this chapter

Light of wavelength \(\lambda\) strikes a photo sensitive surface and electrons are ejected with kinetic energy \(E\). If the \(\mathrm{KE}\) is to be increased to \(2 E\), the wavelength must be changed to \(\lambda^{\prime}\) where (a) \(\lambda^{\prime}=\frac{\lambda}{2}\) (b) \(\lambda^{\prime}=2 \lambda\) (c) \(\frac{\lambda}{2}<\lambda^{\prime}<\lambda\) (d) \(\lambda^{\prime}>\lambda\)

Consider a metal exposed to light of wavelength \(600 \mathrm{~nm}\). The maximum energy of the electron doubles when light of wavelength \(400 \mathrm{~nm}\) is used. The work function in eV is \(\quad\) [NCERT Exemplar] (a) \(1.50 \mathrm{eV}\) (b) \(1.02 \mathrm{eV}\) (c) \(1.94 \mathrm{eV}\) (d) \(2.76 \mathrm{eV}\)

The work function for a certain metal is \(4.2 \mathrm{eV}\). Will this metal give photoelectric emission for incident radiation of wavelength \(330 \mathrm{~nm} ?\) (a) Yes (b) No (c) Cannot be said (d) may be yes or no

Which one of the following statements regarding photo-emission of electrons is correct? (a) Kinetic energy of electrons increases with the intensity of incident light (b) Electrons are emitted when the wavelength of the incident light is above a certain threshold wavelength (c) Photoelectric emission is instantaneous with the incidence of light (d) Photo electrons are emitted whenever a gas is irradiated with ultraviolet light

There are two sources of light each emitting with a power of \(100 \mathrm{~W}\). One emits X-rays of wavelength \(1 \mathrm{~nm}\) and the other visible light of wavelength \(500 \mathrm{~nm} .\) Find the ratio of number of photons of X-rays in the photons of visible light of the given wavelength? (a) \(1: 500\) (b) \(1: 250\) (c) \(1: 20\) (d) 100
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