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Chapter 3: Problem 32

What is the minimum X-ray wavelength produced in bremsstrahlung by electrons that have been accelerated through \(7.5 \times 10^{4}\) V?

Short Answer

Expert verified
The minimum X-ray wavelength is \( 1.6565 \times 10^{-11} \) meters.

Step by step solution

01

Understand the relationship

Use the formula that relates the energy of the electron to the wavelength of the emitted photon. The energy of the electron after being accelerated through a voltage V is given by the equation: \[ E = eV \] where \( e \) is the charge of the electron (\( 1.6 \times 10^{-19} \text{ C} \)) and \( V \) is the voltage.
02

Calculate the energy of the electron

Calculate the energy (in joules) that the electron gains when accelerated through the given voltage using the equation from Step 1: \[ E = 1.6 \times 10^{-19} \text{ C} \times 7.5 \times 10^{4} \text{ V} \] This simplifies to: \[ E = 1.2 \times 10^{-14} \text{ J} \]
03

Use the energy-wavelength relation

Convert the energy to the wavelength using the equation from photon energy: \[ E = \frac{hc}{\text{wavelength}} \] Rearrange to solve for the wavelength: \[ \text{wavelength} = \frac{hc}{E} \] where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ J路s} \)) and \( c \) is the speed of light (\( 3.0 \times 10^{8} \text{ m/s} \)).
04

Plug in the values

Substitute the known values to find the wavelength: \[ \text{wavelength} = \frac{6.626 \times 10^{-34} \text{ J路s} \times 3.0 \times 10^{8} \text{ m/s}}{1.2 \times 10^{-14} \text{ J}} \] This simplifies to: \[ \text{wavelength} = 1.6565 \times 10^{-11} \text{ m} \]

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Key Concepts

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

Electron Acceleration
When electrons are accelerated, they gain energy due to the applied voltage. The relationship between the voltage and the energy of the electron is given by the equation:

  • \( E = eV \)
In this formula, \( e \) represents the charge of an electron and has a value of \(1.6 \times 10^{-19} \text{ C} \), while \( V \) is the voltage through which the electron is accelerated. The gained energy allows electrons to reach higher speeds, which is crucial in generating X-rays through bremsstrahlung.
Photon Energy
Photon energy is directly tied to the frequency and wavelength of the photon. The energy of a photon is given by the equation:

  • \( E = \frac{hc}{\text{wavelength}} \)
Here, \( h \) is Planck's constant, \( c \) is the speed of light, and the wavelength is the distance between successive peaks of the photon wave. In the context of X-rays, when an electron decelerates, the energy lost is emitted as X-ray photons, and this energy can be calculated using the aforementioned formula.
Planck's Constant
Planck's constant (\( h \)) is a fundamental constant in physics and plays a critical role in quantum mechanics. It has a value of \( 6.626 \times 10^{-34} \text{ J路s} \).

  • Planck's constant relates the energy of a photon to its frequency via the equation: \( E = h u \)
  • In the energy-wavelength equation, \( h \) and the speed of light \( c \) are used to determine the energy from the wavelength: \( E = \frac{hc}{\text{wavelength}} \).
This demonstrates how the microscopic world of photons and electrons adheres to precise quantitative rules, foundational to understanding X-ray production.
Speed of Light
The speed of light (\( c \)) is a constant and is approximately \( 3.0 \times 10^{8} \text{ m/s} \). It plays a significant role in the equations involving photons.

  • In the photon energy equation, it serves to connect the energy and the wavelength: \( E = \frac{hc}{\text{wavelength}} \).
  • Knowing the speed of light is essential for calculations involving electromagnetic waves, including the determination of frequencies and wavelengths in X-ray production.
This unity of constants, \( h \) and \( c \), helps bridge the gap between the wave and particle natures of light and electrons.
Voltage to Energy Conversion
Converting voltage to energy is a necessary part of understanding electron acceleration. The energy acquired by an electron accelerated through a voltage can be expressed as:

  • \( E = eV \)
Where:

  • \( e \) is the elementary charge (\( 1.6 \times 10^{-19} \text{ C} \)),
  • \( V \) is the applied voltage (e.g., \( 7.5 \times 10^{4} \text{ V} \)).
By plugging in these values, we can determine the energy in joules, equivalent to the kinetic energy imparted to the electron. This converted energy is what contributes to the emission of X-ray photons during the bremsstrahlung process.

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Most popular questions from this chapter

A hydrogen atom is moving at a speed of \(125.0 \mathrm{m} / \mathrm{s}\). It absorbs a photon of wavelength \(97 \mathrm{nm}\) that is moving in the opposite direction. By how much does the speed of the atom change as a result of absorbing the photon?

You have been hired by NASA to analyze a solar sail that uses the momentum of sunlight for interplanetary travel. A prototype sail of area \(1 \mathrm{km}^{2}\) has been developed and is made from a thin lightweight polymer that has a highly reflective aluminum coating on one side. This material has thickness \(2 \mu \mathrm{m}\) and density \(0.29 \mathrm{g} / \mathrm{cm}^{2}\). You are limited by the design which requires that the supporting frame and the cargo weigh no more than the film itself. Make any necessary assumptions about the parameters of the calculation, and estimate the travel time of this spacecraft from Earth to Mars.

You have been hired as an engineer on a NASA project to design a microwave spectrometer for an orbital mission to measure the cosmic background radiation, which has a black body spectrum with an effective temperature of \(2.725 \mathrm{K}\). (a) The spectrometer is to scan the sky between wavelengths of \(0.50 \mathrm{mm}\) and \(5.0 \mathrm{mm},\) and at each wavelength it accepts radiation in a wavelength range of \(3.0 \times 10^{-4} \mathrm{mm} .\) What maximum and minimum radiation intensity do you expect to find in this region? \((b)\) The photon detector in the spectrometer is in the form of a disk of diameter \(0.86 \mathrm{cm} .\) How many photons per second will the spectrometer record at its maximum and minimum intensities?

When sodium metal is illuminated with light of wavelength \(4.20 \times 10^{2} \mathrm{nm},\) the stopping potential is found to be \(0.65 \mathrm{V} ;\) when the wavelength is changed to \(3.10 \times 10^{2} \mathrm{nm},\) the stopping potential is \(1.69 \mathrm{V}\). Using only these data and the values of the speed of light and the electronic charge, find the work function of sodium and a value of Planck's

A photon of wavelength \(6.13 \mathrm{pm}\) scatters from a free electron at rest. After the interaction, the electron is observed to be moving in the direction of the original photon. Find the momentum of the electron.
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