/*! 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 35 X Rays from Television Screens. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 38: Problem 35

X Rays from Television Screens. Accelerating voltages in cathode-ray-tube (CRT) TVs are about 25.0 \(\mathrm{kV}\) . What are (a) the highest frequency and (b) the shortest wavelength (in nm) of the x rays that such a TV screen could produce? (c) What assumptions did you need to make? (CRT televisions contain shielding to absorb these x rays.)

Short Answer

Expert verified
The highest frequency is approximately \( 6.03 \times 10^{18} \text{ Hz} \), and the shortest wavelength is about \( 0.0498 \text{ nm} \). Assumptions include no energy losses and complete energy conversion.

Step by step solution

01

Identify the Relevant Formula

To find the highest frequency and the shortest wavelength of the X rays, we can use the formula for the energy of photons, which is related to the frequency and wavelength by the equations: \[ E = h \cdot f \] and \[ \lambda = \frac{c}{f} \] where \( E \) is the energy, \( h \) is Planck's constant, \( f \) is the frequency, \( \lambda \) is the wavelength, and \( c \) is the speed of light.
02

Calculate Maximum Photon Energy

The maximum energy of the X rays can be equated to the kinetic energy of electrons accelerated by the voltage. This is given by \[ E = e \cdot V \] where \( e \) is the charge of an electron (\(1.6 \times 10^{-19} \text{ C} \)) and \( V \) is the potential (\( 25.0 \text{ kV} = 25,000 \text{ V} \)). Therefore, \[ E = 1.6 \times 10^{-19} \times 25,000 = 4 \times 10^{-15} \text{ J} \].
03

Calculate the Highest Frequency

Using the energy formula \( E = h \times f \), solve for the frequency \( f \):\[ f = \frac{E}{h} = \frac{4 \times 10^{-15} \text{ J}}{6.626 \times 10^{-34} \text{ J}\cdot\text{s}} \approx 6.03 \times 10^{18} \text{ Hz} \].
04

Calculate the Shortest Wavelength

Substitute the frequency \( f \) found in Step 3 into the wavelength formula \( \lambda = \frac{c}{f} \):\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{6.03 \times 10^{18} \text{ Hz}} \approx 4.98 \times 10^{-11} \text{ m} \]. The shortest wavelength is approximately \( 0.0498 \text{ nm} \).
05

State the Assumptions

Assumptions include: 1) The energy from the voltage is entirely converted into photon energy without losses. 2) The electrons are solely contributing to generating photons with their maximum available energy. 3) The surrounding environment (shielding or other materials) does not absorb or scatter significant energy before photon creation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Photon Energy Formula
In physics, understanding the energy of a photon is crucial when studying electromagnetic waves like X-rays. The photon energy formula links energy to the frequency of the wave using Planck's constant. This relationship is given by the equation: \[ E = h \cdot f \] where:
  • \( E \) represents the energy of the photon,
  • \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \)), and
  • \( f \) is the frequency of the photon.
This equation shows that the energy of a photon increases with its frequency. In CRT television screens, electrons accelerated through a high voltage gain kinetic energy, which is directly translated into the energy of X-rays produced. Assumptions like no energy loss during conversion are made to simplify calculations. This allows the entire kinetic energy to be considered as photon energy.
Maximum Frequency Calculation
Once we have the energy of the photon, we can find the maximum frequency using the photo energy formula described above. To calculate this, simply rearrange the energy formula to solve for frequency \( f \):\[ f = \frac{E}{h} \]Let's break down what happens:
  • You have the total energy \( E \) of the electrons after acceleration under the potential difference of a CRT TV, calculated as \( 4 \times 10^{-15} \, \text{J} \).
  • Substituting \( E \) and Planck鈥檚 constant \( h \) into the formula allows you to calculate \( f \), the highest frequency: \( f \approx 6.03 \times 10^{18} \, \text{Hz} \).
A higher frequency corresponds to higher energy levels, meaning X-rays from these setups have relatively high energy. This fundamental calculation relies on accurately equating energy forms under ideal conditions, assuming no energy losses occur in the process.
Wavelength and Frequency Relationship
Now, to understand how frequency relates to wavelength, we use another fundamental formula:\[ \lambda = \frac{c}{f} \]Here, \( \lambda \) denotes wavelength, \( c \) the speed of light (\(3 \times 10^8 m/s\)), and \( f \) the frequency.
  • Frequency and wavelength are inversely proportional: as frequency increases, the wavelength decreases.
  • Using the maximum frequency for the X-rays from the previous calculation, \( 6.03 \times 10^{18} \, \text{Hz} \), we find the shortest wavelength: about \( 0.0498 \, \text{nm} \).
This concept is essential in understanding electromagnetic waves and provides insight, particularly for X-rays where the ability to penetrate materials often requires having a shorter wavelength. Understanding this relationship helps when designing and utilizing devices like CRT TVs, where managing wavelength and frequency can affect device functionality and safety. Always assume perfect conditions in calculations, unless otherwise specified.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How many photons per second are emitted by a \(7.50-\mathrm{mW}\) \(\mathrm{CO}_{2}\) laser that has a wavelength of 10.6\(\mu \mathrm{m} ?\)

(a) Using the Bohr model, calculate the speed of the electron in a hydrogen atom in the \(n=1,2\) and 3 levels. (b) Calculate- the orbital period in each of these levels. (c) The average lifetime of the first excited level of a hydrogen atom is \(1.0 \times 10^{-8} \mathrm{s}\) . In the Bohr model, how many orbits does an electron in the \(n=2\) level complete before reurning to the ground level?

Protons are accelerated from rest by a potential difference of 4.00 \(\mathrm{kV}\) and strike a metal target. If a proton produces one photon on impact, what is the minimum wavelength of the resulting x rays? How does your answer compare to the minimum wave-length if \(4.00-\mathrm{keV}\) electrons are used instead? Why do x-ray tubes use electrons rather than protons to produce x rays? use electrons rather than protons to produce x rays?

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the \(n=2\) level to the \(n=1\) level?

Two stars, both of which behave like ideal blackbodies, radiate the same total energy per second. The cooler one has a surface temperature \(T\) and 3.0 times the diameter of the hotter star. (a) What is the temperature of the hotter star in terms of \(T\) ? (b) What is the ratio of the peak-intensity wavelength of the hot star to the peak-intensity wavelength of the cool star?
See all solutions

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Particle Model of Matter

Read Explanation

Wave Optics

Read Explanation

Space Physics

Read Explanation

Energy Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.