Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 41

If an electron and a proton are traveling at the same speed, which one has the shorter de Broglie wavelength?

Short Answer

Expert verified
The proton has the shorter de Broglie wavelength when both the electron and proton are traveling at the same speed. This is because the mass of a proton is greater than the mass of an electron, and the de Broglie wavelength is inversely proportional to the mass of the particle.

Step by step solution

01

Recall the de Broglie wavelength formula

The de Broglie wavelength formula is given by: \( λ = \frac{h}{p} \) where λ is the de Broglie wavelength, h is the Planck constant, and p is the momentum of the particle.
02

Recall the momentum formula

Momentum is given by: \( p = mv \) where m is the mass of the particle, and v is the velocity of the particle.
03

Combine the de Broglie wavelength formula with the momentum formula

Substitute the momentum formula into the de Broglie wavelength formula: \( λ = \frac{h}{mv} \)
04

Calculate the de Broglie wavelengths for both particles

Now, we can calculate the de Broglie wavelength for both particles by using their mass and given that they have the same velocity: For electron: \( λ_\text{electron} = \frac{h}{m_\text{electron}v} \) For proton: \( λ_\text{proton} = \frac{h}{m_\text{proton}v} \)
05

Compare the de Broglie wavelengths

We know that the mass of a proton is greater than the mass of an electron, as \( m_\text{proton} > m_\text{electron}\). Since the velocity is the same for both particles, and the Planck constant h is also the same, we can deduce that: As \(m_\text{electron} < m_\text{proton}\), \( λ_\text{electron} > λ_\text{proton} \) So, the electron has a longer de Broglie wavelength than the proton when they are travelling at the same speed. Therefore, the proton has the shorter de Broglie wavelength.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Speculate as to how the diffraction patterns of a typical crystal would be affected if \(\gamma\) -rays were used instead of \(\mathrm{X}\) - rays.

Find the de Broglie wavelength of an electron accelerated from rest in an X-ray tube in the potential difference of \(100 \mathrm{keV}\). (Rest mass energy of an electron is \(E_{0}=511 \mathrm{keV}\).

(a) For what temperature is the peak of blackbody radiation spectrum at \(400 \mathrm{nm} ?\) (b) If the temperature of a blackbody is \(800 \mathrm{K},\) at what wavelength does it radiate the most energy?

The HCl molecule oscillates at a frequency of 87.0 THz. What is the difference (in eV) between its adjacent energy levels?

Do the photons of red light produce better resolution in a microscope than blue light photons? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Work Energy and Power

Read Explanation

Linear Momentum

Read Explanation

Space Physics

Read Explanation

Oscillations

Read Explanation

Kinematics Physics

Read Explanation

Force

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.

Sign-up for free