Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 86

Show that the energy \(E\) in eV of a photon is given by \(E=1.241 \times 10^{-6} \mathrm{eV} \cdot \mathrm{m} / \lambda, \quad\) where \(\lambda\) is its wavelength in meters.

Short Answer

Expert verified
The rearranged equation \(E = 1.241 × 10^{-6} eV.m/λ\) matches the given formula, confirming the relationship between the energy of a photon and its wavelength.

Step by step solution

01

Base Equation

First, start off with the original Planck's equation, which relates the energy of a photon to its frequency. Planck's equation is given by \(E=h \cdot f\), where \(E\) refers to the energy, \(h\) is Planck's constant \((6.62607015 × 10^{-34} \, \mathrm{m}^2\mathrm{kg} / \mathrm{s})\), and \(f\) is the frequency of the photon.
02

Substituting Frequency with Wavelength

In optics, the frequency \(f\) of a wave is inversely related to its wavelength \(λ\), linking them through the speed of light \(c\). This relationship is expressed as \(f=c / λ\). Substituting this into Planck's equation gives \(E = h \cdot c / λ \).
03

Converting units

Next, convert the units of the energy \(E\) in joules to electron volts. 1 electron volt is approximately \(1.602176634 × 10^{-19}\) joules. Therefore, \(E = (h \cdot c / λ) / 1.60217663 \times 10^{−19}\) eV.
04

Simplification

Substitute the known values for Planck's constant \(h = 6.62607015 × 10^{-34} \mathrm{m}^2\mathrm{kg}/\mathrm{s}\), speed of light \(c = 2.998 × 10^8 \, \mathrm{m/s}\), and 1 electron volt in joules \(1.602176634 × 10^{-19}\) into the equation. This simplifies to approximately \(E = 1.241 × 10^{-6}\) eV.m/λ. Verify this matches with the given formula.

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

A sodium lamp emits 2.0 W of radiant energy, most of which has a wavelength of about \(589 \mathrm{nm} .\) Estimate the number of photons emitted per second by the lamp.

(a) Calculate the ionization energy for \(\mathrm{He}^{+}\). (b) What is the minimum frequency of a photon capable of ionizing \(\mathrm{He}^{+}\) ?

Show that \(p=h / \lambda\) and \(E_{f}=h f\) are consistent with the relativistic formula \(E^{2}=p^{2} c^{2}+m_{0}^{2} c^{2}\).

Find the velocity and kinetic energy of a 6.0 -fm neutron. (Rest mass energy of neutron is \(E_{0}=940 \mathrm{MeV}\).)

Does the Heisenberg uncertainty principle allow a particle to be at rest in a designated region in space?
See all solutions

Recommended explanations on Physics Textbooks

Astrophysics

Read Explanation

Modern Physics

Read Explanation

Turning Points in Physics

Read Explanation

Scientific Method Physics

Read Explanation

Famous Physicists

Read Explanation

Classical Mechanics

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