Bucknell Physics 212 Supplement

1.

Calculate:

1. the energy (in eV) of a $500\mathrm{nm}$ wavelength photon, and

2. the non-relativistic kinetic energy for a $500\mathrm{nm}$ electron.

2.

A certain electron has the same wavelength as orange light, $\lambda \simeq 600\mathrm{nm}\text{.}$ Calculate the speed of this electron.

3.

The electron binding energy for a particular metal is $2.25\mathrm{eV}\text{.}$ Calculate the cutoff frequency for the photoelectric effect in this metal.

4.

1. Let's say that you want to probe the structure of a bacteriophage T4 virus.¹ One way to probe the virus is to look at it with electromagnetic waves. If one wants to “see” the T4 structure with a resolution of about $1\mathrm{nm}\text{,}$ what must be the wavelength of the EM wave used? What is the energy of a single photon with this wavelength?

2. Another way to probe the T4 is with an electron microscope. What must the wavelength and energy of an electron used by the microscope if it is to resolve the T4 virus down to the same resolution of ${10}^{-9}\mathrm{m}\text{?}$

5.

Use the de Broglie relation to calculate the momentum of an X-ray photon of frequency $f=1.0\times {10}^{18}\mathrm{Hz}\text{.}$

6.

Photons of frequency $6.0\times {10}^{14}\mathrm{Hz}$ are directed towards a metal. As a result, electrons are ejected with kinetic energies up to $1.4\mathrm{eV}\text{.}$ Determine the binding energy for this metal.

7.

A few years ago there was a flurry of attention given to the potential hazards of electromagnetic fields from overhead power lines. The concern is that the alternating current (AC) in these power lines was emitting radiation that could cause cancer.

1. The AC current in power lines alternates with a frequency of $60\mathrm{Hz}\text{.}$ Use this to determine the energy of photons emitted from the power lines (express your answer in eV).

2. The weakest molecular bonds have binding energies around $0.1\mathrm{eV}\text{.}$ Use this explain why the scientific community is highly skeptical of the claims of cancer dangers.

8.

When a particular metal is illuminated with infrared radiation of wavelength $700\mathrm{nm}\text{,}$ electrons are emitted with kinetic energies that range up to $0.25\mathrm{eV}\text{.}$ Calculate the largest kinetic energy for ejected electrons if the same surface is illuminated with light of wavelength $400\mathrm{nm}\text{.}$

9.

A one-dimensional cavity of length $L$ is filled with electromagnetic standing waves. Show that the frequency of the $j^{\text{ th }}$ longest wavelength mode is given by $f_j=cj/(2L)\text{.}$

10.

1. Given your results from Problem 2.7.9, calculate the energy of a single photon for the $j^{\text{ th }}$ longest wavelength mode in a one-dimensional (1D) cavity with length $L\text{.}$

2. Considering that the Equipartition Theorem predicts an average total energy of $\frac{1}{2}{k}_{B}T$ for each mode, calculate the number of photons that you would expect to find (on average) for the $j^{\text{ th }}$ longest wavelength mode in a 1D cavity with length $L\text{.}$

3. Based on your answer to (b), what would be the largest mode number $j$ for which you would expect to find (on average) one or more photons in the cavity?

4. In one or two sentences, explain why the photon nature of light resolves the problem of the UV catastrophe (i.e., the prediction of an infinite total energy).

11.

As discussed in the reading, the Equipartition Theorem says that, classically, each electromagnetic wave mode should have an average energy of $\frac{1}{2}{k}_{B}T\text{.}$ For simplicity, we'll assume a one-dimensional cavity with a length of $2.0\mathrm{cm}\text{.}$

1. Calculate the value of $\frac{1}{2}{k}_{B}T$ at room temperature of ${22}^{\circ }\mathrm{C}\text{.}$

2. Calculate the wavelength and frequency of the lowest frequency normal mode electromagnetic wave for this cavity.

3. Calculate the energy of one photon of light with this wavelength.

4. Calculate the number of photons that you would expect (on the average) for the lowest frequency mode in this cavity.

5. How many photons would you expect (on average) for the second lowest frequency mode in this cavity?

6. How many photons would you expect for the ${10}^{\text{ th }}$ lowest frequency mode in this cavity? The ${100}^{\text{ th }}$ lowest mode? The ${1000}^{\text{ th }}$ mode? The ${10000}^{\text{ th }}$ lowest mode?

7. Do you think that the equipartition theorem holds for all of your answers in part (f)? Why or why not?

12.

The electromagnetic field in a one-dimensional cavity is in thermal equilibrium, and the longest wavelength mode contains $4500$ photons.

1. Calculate the number of photons in the second-longest wavelength mode.

2. Calculate the number of photons in the third-longest wavelength mode.

3. Calculate $j_{\text{ max }}\text{,}$ the largest mode number.

13.

An X-ray photon of wavelength $60\mathrm{nm}$ ionizes a hydrogen atom with an electron initially in its ground state (with energy $-13.6\mathrm{eV}$). Calculate the kinetic energy of the resulting free electron.

14.

Wave-particle duality means that all the fundamental building blocks in the quantum microscopic world have both wave and particle properties. This is demonstrated in the electron double slit experiment shown in Figure 2.6.

1. Describe an aspect of the experiment that involves electrons acting as particles.

2. Describe an aspect of the experiment that involves the same electrons acting as waves.

15.

A He-Ne laser emits light of wavelength $633\mathrm{nm}\text{.}$ Calculate how many photons per second are emitted by a $5\mathrm{mW}$ He-Ne laser.