Problems & Exercises

A LiBr molecule oscillates with a frequency of $1\text{.}7\times {\text{10}}^{\text{13}}\text{Hz.}$ (a) What is the difference in energy in eV between allowed oscillator states? (b) What is the approximate value of $n$ for a state having an energy of 1.0 eV?

A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized. (a) What is the difference in energy in joules between allowed oscillator states? (b) What is the value of $n$ for a state where the energy is 5.00 J? (c) Can the quantization be observed?

What is the longest-wavelength EM radiation that can eject a photoelectron from silver, given that the binding energy is 4.73 eV? Is this in the visible range?

What is the binding energy in eV of electrons in magnesium, if the longest-wavelength photon that can eject electrons is 337 nm?

What is the maximum kinetic energy in eV of electrons ejected from sodium metal by 450-nm EM radiation, given that the binding energy is 2.28 eV?

Violet light of wavelength 400 nm ejects electrons with a maximum kinetic energy of 0.860 eV from sodium metal. What is the binding energy of electrons to sodium metal?

What is the wavelength of EM radiation that ejects 2.00-eV electrons from calcium metal, given that the binding energy is 2.71 eV? What type of EM radiation is this?

What is the maximum velocity of electrons ejected from a material by 80-nm photons, if they are bound to the material by 4.73 eV?

A laser with a power output of 2.00 mW at a wavelength of 400 nm is projected onto calcium metal. (a) How many electrons per second are ejected? (b) What power is carried away by the electrons, given that the binding energy is 2.71 eV?

Unreasonable Results

Red light having a wavelength of 700 nm is projected onto magnesium metal to which electrons are bound by 3.68 eV. (a) Use ${\text{KE}}_e=\text{hf}–\text{BE}$ to calculate the kinetic energy of the ejected electrons. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

What is the energy in joules and eV of a photon in a radio wave from an AM station that has a 1530-kHz broadcast frequency?

Calculate the frequency in hertz of a 1.00-MeV $\gamma$-ray photon.

Do the unit conversions necessary to show that $\text{hc}=\text{1240 eV}\cdot \text{nm,}$ as stated in the text.

(a) Calculate the energy in eV of an IR photon of frequency $\text{2.00}\times {\text{10}}^{\text{13}}\text{Hz.}$ (b) How many of these photons would need to be absorbed simultaneously by a tightly bound molecule to break it apart? (c) What is the energy in eV of a $\gamma$ ray of frequency $3\text{.}\text{00}\times {\text{10}}^{\text{20}}\text{Hz?}$ (d) How many tightly bound molecules could a single such $\gamma$ ray break apart?

(a) What is the maximum energy in eV of photons produced in a CRT using a 25.0-kV accelerating potential, such as a color TV? (b) What is their frequency?

(a) What is the ratio of power outputs by two microwave ovens having frequencies of 950 and 2560 MHz, if they emit the same number of photons per second? (b) What is the ratio of photons per second if they have the same power output?

Some satellites use nuclear power. (a) If such a satellite emits a 1.00-W flux of $\gamma$ rays having an average energy of 0.500 MeV, how many are emitted per second? (b) These $\gamma$ rays affect other satellites. How far away must another satellite be to only receive one $\gamma$ ray per second per square meter?

How many x-ray photons per second are created by an x-ray tube that produces a flux of x rays having a power of 1.00 W? Assume the average energy per photon is 75.0 keV.

Assuming that 10.0% of a 100-W light bulb’s energy output is in the visible range (typical for incandescent bulbs) with an average wavelength of 580 nm, and that the photons spread out uniformly and are not absorbed by the atmosphere, how far away would you be if 500 photons per second enter the 3.00-mm diameter pupil of your eye? (This number easily stimulates the retina.)

(a) Find the momentum of a 4.00-cm-wavelength microwave photon. (b) Discuss why you expect the answer to (a) to be very small.

(a) What is the wavelength of a photon that has a momentum of $5\text{.}\text{00}\times {\text{10}}^{-\text{29}}\text{kg}\cdot \text{m/s}$? (b) Find its energy in eV.

(a) Calculate the momentum of a photon having a wavelength of $2\text{.}\text{50 μm}$. (b) Find the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and how does it compare with that of the photon?

(a) Calculate the wavelength of a photon that has the same momentum as a proton moving at 1.00% of the speed of light. (b) What is the energy of the photon in MeV? (c) What is the kinetic energy of the proton in MeV?

Take the ratio of relativistic rest energy, $E={\mathrm{\gamma mc}}^2{\text{mc}}^2$, to relativistic momentum, $p=\gamma \text{mu}$, and show that in the limit that mass approaches zero, you find $E/p=c$.

Unreasonable Results

A car feels a small force due to the light it sends out from its headlights, equal to the momentum of the light divided by the time in which it is emitted. (a) Calculate the power of each headlight, if they exert a total force of $2\text{.}\text{00}\times {\text{10}}^{-2}\text{N}$ backward on the car. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

At what velocity will an electron have a wavelength of 1.00 m?

At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = ${\text{10}}^{-\text{15}}\text{m.}$)

Find the wavelength of a proton moving at 1.00% of the speed of light.

(a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic. (b) What is the neutron’s kinetic energy in MeV?

What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?

The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light. (a) What is the proton’s wavelength? (b) What is its kinetic energy, assuming it is nonrelativistic? (c) What was the equivalent voltage through which it was accelerated?

Unreasonable Results

(a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus). (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) If the position of an electron in a membrane is measured to an accuracy of $1\text{.}\text{00 μm}$, what is the electron’s minimum uncertainty in velocity? (b) If the electron has this velocity, what is its kinetic energy in eV? (c) What are the implications of this energy, comparing it to typical molecular binding energies?

Suppose the velocity of an electron in an atom is known to an accuracy of $2\text{.}0\times {\text{10}}^3\text{m/s}$ (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?

A relatively long-lived excited state of an atom has a lifetime of 3.00 ms. What is the minimum uncertainty in its energy?

The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?

What is the approximate uncertainty in the mass of a muon, as determined from its decay lifetime?

Integrated Concepts

The 54.0-eV electron in Example 29.7 has a 0.167-nm wavelength. If such electrons are passed through a double slit and have their first maximum at an angle of $\text{25}\text{.}\mathrm{0º}$, what is the slit separation $d$?

Integrated Concepts

A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength. (a) What is the average photon energy in joules? (b) How many of these photons are required to increase the temperature of a person’s shoulder by $2\text{.}\mathrm{0º}\text{C}$, assuming the affected mass is 4.0 kg with a specific heat of $0\text{.83 kcal}\text{/kg}\cdot \text{ºC}$. Also assume no other significant heat transfer. (c) How long does this take?

Integrated Concepts

(a) Calculate the amount of microwave energy in joules needed to raise the temperature of 1.00 kg of soup from $\text{20}\text{.}\mathrm{0º}\text{C}$ to $\text{100}\text{ºC}$. (b) What is the total momentum of all the microwave photons it takes to do this? (c) Calculate the velocity of a 1.00-kg mass with the same momentum. (d) What is the kinetic energy of this mass?

Integrated Concepts

(a) What is $\gamma$ for a proton having an energy of 1.00 TeV, produced by the Fermilab accelerator? (b) Find its momentum. (c) What is the proton’s wavelength?

Integrated Concepts

(a) Calculate the velocity of electrons that form the same pattern as 450-nm light when passed through a double slit. (b) Calculate the kinetic energy of each and compare them. (c) Would either be easier to generate than the other? Explain.

Integrated Concepts

(a) What is the separation between double slits that produces a second-order minimum at $\text{45}\text{.}\mathrm{0º}$ for 650-nm light? (b) What slit separation is needed to produce the same pattern for 1.00-keV protons.

Integrated Concepts

One problem with x rays is that they are not sensed. Calculate the temperature increase of a researcher exposed in a few seconds to a nearly fatal accidental dose of x rays under the following conditions. The energy of the x-ray photons is 200 keV, and $4\text{.}\text{00}\times {\text{10}}^{\text{13}}$ of them are absorbed per kilogram of tissue, the specific heat of which is $0\text{.}\text{830 kcal/kg}\cdot \text{ºC}$. (Note that medical diagnostic x-ray machines cannot produce an intensity this great.)

Integrated Concepts

The momentum of light is exactly reversed when reflected straight back from a mirror, assuming negligible recoil of the mirror. Thus the change in momentum is twice the photon momentum. Suppose light of intensity $1\text{.}{\text{00 kW/m}}^2$ reflects from a mirror of area $2\text{.}{\text{00 m}}^2$. (a) Calculate the energy reflected in 1.00 s. (b) What is the momentum imparted to the mirror? (c) Using the most general form of Newton’s second law, what is the force on the mirror? (d) Does the assumption of no mirror recoil seem reasonable?

Critical Thinking A photon of light with a wavelength of 550.0 nm is involved in a collision with an electron. Use $6.63\times {10}^{\mathrm{-34}}{\text{m}}^2\text{kg/s}$ for Planck’s constant. (a) How much momentum does the photon have before the collision? (b) If the wavelength of the photon leaving the collision is 650.0 nm, how much momentum does the photon have? (c) Does the momentum have to be conserved in every direction? (d) Does a longer wavelength imply more or less momentum than a shorter wavelength?