### Additional Problems

Determine the power intensity of radiation per unit wavelength emitted at a wavelength of 500.0 nm by a blackbody at a temperature of 10,000 K.

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

A quantum mechanical oscillator vibrates at a frequency of 250.0 THz. What is the minimum energy of radiation it can emit?

In about 5 billion years, the sun will evolve to a red giant. Assume that its surface temperature will decrease to about half its present value of 6000 K, while its present radius of $7.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\text{m}$ will increase to $1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{11}\text{m}$ (which is the current Earth-sun distance). Calculate the ratio of the total power emitted by the sun in its red giant stage to its present power.

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

Photoelectrons are ejected from a photoelectrode and are detected at a distance of 2.50 cm away from the photoelectrode. The work function of the photoelectrode is 2.71 eV and the incident radiation has a wavelength of 420 nm. How long does it take a photoelectron to travel to the detector?

If the work function of a metal is 3.2 eV, what is the maximum wavelength that a photon can have to eject a photoelectron from this metal surface?

The work function of a photoelectric surface is 2.00 eV. What is the maximum speed of the photoelectrons emitted from this surface when a 450-nm light falls on it?

A 400-nm laser beam is projected onto a calcium electrode. The power of the laser beam is 2.00 mW and the work function of calcium is 2.31 eV. (a) How many photoelectrons per second are ejected? (b) What net power is carried away by photoelectrons?

(a) Calculate the number of photoelectrons per second that are ejected from a 1.00-mm^{2} area of sodium metal by a 500-nm radiation with intensity $1.30{\text{kW/m}}^{2}$ (the intensity of sunlight above Earth’s atmosphere). (b) Given the work function of the metal as 2.28 eV, what power is carried away by these photoelectrons?

A laser with a power output of 2.00 mW at a 400-nm wavelength is used to project a beam of light onto a calcium photoelectrode. (a) How many photoelectrons leave the calcium surface per second? (b) What power is carried away by ejected photoelectrons, given that the work function of calcium is 2.31 eV? (c) Calculate the photocurrent. (d) If the photoelectrode suddenly becomes electrically insulated and the setup of two electrodes in the circuit suddenly starts to act like a 2.00-pF capacitor, how long will current flow before the capacitor voltage stops it?

The work function for barium is 2.48 eV. Find the maximum kinetic energy of the ejected photoelectrons when the barium surface is illuminated with: (a) radiation emitted by a 100-kW radio station broadcasting at 800 kHz; (b) a 633-nm laser light emitted from a powerful He-Ne laser; and (c) a 434-nm blue light emitted by a small hydrogen gas discharge tube.

(a) Calculate the wavelength of a photon that has the same momentum as a proton moving with 1% of the speed of light in a vacuum. (b) What is the energy of this photon in MeV? (c) What is the kinetic energy of the proton in MeV?

(a) Find the momentum of a 100-keV X-ray photon. (b) Find the velocity of a neutron with the same momentum. (c) What is the neutron’s kinetic energy in eV?

The momentum of light, as it is for particles, is exactly reversed when a photon is reflected straight back from a mirror, assuming negligible recoil of the mirror. The change in momentum is twice the photon’s incident momentum, as it is for the particles. Suppose that a beam of light has an intensity $1.0\text{kW}\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}{\text{m}}^{2}$ and falls on a $\mathrm{-2.0}{\text{-m}}^{2}$ area of a mirror and reflects from it. (a) Calculate the energy reflected in 1.00 s. (b) What is the momentum imparted to the mirror? (c) Use Newton’s second law to find the force on the mirror. (d) Does the assumption of no-recoil for the mirror seem reasonable?

A photon of energy 5.0 keV collides with a stationary electron and is scattered at an angle of $60\text{\xb0}.$ What is the energy acquired by the electron in the collision?

A 0.75-nm photon is scattered by a stationary electron. The speed of the electron’s recoil is $1.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\text{m/s}.$ (a) Find the wavelength shift of the photon. (b) Find the scattering angle of the photon.

Find the maximum change in X-ray wavelength that can occur due to Compton scattering. Does this change depend on the wavelength of the incident beam?

A photon of wavelength 700 nm is incident on a hydrogen atom. When this photon is absorbed, the atom becomes ionized. What is the lowest possible orbit that the electron could have occupied before being ionized?

What is the maximum kinetic energy of an electron such that a collision between the electron and a stationary hydrogen atom in its ground state is definitely elastic?

Singly ionized atomic helium ${\text{He}}^{+1}$ is a hydrogen-like ion. (a) What is its ground-state radius? (b) Calculate the energies of its four lowest energy states. (c) Repeat the calculations for the ${\text{Li}}^{2+}$ ion.

A triply ionized atom of beryllium ${\text{Be}}^{3+}$ is a hydrogen-like ion. When ${\text{Be}}^{3+}$ is in one of its excited states, its radius in this *n*th state is exactly the same as the radius of the first Bohr orbit of hydrogen. Find *n* and compute the ionization energy for this state of ${\text{Be}}^{3+}.$

In extreme-temperature environments, such as those existing in a solar corona, atoms may be ionized by undergoing collisions with other atoms. One example of such ionization in the solar corona is the presence of ${\text{C}}^{5+}$ ions, detected in the Fraunhofer spectrum. (a) By what factor do the energies of the ${\text{C}}^{5+}$ ion scale compare to the energy spectrum of a hydrogen atom? (b) What is the wavelength of the first line in the Paschen series of ${\text{C}}^{5+}$ ? (c) In what part of the spectrum are these lines located?

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

Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. Find the wavelength of such an ultracold neutron and its kinetic energy.

Find the velocity and kinetic energy of a 6.0-fm neutron. (Rest mass energy of neutron is ${E}_{0}=940\phantom{\rule{0.2em}{0ex}}\text{MeV}\text{.)}$

The spacing between crystalline planes in the NaCl crystal is 0.281 nm, as determined by X-ray diffraction with X-rays of wavelength 0.170 nm. What is the energy of neutrons in the neutron beam that produces diffraction peaks at the same locations as the peaks obtained with the X-rays?

What is the wavelength of an electron accelerated from rest in a 30.0-kV potential difference?

Calculate the velocity of a $1.0\text{-\mu}\text{m}$ electron and a potential difference used to accelerate it from rest to this velocity.

In a supercollider at CERN, protons are accelerated to velocities of 0.25*c*. What are their wavelengths at this speed? What are their kinetic energies? If a beam of protons were to gain its kinetic energy in only one pass through a potential difference, how high would this potential difference have to be? (Rest mass energy of a proton is ${E}_{0}=938\phantom{\rule{0.2em}{0ex}}\text{MeV).}$

Find the de Broglie wavelength of an electron accelerated from rest in an X-ray tube in the potential difference of 100 keV. (Rest mass energy of an electron is ${E}_{0}=511\phantom{\rule{0.2em}{0ex}}\text{keV}\text{.)}$

The cutoff wavelength for the emission of photoelectrons from a particular surface is 500 nm. Find the maximum kinetic energy of the ejected photoelectrons when the surface is illuminated with light of wavelength 450 nm.

Compare the wavelength shift of a photon scattered by a free electron to that of a photon scattered at the same angle by a free proton.

The spectrometer used to measure the wavelengths of the scattered X-rays in the Compton experiment is accurate to $5.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\text{nm}.$ What is the minimum scattering angle for which the X-rays interacting with the free electrons can be distinguished from those interacting with the atoms?

Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge $q=+Ze.$ Derive the formulas for the energy ${E}_{n}$ of the electron in *n*th orbit and the orbital radius ${r}_{n}.$

Assume that a hydrogen atom exists in the $n=2$ excited state for ${10}^{\mathrm{-8}}\text{s}$ before decaying to the ground state. How many times does the electron orbit the proton nucleus during this time? How long does it take Earth to orbit the sun this many times?

An atom can be formed when a negative muon is captured by a proton. The muon has the same charge as the electron and a mass 207 times that of the electron. Calculate the frequency of the photon emitted when this atom makes the transition from $n=2$ to the $n=1$ state. Assume that the muon is orbiting a stationary proton.