35. Special Relativity

Electrons in a photoelectric-effect experiment emerge from an aluminum surface with a maximum kinetic energy of 1.30 eV. What is the wavelength of the light?

Photoelectrons are observed when a metal is illuminated by light with a wavelength less than 388 nm. What is the metal's work function?

A photoelectric-effect experiment finds a stopping potential of 1.56 V when light of 200 nm is used to illuminate the cathode. From what metal is the cathode made?

A 100 W incandescent lightbulb emits about 5 W of visible light. (The other 95 W are emitted as infrared radiation or lost as heat to the surroundings.) The average wavelength of the visible light is about 600 nm, so make the simplifying assumption that all the light has this wavelength. How many visible-light photons does the bulb emit per second?

A ruby laser emits an intense pulse of light that lasts a mere 10 ns. The light has a wavelength of 690 nm, and each pulse has an energy of 500 mJ. What is the rate of photon emission, in photons per second, during the 10 ns that the laser is 'on'?

The graph in FIGURE P38.42 was measured in a photoelectric-effect experiment. What is the work function (in eV) of the cathode?

A beam of alpha particles is incident on a target of lead. A particular alpha particle comes in 'head-on' to a particular lead nucleus and stops $6.50\times {10}^{-14}$ m away from the center of the nucleus. (This point is well outside the nucleus.) Assume that the lead nucleus, which has $82$ protons, remains at rest. The mass of the alpha particle is $6.64\times {10}^{-27}$ kg.

(a) Calculate the electrostatic potential energy at the instant that the alpha particle stops. Express your result in joules and in MeV.

(b) What initial kinetic energy (in joules and in MeV) did the alpha particle have?

(c) What was the initial speed of the alpha particle?

A $4.78$-MeV alpha particle from a $226$Ra decay makes a head-on collision with a uranium nucleus. A uranium nucleus has $92$ protons.

(a) What is the distance of closest approach of the alpha particle to the center of the nucleus? Assume that the uranium nucleus remains at rest and that the distance of closest approach is much greater than the radius of the uranium nucleus.

(b) What is the force on the alpha particle at the instant when it is at the distance of closest approach?

Through what potential difference must electrons be accelerated if they are to have:

(a) the same wavelength as an x ray of wavelength $0.220$ nm; and

(b) the same energy as the x ray in part (a)?

Calculate the de Broglie wavelength of a $5.00$-g bullet that is moving at $340$ m/s. Will the bullet exhibit wavelike properties?

An electron is moving with a speed of $8.00\times {10}^6$ m/s. What is the speed of a proton that has the same de Broglie wavelength as this electron?

An alpha particle ($m=6.64\times {10}^{-27}$ kg) emitted in the radioactive decay of uranium-$238$ has an energy of $4.20$ MeV. What is its de Broglie wavelength?

An electron has a de Broglie wavelength of $2.80\times {10}^{-10}$ m. Determine (a) the magnitude of its momentum and (b) its kinetic energy (in joules and in electron volts).

For crystal diffraction experiments (discussed in Section $39.1$), wavelengths on the order of $0.20$ nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is a photon.

(a) An electron moves with a speed of $4.70\times {10}^6$ m/s. What is its de Broglie wavelength?

(b) A proton moves with the same speed. Determine its de Broglie wavelength.