|| INT What is the total energy, in MeV, of

1. A proton traveling at 99% of the speed of light?

2. An electron traveling at 99% of the speed of light?

Hint: This problem uses relativity.

|| INT What is the velocity, as a fraction of c, of

1. A proton with 500 GeV total energy?

2. An electron with 2.0 GeV total energy?

Hint: This problem uses relativity.

||| INT The Large Hadron Collider accelerates two beams of protons, which travel around the collider in opposite directions, to a total energy of 6.5 TeV per proton. $(1\text{ TeV}=1\text{ teraelectron volt }={10}^{12}\text{ eV}\text{.})$ The beams cross at several points, and a few protons undergo head-on collisions. Such collisions usually produce many subatomic particles, but in principle the colliding protons could produce a single subatomic particle at rest. (It would be unstable and would almost instantly decay into other subatomic particles.) What would be the mass, as a multiple of the proton’s mass, of such a particle?

Hint: This problem uses relativity.

|| INT The Large Hadron Collider accelerates protons to a total energy of 6.5 TeV per proton. $(1\text{TeV}=1\text{teraelectron volt}={10}^{12}\text{eV}\text{.})$ What is the speed of a proton that has this energy? Give your answer as a fraction of c, using as many significant figures as needed.

Hint: This problem uses relativity.

| INT You learned in Chapter 36 that mass has an equivalent amount of energy. What are the energy equivalents in MeV of the rest masses of an electron and a proton?

|| INT The factor $\gamma$ appears in many relativistic expressions. A value $\gamma =1.01$ implies that relativity changes the Newtonian values by approximately 1% and that relativistic effects can no longer be ignored. At what kinetic energy, in MeV, is $\gamma =1.01$ for (a) an electron, (b) a proton, and (c) an alpha particle?

| INT The fission process $\text{n}+{}^{235}\text{U}\to {}^{236}\text{U}\to {}^{144}\text{Ba}+{}^{89}\text{Kr}+3\text{n}$ converts 0.185 u of mass into the kinetic energy of the fission products. What is the total kinetic energy in MeV?

|| INT An electron in a cathode-ray beam passes between 2.5-cm-long parallel-plate electrodes that are 5.0 mm apart. A 1.0 mT, 2.5-cm-wide magnetic field is perpendicular to the electric field between the plates. If the potential difference between the plates is 150 V, the electron passes through the electrodes without being deflected. If the potential difference across the plates is set to zero, through what angle is the electron deflected as it passes through the magnetic field?

| INT The two 5.0-cm-long parallel electrodes in FIGURE P37.34 are spaced 1.0 cm apart. A proton enters the plates from one end, an equal distance from both electrodes. A potential difference $\text{Δ}V=500\text{V}$ across the electrodes deflects the proton so that it strikes the outer end of the lower electrode. What magnetic field strength and direction will allow the proton to pass through undeflected while the 500 V potential difference is applied? Assume that both the electric and magnetic fields are confined to the space between the electrodes.

FIGURE P37.34

|| INT An unknown charged particle passes without deflection through crossed electric and magnetic fields of strengths $187\text{,}500\text{V}\text{/}\text{m}$ and 0.1250 T, respectively. The particle passes out of the electric field, but the magnetic field continues, and the particle makes a semicircle of diameter 25.05 cm. What is the particle’s charge-to-mass ratio? Can you identify the particle?

|| INT In one of Thomson’s experiments he placed a thin metal foil in the electron beam and measured its temperature rise. Consider a cathode-ray tube in which electrons are accelerated through a 2000 V potential difference, then strike a 10 mg copper foil. What is the electron-beam current if the foil temperature rises 6.$0\text{°C}$ in 10 s? Assume no loss of energy by radiation or other means. The specific heat of copper is $385\text{J}\text{/}\text{k}\text{g}\text{K}$.

|| INT A neutral lithium atom has three electrons. As you will discover in Chapter 41, two of these electrons form an “inner core,” but the third—the valence electron—orbits at much larger radius. From the valence electron’s perspective, it is orbiting a spherical ball of charge having net charge $+1e$ (i.e., the three protons in the nucleus and the two inner-core electrons). The energy required to ionize a lithium atom is 5.14 eV. According to Rutherford’s nuclear model of the atom, what are the orbital radius and speed of the valence electron?

Hint: Consider the energy needed to remove the electron and the force needed to give the electron a circular orbit.

|| INT A hydrogen atom ${}^1\text{H}$ with 200 eV of kinetic energy has a head-on, perfectly elastic collision with a ${}^{12}\text{C}$ atom at rest. Afterward, what is the kinetic energy, in eV, of each atom?

|| The diameter of an atom is $1.2\times {10}^{-10}\text{m}$ and the diameter of its nucleus is $1.0\times {10}^{-14}\text{m}$. What percent of the atom’s volume is occupied by mass and what percent is empty space?

||| The diameter of an aluminum atom of mass 27 u is approximately $1.2\times {10}^{-10}\text{ m}$. The diameter of the nucleus of an aluminum atom is approximately $8\times {10}^{-15}\text{m}$. The density of solid aluminum is $2700\text{kg}\text{/}{\text{m}}^3$.

1. What is the average density of an aluminum atom?

2. Your answer to part a was larger than the density of solid aluminum. This suggests that the atoms in solid aluminum have spaces between them rather than being tightly packed together. What is the average volume per atom in solid aluminum? If this volume is a sphere, what is the radius?

   Hint: The volume per atom is not the same as the volume of an atom.

3. What is the density of the aluminum nucleus? By what factor is the nuclear density larger than the density of solid aluminum?

|| INT A ${}^{222}\text{Rn}$ atom (radon) in a 0.75 T magnetic field undergoes radioactive decay, emitting an alpha particle in a direction perpendicular to $\vec{B}$. The alpha particle begins cyclotron motion with a radius of 45 cm. With what energy, in MeV, was the alpha particle emitted?

|| INT In a head-on collision, the closest approach of a 6.24 MeV alpha particle to the center of a nucleus is 6.00 fm. The nucleus is in an atom of what element? Assume the nucleus remains at rest.

|| INT Through what potential difference would you need to accelerate an alpha particle, starting from rest, so that it will just reach the surface of a 15-fm-diameter ${}^{238}\text{U}$ nucleus?

|| INT The oxygen nucleus ${}^{16}\text{O}$ has a radius of 3.0 fm.

1. With what speed must a proton be fired toward an oxygen nucleus to have a turning point 1.0 fm from the surface? Assume the nucleus remains at rest.

2. What is the proton’s kinetic energy in MeV?

|| INT To initiate a nuclear reaction, an experimental nuclear physicist wants to shoot a proton into a 5.50-fm-diameter ${}^{12}\text{C}$ nucleus. The proton must impact the nucleus with a kinetic energy of 3.00 MeV. Assume the nucleus remains at rest.

1. With what speed must the proton be fired toward the target?

2. Through what potential difference must the proton be accelerated from rest to acquire this speed?

|| INT A classical atom that has an electron orbiting at frequency f would emit electromagnetic waves of frequency f because the electron’s orbit, seen edge-on, looks like an oscillating electric dipole.

1. At what radius, in nm, would the electron orbiting the proton in a hydrogen atom emit light with a wavelength of 600 nm?

2. What is the total mechanical energy of this atom?

||| An alpha particle approaches a ${}^{197}\text{Au}$ nucleus with a speed of $1.50\times {10}^7\text{m}\text{/}\text{s}$. As FIGURE CP37.47 shows, the alpha particle is scattered at a $49\text{°}$ angle at the slower speed of $1.49\times {10}^7\text{ m/s}$. In what direction does the nucleus recoil, and with what speed?

FIGURE CP37.47

||| Physicists first attempted to understand the hydrogen atom by applying the laws of classical physics. Consider an electron of mass m and charge $-e$ in a circular orbit of radius r around a proton of charge $+e$.

1. Use Newtonian physics to show that the total energy of the atom is $E=-e^2\text{/}8\pi {ϵ}_{0}r$.

2. Show that the potential energy is $-2$ times the electron’s kinetic energy. This result is called the virial theorem.

3. The minimum energy needed to ionize a hydrogen atom (i.e., to remove the electron) is found experimentally to be 13.6 eV. From this information, what are the electron’s speed and the radius of its orbit?

||| Consider an oil droplet of mass m and charge q. We want to determine the charge on the droplet in a Millikan-type experiment. We will do this in several steps. Assume, for simplicity, that the charge is positive and that the electric field between the plates points upward.

1. An electric field is established by applying a potential difference to the plates. It is found that a field of strength $E_0$ will cause the droplet to be suspended motionless. Write an expression for the droplet’s charge in terms of the suspending field $E_0$ and the droplet’s weight mg.

2. The field $E_0$ is easily determined by knowing the plate spacing and measuring the potential difference applied to them. The larger problem is to determine the mass of a microscopic droplet. Consider a mass m falling through a viscous medium in which there is a retarding or drag force. For very small particles, the retarding force is given by $F_{\text{drag}}=-bv$ where b is a constant and v the droplet’s velocity. The sign recognizes that the drag force vector points upward when the droplet is falling (negative v). A falling droplet quickly reaches a constant speed, called the terminal speed. Write an expression for the terminal speed $v_{\text{term}}$ in terms of m, g, and b.

3. A spherical object of radius r moving slowly through the air is known to experience a retarding force $F_{\text{drag}}=-6\pi \eta rv$ where $\eta$ is the viscosity of the air. Use this and your answer to part b to show that a spherical droplet of density $\rho$ falling with a terminal velocity $v_{\text{term}}$ has a radius

   $$r=\sqrt{{\displaystyle \frac{9\eta v_{\text{term}}}{2\rho g}}}$$

4. Oil has a density $860\text{kg}\text{/}{\text{m}}^3$. An oil droplet is suspended between two plates 1.0 cm apart by adjusting the potential difference between them to 1177 V. When the voltage is removed, the droplet falls and quickly reaches constant speed. It is timed with a stopwatch, and falls 3.00 mm in 7.33 s. The viscosity of air is $1.83\times {10}^{-5}\text{Pa}\text{s}$. What is the droplet’s charge?

5. How many units of the fundamental electric charge does this droplet possess?

||| The polonium isotope ${}^{211}\text{Po}$ is radioactive and undergoes alpha decay. In the decay process, a ${}^{211}\text{Po}$ nucleus at rest explodes into an alpha particle (a ${}^4\text{He}$ nucleus) and a ${}^{207}\text{Pb}$ lead nucleus. The lead nucleus is found to have 0.14 MeV of kinetic energy. The energy released in a nuclear decay is the total kinetic energy of all the decay products. How much energy is released, in MeV, in a ${}^{211}\text{Po}$ decay?