Questions and Answers of The Physics Energy

Estimate the gravitational energy in the Sun, using eqs. (22.1) and (22.2) assuming that the mass is distributed uniformly. Can you confirm the statement in the text that gravitational potential
Compute the frequency ωmax corresponding to the maximum power of radiation for a blackbody at temperature T = 280 K. This is roughly the average temperature of Earth’s surface. Compute the
The incandescent light bulb is a notoriously inefficient way to convert electric power into visible light. The tungsten filament emits blackbody radiation at a temperature that is limited by its
Estimate the rate of thermal radiation from a household hot-water radiator with a surface area of 1 m2 and a temperature of 80℃.
Compute the power radiated in the visible range from the toaster oven coil radiating with a total power of 1500 W, as described in Example 22.1.
Compute or estimate the fraction of radiated thermal energy that is in the visible range (wavelength 400–750 nm) for the following two radiation spectra: (a) solar radiation, assumed to be perfect
An object moving at the speed of light along thex-axis in the metric (21.35) of an inflating universe satisfies x?(t)eHt = c. Assuming x(0) = 0, compute x(t).Show that x(t) never exceeds the value
Show that two space-time points (0, 0, 0, 0) and(t, x = vt, 0, 0) that lie on the trajectory of an object traveling at speed v along the x-axis are separated by a distance satisfying Δ2 = 0 if and
Consider the circumstances described in Problem 21.8, but now suppose that Δ2 < 0. Use the transformation defined in Problem 21.7 to show that there is a reference frame traveling at a speed v
Consider two events seen by observer (1), who chooses her coordinates so that the first event occurs at the space-time origin x?1 = (0, 0, 0, 0), while the second event occurs at x??1 = ( c? t1, x?1,
The relativistic transformation law that relates the coordinates seen by an observer (2) moving at velocity v relative to an observer (1) is x2 = (cosh ?)x1 ?(sinh ?)ct1, ct2 = (cosh ?)ct1 ? (sinh
According to special relativity, the speed of light should be the same as seen by two observers moving at different relative velocities. In Newtonian physics, an observer (2) moving at a velocity v
Consider a particle of mass m moving in two dimensions, bound in a central potential V(r), so that its Hamiltonian is H = T(p) + V(r), where T(p) = (p2x+ p2y)/2m. Show that the quantity D = xpx + ypy
Consider motion in the xy-plane governed by a Hamiltonian H(px , py, x, y). Introduce polar coordinates x = r cos?, y = r sin ?, px = p cos ?p, py = p sin ?p, and use the chain rule for partial
Confirm by explicit computations the assertions made in Example 21.1 that the Hamiltonian is invariant under translation and rotation so that total momentum and angular momentum are conserved.
The equation of motion of a classical harmonic oscillator subject to an external time-dependent force F(t) is m̈x = −kx + F(t). Show that the oscillator’s energy is not conserved, but that its
Consider a situation in which, through accident or ill-intention, all the spent nuclear fuel from a year’s operation of a 1 GWe light water reactor were uniformly distributed across one square
In §20.6.1 it was stated that approximately 21 t of SNF are removed yearly from a 1 GWe reactor operating at 33% efficiency. Assuming that the spent fuel contains 0.9% unconsumed 235U and that the
Using the data from Table 18.2, compute the spontaneous fission rate (in Bq/kg) for 240Pu . Compare this with the rate in 239Pu and in uranium enriched to 85% 235U. (See Problem 20.10.) Table 18.2?
40K accounts for 0.0117% of naturally occurring potassium and potassium accounts for about 0.2% of the human body mass.Considering only the dominant decay of 40K, which yields a β-decay electron of
Particle physics experiments are shielded from cosmic ray muons by placement underground, often in old mines. Cosmic ray muons have been detected at 2000 feet below ground at the Home stake Gold Mine
The flux of cosmic ray muons at sea level is about one per square centimeter per minute. Given their mass stopping power of2 MeV cm2 /gm, and their weighting factor , wμ = 1 estimate an average
According to the text, roughly 10–15% of lung cancer deaths in the US are due to radon exposure. Use the LNT model and average yearly effective dose from radon to estimate the cancer rate from this
Rubidium-87, 87 Rb is a relatively common, very long-lived nuclide. Look up and compare its abundance and lifetime to 40K. How does 87 Rb decay? Why do you think it was omitted from the
131I is a dangerous fission product because iodine is preferentially absorbed by the thyroid gland in children. It has a half-life of about 8 days and emits a β-particle with average energy 180
A nuclear accident has exposed a 70 kg worker to 1 Ci of 2 MeV photons for about one minute. Assuming that all the energy of absorbed photons is deposited in the body, with uniform distribution,what
Thermal neutrons wandering through biological tissue are most likely to be absorbed by 1H, 12C, or 16O. If absorption is accompanied by emission of a single γ-ray, what are the γ-ray energies
In §20.3.1 it is stated that activity a is often used as measure of the mass of radio nuclide in a sample m. Show that for a radio nuclide with half-life t1/2 and atomic mass A, this
The mass attenuation coefficients (μ/ρ) for 1.5 MeV photons inconcrete (ρ = 2.35 g/cm3) and lead (ρ=11.4 g/cm3) are both about 0.5 cm2/g. How thick must concrete or lead shielding be in order to
To a first approximation for computing stopping power, living tissue can be approximated as water. Using data on the range of protons in water, estimate the energy of a proton required to place its
Using the results of Problem 20.3, compare the range of a 2 MeV α-particle in water, air, and bone. (The density of bone is approximately 1100 kg/m3).
Show that in any material, the range of an ?-particle and a proton with the same speed are the same (in the approximation that m? = 4mp). Using Figure 20.3 find the range of a 2 MeV proton in
Show that a particle’s range (see §20.2.1) expressed as a function of its initial velocity is proportional to its mass and inversely proportional to the square of its charge and to the electron
Estimate a lower limit on the number of collisions a 1 MeV proton must make with atomic electrons before its energy is reduced to 10 eV. Why is it reasonable to ignore the atomic binding energy of
Show that the maximum energy transfer by a heavy particle with mass M and speed v to an electron initially at rest is 2mev2, hence the appearance of this factor in the formula for the stopping power.
Suppose a fusion power plant operates at the pressure (≈ 7 atm) and temperature (≈ 150 × 106 K) planned for ITER, and suppose the plasma volume equals that of ITER (840 m3) as well. Assuming
A dt plasma fusion reactor is operating in steady state at a temperature kBT = 15 keV and a pressure of 7 atm, where 〈σfv〉 = 3 × 10−22 m3/s. What is the energy confinement time τE
Show that a charged particle moving in the plane perpendicular to a constant magnetic field (eq. (19.29) moves in a circle with Br = mv/q. What is the radius of the circle for an electron with
A thermal breeder reactor is fueled with a mixture of a fertile nucleus A (fraction 1 − x) which breeds a fissile nucleus B (fraction x). To breed more of B it is essential that the reproduction
A thermal-neutron reactor is charged every year with enriched uranium containing 1.6 t of 235U. When the spent fuel is removed, it contains 400 kg of 235U and 560 kg of 239Pu. Assuming a conversion
A uranium enrichment facility produces nuclear reactor fuel enriched to 4% 235U to fuel the reactor whose power to fuel consumption ratio is described by eq. (19.15)For each tonne of natural uranium,
When a reactor is turned off, the amount of 135Xe decreases as it decays, but grows as 135I decays to it. Write a pair of differential equations that describe the time rate of change of the numbers
The analysis of prompt and delayed neutrons in ?19.1.6 leads to the following formula for the time dependence of the neutron density, Analyze the time dependence of n(t) when ? d. For example,
Suppose a pressurized water reactor is loaded with 200 tons of 3% enriched uranium. The reactor has been designed to run with an average thermal-neutron flux of ??n??= 1.5?1013 cm?2 s?1 for a period
An approximate description of the Oklo uranium deposit is 90% uraninite (UO2) by mass, saturated with water that acted as the moderator. Assume that the Oklo deposit was homogeneously saturated with
It is estimated that the Oklo reactor occurred when the ratio of 235U to 238U was 3.67%. Given the present ratio of 0.72% and the half-lives of both isotopes, estimate the time when the Oklo reactor
Show that an infinite, homogeneous reactor fueled with natural uranium (0.72% 235U) and moderated by heavy water (D2O) can sustain a fission chain reaction. What is the optimal ratio of moderator to
Consider an infinite, homogeneous reactor fueled with uranium enriched to 3% and moderated by graphite with a ratio of graphite : uranium of 800 : 1. Find k?. To find p, you can ignore the uranium
Show from eq. (19.9) that in an infinite, homogeneous graphite-moderated reactor, the resonance escape probability is given by p(y) = exp(?17.28 (1/(9.3 + 4.9y))0.514), where y is the ratio of
Starting from the Maxwell–Boltzmann distribution, eq. (18.10)for the probability of finding a particle with energy E in a gas with temperature T, show that the most probable speed for a particle is
Tritium is just barely unstable. How much more tightly would tritium have to be bound in order for it to bestable? Write the nuclear reaction describing tt fusion.How much energy is emitted in tt
The energy released per nucleon for (a) fission of 235U; (b) dt fusion.
Estimate the quantity of tritium (kg/year) required to fuel a 1 GWe fusion power plant assuming 20% overall efficiency.
Show that in d+t → n+4He fusion reaction, the ratio of the kinetic energy on the neutron to that on the 4He is 4:1 if you neglect the initial kinetic energies of the reactants.Why is this a good
The Coulomb barrier keeping two nuclei apart is given by V(r) = Z1Z2e2/(4??0r), where r is the relative separation of the two nuclei. Assume that V reaches its maximum value VC when r ? R1 + R2,
The net reaction that accounts for the Sun’s power takes four neutral hydrogen atoms and fuses them to make a neutral helium atom, 4 11H→ 42He. Compute the energy released per hydrogen atom. The
Show that the probability of finding a particle with energy greater than 0.2 MeV in a gas at temperature 15 × 106 K is ∼10−69.
Estimate the height of the Coulomb barrier between two protons as follows: the strong nuclear interaction potential between two protons at large separation is given by eq. (14.5) Add to this the
(a)?Derive the large A approximation to the logarithmic energy decrement given in eq. (18.5) (b) Show that the number of collisions with a nucleus with atomic mass A required to decrease the energy
Derive the expression eq. (18.7)? for ?(A) by averaging over angles in the center-of-mass reference? frame.Take E/E? from eq. (18.4) and remember that the measure for averaging over ? is 1/2 sin
Verify the classical energy loss formula (18.4) For an elastic two-body collision. Use the notation in the Figure 18.8 Use the fact that the relationship between a velocity vector V in the target
Two sources of delayed neutrons in 235U fission are the fission fragments 87Br and 137I. Find the fraction of their decays that yield a neutron, and the half-life of each. Verify that the Q-value for
When 235U fissions by absorbing a slow neutron, 93Zr is one of the most common fission fragments. Suppose three neutrons are given off in the fission. What is the other nucleus produced in this
An example of a thermal-neutron fission reaction isnth + 235U → 139Ba + 94Kr + 3n. Perform the analysis of Example 18.1 for this case.
Verify the statements made after eq. (18.2) About the energy released for the case of thermal-neutron induced fission of 235U. First, use measured mass excesses to show that the asymmetric fission
In §18.3.3 it is stated that a neutron in uranium is bound by ∼ 6 MeV. Check this by using data from [70] to compute the binding energy of a neutron
Given the data in Table 18.2 estimate the number of neutrons emitted by spontaneous fission per second per kilogram of naturally occurring uranium. Nucleus Natural Half-life Fraction of decays by
A beam of thermal neutrons with current? is incident on a uniform (thin) target of thickness 1 cm, consisting of 20% 235U and 80% 238U. Table 18.3? ? and Problem 18.1. What is the fission rate?
Assuming that the particles in Problem 18.1 that react are removed from the beam, use the result of that problem to show that the beam current falls exponentially I(z) = I0e−z/λ through the
Suppose a beam of particles with flux Φ is incident on a uniform target of thickness T and density ρ(mass/volume). Assuming that the atoms in the target do not block one another (the thin-target
The uranium isotope 234U accounts for 0.0054% of naturally occurring uranium. Its half-life τ1/2 (234U) =2.455 × 105 y is much too short for any primordial 234U to still exist on Earth.
In light of the results of Problem 17.22, what is the rate of energy emission (J/kg) of a sample of 238U that has been around long enough to come into decay equilibrium?
When one of the naturally occurring radioactive heavy elements like 238U decays, a decay chain follows, leading to the creation of various unstable,radioactive descendant nuclides. Consider a sample
97% of naturally occurring calcium is calcium-40,4020Ca. This may seem surprising, since if we use the semi-empirical mass formula to estimate the most stable nuclide with A = 40 we find Z ≈ 18.
Ignoring small electron binding energies and the very small mass of the neutrino, show that the mass of a nucleus increases when it decays by electron capture if the Q-value of the decay is less than
Verify the expressions for the Q-value in ??-decay and electron capture, eq. (17.34)? Q(B¯) = A(Z, A) – A(Z + 1, A), Q(B+) = A(Z, A) – A(Z – 1, A) – 2Ae Q(EC) = A(Z, A) - A(Z – 1, A),
Check that 3He is less tightly bound than 3H. Explain why 3H decays to 3He, and not vice versa.
Examine the derivation of Gamow’s formula for α-decay lifetimes, and then derive the equivalent expression for decay by emission of a (small) nucleus of charge z and mass number a. Assume a/A is
Radon gas 22286Rn is a serious environmental hazard(see §20). It is a decay product of 23892U, which is a relatively common constituent of rock. 22286Rn undergoesα-decay to 21884Po. When 22286Rn is
235U can decay by emitting a neon nucleus. How many such decays occur per second per mole of 235U?
Show that the longest-lived isotope of radium, 22688Ra,is energetically allowed to decay by emission of a146C nucleus. This decay has been observed with aprobability of 3.2 × 10−9.
Explain how to read from Figure 17.8? the Q-value for?-decay predicted by the SEMF.? 10 dbmax bmax (A) + A dA 4He 3H 'H or n 50 100 150 200 250 A MeV
The maximum value of A for which nuclei are stable against instantaneous fission can be estimated as follows. When a charged, spherical nucleus, AZ, with surface tension is deformed into an ellipsoid
Check the assertion in the text that the SEMF predicts that nuclei have positive binding energy up to A ∼ 3150.
Derive eq. (17.20) ?by differentiating the SEMF expression for b(Z, A) with respect to Z at fixed A.? A/2 1+7.52 x 10-3 A²/3 A/2 Zmin (A) 1+CA2/3 4E sym
Find the SEMF prediction for Zmin(A = 200) from eq. (17.20) Find the actual value of Z that gives the greatest binding energy for A = 200. How well does the SEMF do?? A/2 1+7.52 x 10-3 A²/3 A/2
Pressure can be defined as the rate of change of energy with volume, p = ∂E/∂V. Similarly, the surface analogue of pressure, surface tension, is defined as the rate of change of energy with
Estimate the density of a nucleus by assuming it to be a sphere of radius R0 = 1.3 A1/3 fm. What would be the radius of a nucleus with the mass of Earth?
The radius of a nucleon is about RN = 0.9 fm. Compute the volume of a nucleus with a moderately large value of A in the liquid drop model (i.e. using the SEMF) and show that roughly 2/3 of the
3He is so rare that?it must be produced artificially.One way to produce it is through a sequence of reactions that can be carried out in a nuclear reactor, where thermal neutrons are quite common.
Helium three is an extremely rare but stable isotope of helium made up of two protons and only one neutron.Because 4He is so tightly bound, considerable energy is given off when 3He absorbs a
There is no stable nucleus with A = 5. This has profound consequences for the way stars burn and form elements. The two candidates for stable nuclei with A = 5 are 52He and 53Li. Look up or figure
A typical fission reaction involving 235U is The kinetic energy of a thermal neutron is ?1/40 eV,small enough to ignore in the energy balance. Show that the energy released in the fission is 179.5
Derive the relation quoted in eq. (17.7) ? between the binding energy of a neutral atom B(Z, A) and its mass excess ?(Z, A). B(Z, A) - ΝΔ n) + ΖΔ(| Η) -Δ(Ζ, A)
Assume that thorium can be converted 100% into 233U and used as a fission fuel. According to world thorium reserves are roughly 6 × 106 tonnes. How long could this amount of thorium supply 100% of
Check the quoted estimate of 4 × 109 tonnes of uranium from seawater quoted in this chapter. For how many years could this amount of uranium supply 100% of the world’s energy needs at the level of
In one scenario, enrichment of 200 t of natural uranium yields 26 t enriched to 4% 235U that is used to power a reactor for a year. How much 235U is lost in the depleted uranium discarded during the
Given the (higher) molar heat of combustion of gasoline (§11.2.2) and knowing its molecular weight, verify that it yields about 1/2 eV per nucleon in complete combustion.
The energy levels of the three-dimensional harmonic oscillator are given by En1,n2,n3 = (n1 +n2 +n3 + 3/2)ћω with n1,2,3 = 0, 1, . . .. Suppose three spin-1/2 fermions are trapped in a
The energy levels of the one-dimensional harmonic oscillator are given by En = (n + 1/2)ћω with n = 0, 1, 2, . . . . Suppose two spin-1/2 fermions are trapped in a one-dimensional oscillator

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