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Chapter: 27 -Problem: 1 >> The GPS satellites are in circular orbits at a height of 20,200 km above Earth’s surface, where their orbital period is 12 sidereal hours. If the ticking rates of the clocks on the satellites were not corrected for the gravitational redshift, roughly how long would it take them to accumulate a time shift, relative to clocks on Earth, large enough to degrade the GPS position accuracy by 10m? by 1 k
Answer Preview: To calculate the time it takes for the clocks on the GPS satellites to accumulate a time shift, we c…

, Chapter: 6 -Problem: 23 >> Consider a classical simple harmonic oscillator (e.g., the nanomechanical oscillator, LIGO mass on an optical spring, L-C-R circuit, or optical resonator briefly discussed in Ex. 6.17). Let the oscillator be coupled weakly to a dissipative heat bath with temperature T . The Langevin equation for the oscillator’s generalized coordinate x is Eq. (6.79). The oscillator’s coordinate x(t) and momentum
Answer Preview: The solution to the harmonic oscillator equation is (14 11)x=A cos(t+) where A is …

, Chapter: 1 -Problem: 1 >> Without introducing any coordinates or basis vectors, show that when a particle with charge q interacts with electric and magnetic fields, its kinetic energy changes at a rate Transcribed Image Text: dE/dt =qv. E. (1.8)
Answer Preview: ANSWER We can start by considering the work done on the particle by the electromagnetic fields. The …

, Chapter: 25 -Problem: 3 >> Show that in an arbitrary coordinate system x?(P) the geodesic equation (25.11c) takes the form of Eq. (25.14). Transcribed Image Text: V? P=0 (25.11c)
Answer Preview: To show that the geodesic equation in an arbitrary coordinate system x^(P) takes the form of Eq. (25 14), which is: d^2x^ / d^2 + ^_ (dx^ / d) (dx^ / …

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, Chapter: 10 -Problem: 9 >> Consider a wave propagating through a dielectric medium that is anisotropic, but not necessarily—for the moment—axisymmetric. Let the wave be sufficiently weak that nonlinear effects are unimportant. Then the nonlinear wave equation (10.22a) takes the linear form (a) Specialize to a monochromatic plane wave with angular frequency ? and wave vector k. Show that the wave equation (10.50) reduces to
Answer Preview: ANSWER a) For a monochromatic plane wave in the anisotropic dielectric medium, we have E = Ex0 e^(i(kx-t)) where Ex0 is a constant vector. Taking the …

, Chapter: 11 -Problem: 25 >> A torsion pendulum is a very useful tool for testing the equivalence principle (Sec. 25.2), for seeking evidence for hypothetical fifth (not to mention sixth!) forces, and for searching for deviations from gravity’s inverse-square law on submillimeter scales, which could arise from gravity being influenced by macroscopic higher spatial dimensions. It would be advantageous to design a torsion pendu
Answer Preview: a) The longitudinal strain is given by: l = l/l where l is the change in length and l is the original length. The change in length is caused by the we…

, Chapter: 16 -Problem: 16 >> Idealize the trumpet as a bent pipe of length 1.2 m from the mouthpiece (a node of the air’s displacement) to the bell (an antinode). The lowest note is a first overtone and should correspond to B flat (233Hz). Does it?
Answer Preview: To determine whether the lowest note of the idealized trumpet corresponds to B flat (233Hz), we need …

, Chapter: 13 -Problem: 20 >> By manipulating the differential forms of the law of rest-mass conservation and the law of energy conservation, derive the constancy of B = (? + P)?/?o along steady flow lines, Eq. (13.88). Transcribed Image Text: dB dt =yoj aB = = 0, axi where B = (p+P)r Po (13.88)
Answer Preview: To derive the constancy of B = ( + P)/o along steady flow lines, we start by manipulating the differential forms of the law of rest-mass conservation …

, Chapter: 9 -Problem: 15 >> Show that the PDH method for locking a laser’s frequency to an optical cavity works for modulations faster than the cavity’s response time, and even work for ? »1/?response.More specifically, show that the reflected power still contains the information needed for feedback to the laser. Transcribed Image Text:
Answer Preview: ANSWER The Pound-Drever-Hall (PDH) method is a technique used for locking a laser's frequency to an optical cavity by measuring the phase shift betwee…

, Chapter: 4 -Problem: 14 >> Show that in the Bose-Einstein condensate discussed in the text, the momentum distribution for the ground-state-mode atoms is Gaussian with rms momentum and that for the classical cloud it is Gaussian with rms momentum Transcribed Image Text: Pcondensate = ?3/2h/oo = ?3hmwo/2
Answer Preview: ANSWER. To show that the momentum distribution for the ground-state-mode atoms in a Bose-Einstein condensate is Gaussian with an rms momentum of P_condensate = (3/2) hbar/(m*w_0), we need to start wit…

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, Chapter: 15 -Problem: 6 >> Compute the width w(x) and velocity deficit uo(x) for the 3-dimensional turbulent wake behind a sphere.
Answer Preview: The wake behind a sphere is a classic example of turbulent flow. In the far wake region, the flow be…

, Chapter: 10 -Problem: 12 >> Derive the solution (10.54) to the evolution equations (10.47) for frequency doubling, and verify that it has the claimed properties. Transcribed Image Text: d'Az ik dz 2 = (A?)², d'A? dz =ik AzA; K=B?, 200703 €0c³n²n3 -dijk fi(¹) f(¹) f(3) (10.47)
Answer Preview: User Consider a cell with volume V , like those of Fig. 5 1, that has imaginary walls and is immersed in a bath of identical, nonrelativistic, classical perfect-gas particles with temperature Tb and c…

, Chapter: 3 -Problem: 19 >> Consider a nonrelativistic fluid that, in the neighborhood of the origin, has fluid velocity with ?ij symmetric and trace-free. As we shall see in Sec. 13.7.1, this represents a purely shearing flow, with no rotation or volume changes of fluid elements; ?ij is called the fluid’s rate of shear. Just as a gradient of temperature produces a diffusive flow of heat, so the gradient of velocity embodie
Answer Preview: (a) To derive Eq. (3 85b) for the shear viscosity, let's begin with an order-of-magnitude analysis. We start with the fluid velocity given by Vj = ijxj. The shear stress tensor is defined as Tij = Viv…

, Chapter: 17 -Problem: 8 >> (a) Almost all equations of state satisfy the condition (?2V/?P2)s > 0. Show that, when this condition is satisfied, the Rankine-Hugoniot relations and the law of entropy increase imply that the pressure and density must increase across a shock and the fluid must decelerate: P2 > P1, V2 < V1, and v2 < v1.(b) Show that in a fluid that violates (?2V/?P2)s > 0, the pressure and density must still inc
Answer Preview: To understand why the conditions (^2V/P^2)s > 0 (where V is specific volume, P is pressure, and s is entropy) lead to an increase in pressure and density across a shock and cause the fluid to decelera…

, Chapter: 13 -Problem: 7 >> There’s a hole in my bucket. How long will it take to empty? (Try an experiment, and if the time does not agree with the estimate, explain why not.)
Answer Preview: Determining how long it will take to empty a bucket with a hole depends on various factors, such as the size of the hole, the flow rate of the liquid, …

, Chapter: 9 -Problem: 7 >> We have defined the degree of coherence ?12(a, ?) for two points in the radiation field separated laterally by a distance a and longitudinally by a time ?. Under what conditions will this be given by the product of the spatial and temporal degrees of coherence? Transcribed Image Text: Y?2(a, T) = y
Answer Preview: ANSWER The degree of coherence between two points in the radiation field is defined as the normalize…

, Chapter: 2 -Problem: 3 >> Show that Eq. (2.19) can be true for all time like, unit-length vectors u(vector) if and only if F is antisymmetric. Transcribed Image Text: F(u, ?) = 0. (2.19)
Answer Preview: \ The equation (2 19) : d/dt (u u) = 2 (u (du/dt)) where u is a unit-length vector and F is a vecto…

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, Chapter: 9 -Problem: 17 >> A Sagnac interferometer is a rudimentary version of a laser gyroscope for measuring rotation with respect to an inertial frame. The optical configuration is shown in Fig. 9.12. Light from a laser L is split by a beam splitter B and travels both clockwise and counterclockwise around the optical circuit, reflecting off three plane mirrors. The light is then recombined at B, and interference fringes
Answer Preview: The difference in the time it takes light to traverse the circuit in the two directions can be calcu…

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, Chapter: 14 -Problem: 6 >> Explain why the pressure and temperature of the core of a wingtip vortex are significantly lower than the pressure and temperature of the ambient air. Under what circumstances will this lead to condensation of tiny water droplets in the vortex core, off which light can scatter, as in Fig. 14.2b?  Fig. 14.2b. Transcribed
Answer Preview: A wingtip vortex is generated due to the pressure difference between the upper and lower surface of …

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, Chapter: 11 -Problem: 23 >> Derive Eqs. (11.76) and (11.77) for the divergence of the vector field ? in cylindrical and spherical coordinates using the connection coefficients (11.70) and (11.71). In Equations Transcribed Image Text: ???? ? > ?? ??? = ? (11.70)
Answer Preview: To derive equations (11 76) and (11 77), we will use the connection coefficients (Christoffel symbols) in cylindrical and spherical coordinates. Let'…

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, Chapter: 28 -Problem: 15 >> Explore nonlinear effects in the growth of perturbations in the gravitational age— when radiation and the cosmological constant can be ignored—by considering the evolution of a sphere in which the matter density is uniform and exceeds the external density by a small quantity. (a) Use the Friedmann equations (28.16), (28.18), and (28.19) to show that the sphere behaves like a universe with density
Answer Preview: To explore the nonlinear effects in the growth of perturbations in the gravitational age, let's consider a sphere with a uniform matter density that exceeds the external density by a small quantity. (…

, Chapter: 9 -Problem: 9 >> The longest radio-telescope separation available in 2016 is that between telescopes on Earth’s surface and a 10-m diameter radio telescope in the Russian RadioAstron satellite, which was launched into a highly elliptical orbit around Earth in summer 2011, with perigee ?10,000 km (1.6 Earth radii) and apogee ?350,000 km (55 Earth radii). (a) Radio astronomers conventionally describe the specific in
Answer Preview: a) The specific intensity of a source in terms of its brightness temperature can be described as: I(…

, Chapter: 12 -Problem: 6 >> Show that the sound speeds for the following types of elastic waves in an isotropic material are in the ratios The elastic waves are  (i) Longitudinal waves along a rod,  (ii) Longitudinal waves along a sheet,  (iii) Longitudinal waves along a rod embedded in an incompressible fluid. (iv) Shear waves in an extended solid. (v) Torsional waves along a rod.
Answer Preview: To show the ratios of sound speeds for different types of elastic waves in an isotropic material, we …

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, Chapter: 8 -Problem: 4 >> Derive and plot the Airy diffraction pattern [Eq. (8.18)] and show that 84% of the light is contained within the Airy disk. Transcribed Image Text: ?(?) ? Disk with diameter D k?? 2 x jinc e-ikxed? (8.18)
Answer Preview: The Airy diffraction pattern can be derived using the following equation: I = (I_0 * (2 * J1(k * r) …

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, Chapter: 27 -Problem: 6 >> (a) Derive the behavior [Eq. (27.31)] of h+ and h× under rotations in the transverse plane. (b) Show that, with the orientations of spatial basis vectors described after Eq. (27.31), h+ and h× are unchanged by boosts. Equation 27.31. Transcribed Image Text: (h+ + ihx) new = (h++ihx) olde²i, when (
Answer Preview: To derive the behavior of h+ and h under rotations in the transverse plane, we start with the equation (27 31): |(h+ + ih)new| = |(h+ + ih)old|. Let's …

, Chapter: 7 -Problem: 7 >> Show that Hamilton’s equations for the standard dispersionless dispersion relation (7.4) imply the same ray equation (7.48) as we derived using Fermat’s principle. Transcribed Image Text: w=2(k)=Ck = C[k], (7.4)
Answer Preview: To show that Hamilton s equations for the standard dispersionless dispersion relation imply the same …

, Chapter: 4 -Problem: 19 >> Consider messages of length L >> 2 constructed from just two symbols (N = 2),which occur with frequencies p and (1? p). Plot the average information per symbol I? (p) in such messages, as a function of p. Explain why your plot has a maximum I? = 1 when p = 1/2, and has I? = 0 when p = 0 and when p = 1.
Answer Preview: The average information per symbol I (p) in a message of length L constructed from two symbols occur…

, Chapter: 14 -Problem: 18 >> Fluid flows down a long cylindrical pipe of length b much larger than radius a, from a reservoir maintained at pressure P0 (which connects to the pipe at x = 0) to a free end at large x, where the pressure is negligible. In this problem, we try to understand the velocity field vx(?? , x) as a function of radius ?? and distance x down the pipe, for a given discharge (i.e., mass flow per unit time)
Answer Preview: (a) Close to the entrance of the pipe (small x), the boundary layer will be thin, and the velocity will be nearly independent of radius. This region is called the fully developed region. In this regio…

, Chapter: 13 -Problem: 15 >> Consider a velocity field with non vanishing curl. Define a locally orthonormal basis at a point in the velocity field, so that one basis vector, ex, is parallel to the vorticity. Now imagine the remaining two basis vectors as being frozen into the fluid. Show that they will both rotate about the axis defined by ex and that the vorticity will be the sum of their angular velocities (i.e., twice the
Answer Preview: Let us consider a velocity field with non-vanishing curl. At any point in the velocity field, we can …

, Chapter: 7 -Problem: 1 >> Derive the group and phase velocities (7.10)–(7.13) from the dispersion relations (7.4)–(7.7). Transcribed Image Text: w = 22 (k)=Ck = C|k], (7.4)
Answer Preview: To derive the group and phase velocities from the dispersion relations (7 4)-(7 7), we can start with the general definition of the group velocity and phase velocity: Group Velocity: The group velocit…

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, Chapter: 28 -Problem: 12 >> Assume that the universe will continue to expand according to Eq. (28.43). (a) Calculate the behavior of the angular diameter distance and the associated volume as a function of the scale factor for the next 20 billion years. (b) Interpret your answer physically. (c) Explain qualitatively what will happen if the universe accelerates even faster than this. Equation 28.43.
Answer Preview: To calculate the behavior of the angular diameter distance and the associated volume as a function of the scale factor, we need to use the appropriate equations from cosmology. Eq. (28 43) is not prov…

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, Chapter: 11 -Problem: 18 >> Allometry is the study of biological scaling laws that relate various features of an animal to its size or mass. One example concerns the ratio of the width to the length of leg bones. Explain why the width to the length of a thigh bone in a quadruped might scale as the square root of the stress that it has to support. Compare elephants with cats in this regard. (The density of bone is roughly 1.5
Answer Preview: The width to length ratio of leg bones in quadrupeds can be explained by allometry, which relates bi…

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, Chapter: 13 -Problem: 5 >> (a) Show that the spatially variable part of the gravitational potential for a uniform density, nonrotating planet can be written as ? = 2?G?r2/3, where ? is the density. (b) Hence argue that the gravitational potential for a slowly spinning planet can be written in the form where A is a constant, and P2 is the Legendre polynomial with argument ? = sin(latitude).What happens to the P1 term? (c) G
Answer Preview: ANSWER (a) The gravitational potential due to a uniform density, nonrotating planet can be expressed as: = - (4/3)Gr where G is the gravitational constant, is the density, and r is the distance from t…

, Chapter: 7 -Problem: 20 >> Consider an elliptical gravitational lens where the potential ? is modeled by Determine the generic form of the caustic surfaces, the types of catastrophe encountered, and the change in the number of images formed when a point source crosses these surfaces. Note that it is in the spirit of catastrophe theory not to compute exact expressions but to determine scaling laws and to understand the qual
Answer Preview: The potential of an elliptical gravitational lens is modeled by: (0) = (1 + A^2 + 2B0x0y + ce3); 0 < q < 1/2 where A, B0, c are constants and q is the …

, Chapter: 2 -Problem: 14 >> Use spacetime diagrams to prove the following:(a) Two events that are simultaneous in one inertial frame are not necessarily simultaneous in another. More specifically, if frame F? moves with velocity v(vector) = ?e(vector)x as seen in frame F, where ? > 0, then of two events that are simultaneous in F? the one farther “back” (with the more negative value of x?) will occur in F before the one fart
Answer Preview: To prove the statements using spacetime diagrams, let's go through each of them one by one. (a) Two events that are simultaneous in one inertial frame are not necessarily simultaneous in another. More …

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, Chapter: 9 -Problem: 11 >> Consider monochromatic electromagnetic waves that propagate from a medium with index of refraction n1 into a medium with index of refraction n2. Let z be a Cartesian coordinate perpendicular to the planar interface between the media.(a) From the Helmholtz equation [??2 + (c2/n2)?2]? = 0, show that both ? and ?,z must be continuous across the interface.(b) Using these continuity requirements, show
Answer Preview: ANSWER (a) Starting from the Helmholtz equation: [-^2 + (c^2/n2)^2] = 0 Let's write the solution of …

, Chapter: 11 -Problem: 6 >> Explain why all animals, from fleas to humans to elephants, can jump to roughly the same height. The field of science that deals with topics like this is called allometry (Ex. 11.18). Data from Exercises 11.18Allometry is the study of biological scaling laws that relate various features of an animal to its size or mass. One example concerns the ratio of the width to the length of leg bones. Explai
Answer Preview: The ability of animals to jump to roughly the same height despite differences in size can be explained by the scaling laws of allometry. Allometry sho…

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, Chapter: 7 -Problem: 4 >> (a) Show that the prototypical scalar wave equation (7.17) follows from the variational principle where L is the lagrangian density (b) For any scalar-field lagrangian density L(?, ??/?t , ??, x, t), the energy density and energy flux can be expressed in terms of the lagrangian, in Cartesian coordinates, as Show, from the Euler-Lagrange equations for L, that these expressions satisfy energy co
Answer Preview: To solve the given problems, we will work with the provided equations and apply the principles of classical field theory. Here are the step-by-step solutions to each part of the problem: (a) To show t…

, Chapter: 11 -Problem: 16 >> Explore numerically the free energy (11.57) of a bent beam with a compressive force F and lateral force Flat. Examine how the extrema (equilibrium states) evolve as F and Flat change, and deduce the physical consequences. Transcribed Image Text: 2 ??? 72(F Fcrit) 1-1()-(F) (0) - () (0) 2 Ferit == 4
Answer Preview: To explore the free energy of a bent beam, we can use the formula: F = (1/2)k(l-l0)^2 - Fc(lcos(theta)) - Fl(lsin(theta)) where F is the free energy, …

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, Chapter: 9 -Problem: 1 >> X-rays with wavelength 8.33?A (0.833 nm) coming from a point source can be reflected at shallow angles of incidence from a plane mirror. The direct ray from a point source to a detector 3m away interferes with the reflected ray to produce fringes with spacing 25 ?m. Calculate the distance of the X-ray source from the mirror plane.
Answer Preview: To calculate the distance of the X-ray source from the mirror plane, we can us…

, Chapter: 6 -Problem: 19 >> Derive Eqs. (6.96) for A and B from the Fokker-Planck equation (6.94), and then from Eqs. (6.96) derive Eqs. (6.97). Equations Transcribed Image Text: ? - P2 ?t = ? ?? -[A(y)Pz] + 1 02 2 ??2 -[B(y) P2]. (6.94)
Answer Preview: To derive Eqs. (6 96) from the Fokker-Planck equation (6 94), we start by considering the Fokker-Planck equation: P/t = -(A(x)P)/x + ^2(B(x)P)/x^2. We want to derive the expressions for A and B in ter…

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, Chapter: 13 -Problem: 16 >> Estimate the collision mean free path of the air molecules around you. Hence verify the estimate for the kinematic viscosity of air given in Table 13.2. Transcribed Image Text: TABLE 13.2: Approximate kinematic viscosity for common fluids Quantity Water Air Glycerine Blood Kinematic viscosity v (m²
Answer Preview: ANSWER The collision mean free path of air molecules can be estimated using the kinetic theory of ga…

, Chapter: 8 -Problem: 7 >> Conceive and carry out an experiment using light diffraction to measure the thickness of a hair from your head, accurate to within a factor of ?2.
Answer Preview: To measure the thickness of a hair from your head using light diffraction, we can use a laser beam a…

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, Chapter: 13 -Problem: 3 >> Use Archimedes’ law to explain qualitatively the conditions under which a boat floating in still water will be stable to small rolling motions from side to side. You might want to define and introduce a center of buoyancy and a center of gravity inside the boat, and pay attention to the change in the center of buoyancy when the boat tilts. See Fig. 13.4. Fig 13.4.
Answer Preview: Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is eq…

, Chapter: 8 -Problem: 2 >> Use the convolution theorem to carry out the calculation of the Fraunhofer diffraction pattern from the grating shown in Fig. 8.6. Fig. 8.6 Transcribed Image Text: ????? ft) Hg(x) -Na 444 transparent 2a m --?1 ?? = www anbudo /(ka)=k/(2a) | | |···> (b) 4n/(ka) =2x/a x/(Na) x/(2a) (d) >à/(Na) 2Na H
Answer Preview: To apply the convolution theorem to calculate the Fraunhofer diffraction pattern from the grating shown in Fig. 8 6, we first need to find the diffrac…

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, Chapter: 16 -Problem: 17 >> Consider the emission of quadrupolar soundwaves by a Kolmogorov spectrum of free turbulence. Show that the power radiated per unit frequency interval has a spectrumAlso show that the total power radiated is roughly a fraction M5 of the power dissipated in the turbulence, where M is the Mach number. Transcribed Image Text:
Answer Preview: To derive the power spectrum of the radiated sound waves from a Kolmogorov spectrum of free turbulence, we'll follow some of the main steps of the derivation. We'll use some key results from turbulenc…

, Chapter: 27 -Problem: 18 >> Consider a particle that is at rest in the TT coordinate system of the gravitational-wave metric (27.80) before the gravitational wave arrives. In the text it is shown that the particle’s 4-velocity has ux = uy = 0 as the wave passes. Show that uz = 0 and ut = 1 as the wave passes, so the components of the particle’s 4-velocity are unaffected by the passing gravitational wave, and the particle rem
Answer Preview: To show that the particle's 4-velocity components remain unaffected by the passing gravitational wav…

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, Chapter: 18 -Problem: 3 >> Use the Rayleigh criterion to estimate the temperature difference that would have to be maintained for 2mm of corn/canola oil, or water, or mercury in a skillet to start convecting. Look up the relevant physical properties and comment on your answers. Do not perform this experiment with mercury.
Answer Preview: The Rayleigh criterion is used to estimate the critical temperature differ…

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, Chapter: 5 -Problem: 12 >> Consider a gigantic container of gas made of identical particles that might or might not interact. Regard this gas as a bath, with temperature Tb and pressure Pb. Pick out at random a sample of the bath’s gas containing precisely N particles, with N >> 1. Measure the volume V of the sample and the temperature T inside the sample. Then pick another sample of N particles, and measure its V and T , a
Answer Preview: (a) The probability distribution dp/dT dV represents the likelihood of observing a particular combination of temperature and volume for a sample of N …

, Chapter: 4 -Problem: 5 >> Consider fully thermalized electromagnetic radiation at temperature T , for which the mean occupation number has the standard Planck (blackbody) form ? = 1/(ex ? 1) with x = h?/(kBT).(a) Show that the entropy per mode of this radiation is (b) Show that the radiation’s entropy per unit volume can be written as the following integral over the magnitude of the photon momentum: (c) By performing the
Answer Preview: (a) To find the entropy per mode, we can start with the definition of entropy as the logarithm of the number of available microstates, which for electromagnetic radiation is related to the number of p…

, Chapter: 16 -Problem: 4 >> What is the maximum size of water droplets that can form by water very slowly dripping out of a syringe? Out of a water faucet (whose opening is far larger than that of a syringe)?
Answer Preview: The maximum size of water droplets that can form depends on several factors, including the surface t…

, Chapter: 19 -Problem: 14 >> Derive the dispersion relation ?2(k) for axisymmetric perturbations of the ?-pinch configuration when the magnetic field is confined to the cylinder’s exterior, and conclude from it that the ?-pinch is stable against axisymmetric perturbations. Repeat your analysis for a general, variable separated perturbation of the form ? ? ei(m?+kz??t), and thereby conclude that the ?-pinch is fully MHD stable
Answer Preview: To derive the dispersion relation for axisymmetric perturbations of the -pinch configuration, we consider the magnetohydrodynamic (MHD) equations in c…

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, Chapter: 2 -Problem: 15 >> Show, using spacetime diagrams and also using frame-independent calculations, that the law of conservation of 4-momentum forbids a photon to be absorbed by an electron, e + ? ? e, and also forbids an electron and a positron to annihilate and produce a single photon, e+ + e? ? ? (in the absence of any other particles to take up some of the 4-momentum); but the annihilation to form two photons, e+ +
Answer Preview: To understand why e + e is forbidden, we can use a spacetime diagram. In a spacetime diagram, time is plotted on the vertical axis and space on the ho…

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, Chapter: 7 -Problem: 2 >> Consider a 1-dimensionalwave packet with dispersion relation ? = ?(k). For concreteness, let A(k)be a narrow Gaussian peaked around (a) Expand ? as ?(k) = ?o ? xo? with xo a constant, and assume for simplicity that higher order terms are negligible. Similarly, expand ? ? ?(k) to quadratic order and explain why the coefficients are related to the group velocity Vg at k = ko by ? = ?o + Vg? + (dVg
Answer Preview: (a) To expand as (k) = o - xo, we can use Taylor expansion around the central wave number k=ko: (k) = (ko) + (d/dk)|ko * (k - ko) + . Since higher order terms are negligible, we neglect the higher ord…

, Chapter: 3 -Problem: 9 >> Derive Eq. (3.43) for the electron pressure in a nonrelativistic, electron-degenerate hydrogen gas. Transcribed Image Text: 5/3 2/3 3 m?c² P. = 2/10 (²) ²0 m² (™. P Pe 23 mp/23) (3.43)
Answer Preview: To derive Eq. (3 43) for the electron pressure in a nonrelativistic, electron-degenerate hydrogen ga…

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, Chapter: 8 -Problem: 12 >> (a) Suppose that you have two thin sheets with transmission functions t = g(x, y) and t = h(x, y), and you wish to compute via Fourier optics the convolution (b) Suppose you wish to convolve a large number of different pairs of 1-dimensional functions {g1h1}, {g2h2}, . . . simultaneously; that is, you want to compute for j = 1, 2, . . . . Devise a way to do this via Fourier optics using appropri
Answer Preview: (a) To compute the convolution gh(xo, yo), we can use the Fourier transform properties of thin lenses. Let F(kx, ky) be the Fourier transform of g(x, …

, Chapter: 3 -Problem: 3 >> (a) Cygnus X-1 is a source of X-rays that has been studied extensively by astronomers. The observations (X-ray, optical, and radio) show that it is a distance r ? 6,000 light-years from Earth. It consists of a very hot disk of X-ray-emitting gas that surrounds a black hole with mass ,and the hole in turn is in a binary orbit with a heavy companion star. Most of the X-ray photons have energies E ?
Answer Preview: (a) The mean occupation number of a photon state is given by Planck's law: n(E) = 1 / [exp(E/kT) - 1] where E is the photon energy, k is the Boltzmann …

, Chapter: 28 -Problem: 22 >> Polarization observations of the CMB provide an extremely important probe of fluctuations in the early universe.(a) By invoking the electromagnetic features of Thomson scattering by free electrons, give a heuristic demonstration of why a net linear polarization signal is expected.(b) Using the Monte Carlo formalism sketched in Sec. 28.6.1, calculate the polarization expected from a single fluctuat
Answer Preview: (a) Heuristic Demonstration of Net Linear Polarization Signal from CMB: The Cosmic Microwave Background (CMB) photons originated from the early universe when it was in a hot and ionized state. As the …

, Chapter: 14 -Problem: 11 >> Rooms are sometimes heated by radiators (hot surfaces) that have no associated blowers or fans. Suppose that, in a room whose air is perfectly still, a radiator is turned on to high temperature. The air will begin to circulate (convect), and that air motion contains vorticity. Explain how the vorticity is generated in terms of the ??dP/? term of Kelvin’s theorem (14.14) and the (??P × ??)/?2 term
Answer Preview: When a radiator is turned on to high temperature, the air in contact with the radiator gets heated a…

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, Chapter: 16 -Problem: 15 >> Consider the G string (196Hz) of a violin. It is ?30 cm from bridge to nut (the fixed endpoints), and the tension in the string is ?40 N.(a) Infer the mass per unit length in the string and estimate its diameter. Hence estimate the strain in the string before being plucked. Estimate the strain’s increase if its midpoint is displaced through 3 mm.(b) Now suppose that the string is released. Estimat
Answer Preview: (a) Mass per unit length and diameter of the string: To infer the mass per unit length () in the string, we can use the formula for wave speed on a string: Wave speed (v) = (T/) Where: T = Tension in …

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, Chapter: 9 -Problem: 3 >> A circularly symmetric light source has an intensity distribution I (?) = I0 exp[??2/(2?02)], where ? is the angular radius measured from the optic axis. Compute the degree of spatial coherence. What is the lateral coherence length? What happens to the degree of spatial coherence and the interference fringe pattern if the source is displaced from the optic axis?
Answer Preview: ANSWER The degree of spatial coherence (r1, r2) for a source with intensity distribution I() is give…

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, Chapter: 14 -Problem: 7 >> At time t = 0, a 2-dimensional barotropic flowhas a velocity field, in circular polar coordinates, v = (j/??)e? (Fig. 14.1b); correspondingly, its vorticity is ? = 2?j?(x)?(y)ez: it is a delta-function vortex. In this exercise you will solve for the full details of the subsequent evolution of the flow. (a) Solve the vorticity evolution equation (14.6) to determine the vorticity as a function of ti
Answer Preview: (a) The vorticity evolution equation in two-dimensional barotropic flow is given by: D/Dt = - (P*)/^…

, Chapter: 8 -Problem: 3 >> Sketch the Fraunhofer diffraction pattern you would expect to see from a diffraction grating made from three groups of parallel lines aligned at angles of 120° to one another (Fig. 8.7). Fig. 8.7.
Answer Preview: The Fraunhofer diffraction pattern from a diffraction grating made from three …

, Chapter: 2 -Problem: 5 >> Derive the relativistic component manipulation rules (2.23e)–(2.23g). Transcribed Image Text: [Contravariant components of T(____)S_?_)] = Tage (2.23e)
Answer Preview: To derive the relativistic component manipulation rules (2 23e)(2 23g), let's start with some background information. In general relativity, tensors are mathematical objects that describe physical qua…

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, Chapter: 4 -Problem: 17 >> Derive Eq. (4.70) for the average number of bits per symbol in a long message constructed from N distinct symbols, where the frequency of occurrence of symbol n is pn. Transcribed Image Text: N 1 = L?-Pn log? Pn³ n=1 (4.70)
Answer Preview: We can derive Eq. (4 70) using the concept of entropy in information theory. Suppose we have a long message composed of N distinct symbols, where the …

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, Chapter: 28 -Problem: 23 >> We have hitherto focused on the statistical properties of the cosmological perturbations as probed by a variety of observations. However, we on Earth occupy a unique location in a specific realization of wave modes that we have argued are drawn from a specific set of waves with particular amplitudes and phases, despite these supposedly being drawn from a statistical distribution (in much the same
Answer Preview: (a) To calculate the comoving radii of recombination, reionization, and the most distant galaxies and quasars, we need to use the standard cosmological model. In this model, the comoving distance is g…

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, Chapter: 2 -Problem: 20 >> By performing a 3 + 1 split on the geometric version of Maxwell’s equations (2.48), derive the elementary, frame-dependent version Data from Equation 2.48 Transcribed Image Text: V.E=4? ?? V.B = 0, V x B - VxE+ aB ?t 2E ?t = = 0. ???, (2.50)
Answer Preview: ANSWER To perform a 3 + 1 split on Maxwell's equations (2 48), we first write them in a form that se…

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, Chapter: 15 -Problem: 14 >> Consider the logistic equation (15.35) for the special case a = 1, which is large enough to ensure that chaos has set in. (a) Make the substitution xn = sin2??n, and show that the logistic equation can be expressed in the form ?n+1= 2?n (mod 1); that is, ?n+1 equals the fractional part of 2?n. (b) Write ?n as a “binimal” (binary decimal). For example, 11/16 = 1/2 + 0/4 + 1/8 + 1/16 has the binary
Answer Preview: (a) Substituting xn = sin2n into the logistic equation, we get: xn+1 = 4xn(1 xn) sin2n+1 = 4sin2n(1 …

, Chapter: 10 -Problem: 1 >> A device much ballyhooed in the United States during the presidency of Ronald Reagan, but thankfully never fully deployed, was a futuristic, superpowerful X-ray laser pumped by a nuclear explosion. As part of Reagan’s Strategic Defense Initiative (“StarWars”), this laser was supposed to shoot down Soviet missiles.How would you design a nuclear powered X-ray laser? The energy for the pump comes fro
Answer Preview: Designing a nuclear-powered X-ray laser is a complex task that requires a deep understanding of both nuclear physics and laser technology. To start, w…

, Chapter: 2 -Problem: 12 >> Show that the matrices (2.37a), with ? and ? satisfying Eq. (2.37b), are the inverses of each other, and that they obey the condition (2.35b) for a Lorentz transformation. Transcribed Image Text: [2] - ? ?? 0 0 ?? ? ? Y ? ? 0 ? 1 0 ? 1 Y -?? ? ? -?? Y ? ? ??? ? > 0 0 10 0 001 (2.37a)
Answer Preview: To show that the matrices (2 37a) are inverses of each other, we need to multiply them and sho…

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, Chapter: 28 -Problem: 26 >> There are many elaborations of standard cosmology either involving new features following from known physics or involving new physics. While no convincing evidence exists for any of the mas of this writing, they are all being actively sought. Explain how to generalize standard cosmology to accommodate these possibilities and to test for them, repeating calculations, where possible. (a) Space curva
Answer Preview: To generalize standard cosmology and accommodate the possibilities you mentioned, we can consider the following approaches: (a) Space curvature: In standard cosmology, space is assumed to be flat on l…

, Chapter: 7 -Problem: 10 >> Consider a simple refracting telescope (Fig. 7.7) that comprises two converging lenses, the objective and the eyepiece. This telescope takes parallel rays of light from distant stars, which make an angle ? ? 1with the optic axis, and converts them into parallel rays making a much larger angle M?. Here M is the magnification with ? negative, |M| >> 1, and |M?|? 1. (The parallel output rays are then
Answer Preview: ANSWER (a) To investigate how the output rays depend on the separation of the two lenses, we can use matrix methods. Let's denote the focal length of …

, Chapter: 4 -Problem: 15 >> Analyze Bose-Einstein condensation in a cubical box with edge lengths L [i.e., for a potential V (x, y, z) that is zero inside the box and infinite outside it]. In particular, using the analog of the text’s simplest approximation, show that the critical temperature at which condensation begins is and the number of atoms in the ground-state condensate, when T c0 , is
Answer Preview: To analyze Bose-Einstein condensation in a cubical box with edge lengths L, we can use the simplest approximation, known as the ideal gas approximation. In this approximation, we treat the particles a…

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, Chapter: 15 -Problem: 11 >> Consider an inviscid (? = 0), incompressible flow near a plane wall where a laminar boundary layer is established. Introduce coordinates x parallel to the wall and y perpendicular to it. Let the components of the equilibrium velocity be vx(y). (a) Show that a weak propagating-wave perturbation in the velocity, ?vy ? exp ik(x ? Ct), with k real and frequency Ck possibly complex, satisfies the diffe
Answer Preview: a) Starting with the given differential equation: d^4vy / dy^4 - k^4 vy = 0 Assume a solution of the …

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, Chapter: 15 -Problem: 13 >> Use a computer to calculate the first five critical parameters aj for the sequence of numbers generated by the logistic equation (15.35). Hence verify that the ratio of successive differences tends toward the Feigenbaum number F quoted in Eq. (15.36). Transcribed Image Text: Xn+1=4axn(1-xn). (15.3
Answer Preview: The logistic equation is given by: x(n+1) = rx(n)(1-x(n)) where x(n) is the population at time n, r …

, Chapter: 6 -Problem: 14 >> (a) Show that for shot noise, y(t) = ?i F(t ? ti), the spectral density Sy(f ) is given by Eq. (6.68b). Show that the relaxation time appearing in the correlation function is approximately the duration ?p of F(t). (b) Suppose the shapes of Fj (t ? tj) are all different instead of being identical but all last for times and all have the same Fourier transform at zero frequency, Show that the shot
Answer Preview: To solve this problem, we'll first derive the spectral density Sy(f) for shot noise in the given form y(t) = i F(t - ti). Then we'll show that the relaxation time appearing in the correlation function …

, Chapter: 3 -Problem: 10 >> Derive the equations of state (3.52) for an electron-degenerate hydrogen gas. Transcribed Image Text: EF = ? = m? cosh(t/4), PF= |C –m?=mesinh(t/4). (3.52a)
Answer Preview: To derive the equations of state for an electron-degenerate hydrogen gas, we need to con…

, Chapter: 18 -Problem: 6 >> Consider a small bubble of air rising slowly in a large expanse of water. If the bubble is large enough for surface tension to be ignored, then it will form an irregular cap of radius r. Show that the speed with which the bubble rises is roughly (gr)1/2. (A more refined estimate gives a numerical coefficient of 2/3.)
Answer Preview: To show that the speed with which the bubble rises is roughly proportional to (gr)^(1/2), we can use some basic principles of fluid mechanics. Let's c…

, Chapter: 28 -Problem: 3 >> Suppose that the universe contained a significant component in the form of isotropic but noninteracting particles with momentum p and rest mass m. Suppose that they were created with a distribution function f(p, a) ? p?q, with 4 < q < 5 extending from p ? m to p ? m.(a) Show that the pressure and the internal energy density (not including rest mass) of these particles are both well defined.(b) Sho
Answer Preview: To address the given questions, we will make use of the distribution function for the isotropic but noninteracting particles: f(p, a) p^(-q) where p is the momentum, a is the scale factor, and q is a …

, Chapter: 11 -Problem: 8 >> A homogeneous, isotropic, elastic solid is in equilibrium under (uniform) gravity and applied surface stresses. Use Eq. (11.30) to show that the displacement inside it, ?(x), is biharmonic, i.e., it satisfies the differential equation Show also that the expansion ? satisfies the Laplace equation Eq.11.30. Transcribed I
Answer Preview: To show that the displacement inside a homogeneous, isotropic, elastic solid is biharmonic and satisfies the differential equation mentioned, we'll start by considering Eq. (11 30): _ij,j + g_i = 0, …

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, Chapter: 11 -Problem: 21 >> By a computation analogous to Eq. (11.72), derive Eq. (11.78) for the components of the gradient of a second-rank tensor in any orthonormal basis. Transcribed Image Text: Vk (Šjej) = (Vkšj)ej + šj (Vkej) = j,kej + Šjljke. (11.72)
Answer Preview: To derive Eq. (11 78) for the components of the gradient of a second-rank tensor in any orthonormal …

, Chapter: 24 -Problem: 17 >> Consider a thin disk with radius R at z = 0 in a Lorentz reference frame. The disk rotates rigidly with angular velocity ?. In the early years of special relativity there was much confusion over the geometry of the disk: In the inertial frame it has physical radius (proper distance from center to edge) R and physical circumference C = 2?R. But Lorentz contraction dictates that, as measured on the
Answer Preview: To explore the issues raised in the exercise, we will analyze the geometry of the rotating disk in a Lorentz reference frame. (a) Consider a family of observers who ride on the edge of the disk. The w…

, Chapter: 9 -Problem: 22 >> Is it possible to construct an intensity interferometer (i.e., a number-flux interferometer) to measure the coherence properties of a beam of electrons? What qualitative differences do you expect there to be from a photon-intensity interferometer? What do you expect Eq. (9.59) to become? Transcribed Image Text:
Answer Preview: An intensity interferometer is a type of interferometer that measures the intensity correlation of l…

, Chapter: 1 -Problem: 2 >> Consider a particle moving in a circle with uniform speed v = |v| and uniform magnitude a = |a| of acceleration. Without introducing any coordinates or basis vectors, do the following.(a) At any moment of time, let n = v/? be the unit vector pointing along the velocity, and let s denote distance that the particle travels in its orbit. By drawing a picture, show that dn/ds is a unit vector that poi
Answer Preview: (a) At any moment of time, the particle moves with uniform speed v along the tangent to the …

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, Chapter: 7 -Problem: 6 >> Derive the quasi-spherical solution (7.42) of the vacuum scalar wave equation ??2?/?t2 + ?2? = 0 from the geometric-optics laws by the procedure sketched in the text. Transcribed Image Text: 4 B(Ct-r,0,0)i(Ct-r,0,0), r (7.42)
Answer Preview: The scalar wave equation in vacuum is given by: ^2/t^2 - ^2 = 0. To derive the quasi-spherical solution (7 42) from the geometric-optics laws, we start by assuming that the wave propagates in a medium …

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, Chapter: 4 -Problem: 21 >> Derive, or at least give a plausibility argument for, Landauer’s theorem.
Answer Preview: Landauer's theorem is a fundamental result in computer science and thermodynamics that relates the amount of energy dissipated by a computing device t…

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, Chapter: 10 -Problem: 15 >> A child, standing in a swing, bends her knees and then straightens them twice per swing period, making the distance ? from the swing’s support to her center of mass oscillate as is the swing’s mean angular frequency. (a) Show that the swing’s angular displacement from vertical, ?, obeys the equation of motion where ?1 = ?g?1, and ? is assumed to be small, ? «1. (b) Write ? = X1 cos ?0t + X2 sin
Answer Preview: (a) To analyze the motion of the child on the swing, let's consider the forces acting on her. The two main forces are the gravitational force mg acting downward and the tension T in the swing's ropes. …

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, Chapter: 11 -Problem: 20 >> Show how to construct a paraboloidal mirror of radius R and focal length f by stress polishing. (a) Adopt a strategy of polishing the stressed mirror into a segment of a sphere with radius of curvature equal to that of the desired paraboloid at its center, r = 0. By comparing the shape of the desired paraboloid to that of the sphere, show that the required vertical displacement of the stressed mir
Answer Preview: To construct a paraboloidal mirror using stress polishing, we'll follow the given strategy step by step. (a) We start by comparing the shape of the desired paraboloid to that of the sphere. The equati…

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, Chapter: 9 -Problem: 6 >> An example of a Michelson interferometer is the Far Infrared Absolute Spectrophotometer (FIRAS) carried by the Cosmic Background Explorer satellite (COBE). COBE studied the spectrum and anisotropies of the cosmic microwave background radiation (CMB) that emerged from the very early, hot phase of our universe’s expansion. One of the goals of the COBE mission was to see whether the CMB spectrum real
Answer Preview: To show the visibility of the fringes for a Wien spectrum in a Michelson interferometer, we can start with the equation for the intensity of the inter…

, Chapter: 2 -Problem: 4 >> In Newtonian theory, the gravitational potential ? exerts a force F = dp/dt = ?m?? on a particle with mass m and momentum p. Before Einstein formulated general relativity, some physicists constructed relativistic theories of gravity in which a Newtonian-like scalar gravitational field ? exerted a 4-force F(vector) = dp(vector)/d? on any particle with rest mass m, 4-velocity u(vector), and 4-moment
Answer Preview: (i) In order to obey the Principle of Relativity, the force law must be a scalar, i e., it must be i…

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, Chapter: 7 -Problem: 23 >> A Martian Rover is equipped with a single gyroscope that is free to pivot about the direction perpendicular to the plane containing its wheels. To climb a steep hill on Mars without straining its motor, it must circle the summit in a decreasing spiral trajectory. Explain why there will be an error in its measurement of North after it has reached the summit. Could it be programmed to navigate corre
Answer Preview: ANSWER The gyroscope in the Martian Rover will measure the rotation rate around the direction perpen…

, Chapter: 11 -Problem: 17 >> Buckling plays a role in many natural and human-caused phenomena. Explore the following examples.(a) Mountain building. When two continental plates are in (very slow) collision, the compressional force near their interface drives their crustal rock to buckle upward, producing mountains. Estimate how high such mountains can be on Earth and on Mars, and compare your estimates with their actual heigh
Answer Preview: (a) Mountain building: Mountain building is a natural phenomenon that occurs when two tectonic plate…

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, Chapter: 5 -Problem: 18 >> Modify your computer program from Ex. 5.17 to deal with the 2-dimensional Ising model augmented by an externally imposed, uniform magnetic field [Eqs. (5.80)]. Compute the magnetization and the magnetic susceptibility for wisely selected values of moB/J and K = J/(kBT ). Data from Exercises 5.17 Write a simple computer program to compute the energy and the specific heat of a 2-dimensional Isin
Answer Preview: Certainly! I can help you modify your computer program to deal with the 2-dimensional Ising model augmented by an externally imposed, uniform magnetic field. Here's an outline of the modifications you …

, Chapter: 2 -Problem: 7 >> (a) Convert the following expressions and equations into geometric, index-free notation (b) Convert T (___, S(R(C(vector), ___), ___), ___) into slot-naming index notation. Transcribed Image Text: Aa Bysi AaBys; SBY = SYBA B A B g B = Y ? B.
Answer Preview: (a) The following expressions and equations can be converted into geometric, index-free notat…

, Chapter: 25 -Problem: 17 >> (a) Derive the Bianchi identity (25.70) in 4-dimensional spacetime. (b) By contracting the Bianchi identity (25.70) on ??????????, derive the contracted Bianchi identity (25.69). Equations. Transcribed Image Text: V.G=0, (25.69)
Answer Preview: The Bianchi identity and its contracted form you mentioned are related to the Riemann curvature tens…

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, Chapter: 4 -Problem: 2 >> Make rough estimates of the entropy of the following systems, assuming they are in statistical equilibrium. (a) An electron in a hydrogen atom at room temperature. (b) A glass of wine. (c) The Pacific ocean. (d) An ice cube. (e) The observable universe.
Answer Preview: (a) An electron in a hydrogen atom at room temperature can be approximated as a system in thermal equilibrium. The energy levels of the hydrogen atom …

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, Chapter: 15 -Problem: 16 >> One of the first discoveries of chaos in a mathematical model was by Lorenz (1963), who made a simple model of atmospheric convection. In this model, the temperature and velocity field are characterized by three variables, x, y, and z, which satisfy the coupled, nonlinear differential equations (The precise definitions of x, y, and z need not concern us here.) Integrate these equations numericall
Answer Preview: To integrate the Lorenz equations numerically, we can use a numerical integration method such as the …

, Chapter: 2 -Problem: 24 >> Consider a fluid with 4-velocity u(vector) and rest-mass density ?o as measured in the fluid’s rest frame.(a) From the physical meanings of u(vector), ?o, and the rest-mass-flux 4-vector S(vector)rm, deduce Eqs. (2.62). (b) Examine the components of S(vector)rm in a reference frame where the fluid moves with ordinary velocity v. Show that Explain the physical interpretation of these formulas in t
Answer Preview: ANSWER. (a) The physical meaning of the 4-velocity u(vector) is that it represents the direction and speed of motion of a fluid element in spacetime. The rest-mass density o is a measure of the amount …

, Chapter: 16 -Problem: 7 >> Consider deep-water gravity waves of short enough wavelength that surface tension must be included, so the dispersion relation is Eq. (16.14). Show that there is a minimum value of the group velocity, and find its value together with the wavelength of the associated wave. Evaluate these for water (? ? 0.07 N m?1= 70 dyne/cm). Try performing a crude experiment to verify this phenomenon.
Answer Preview: To find the minimum value of the group velocity for deep-water gravity waves considering surface tension, we need to use the dispersion relation: = gk …

, Chapter: 5 -Problem: 16 >> Estimate how long it would take a personal computer to calculate the partition function for a 32 × 32 Ising lattice by evaluating every possible state.
Answer Preview: The partition function for an Ising lattice represents the sum of the Boltzmann factors of every pos…

, Chapter: 12 -Problem: 11 >> The magnitude M of an earthquake, on modern variants of the Richter scale, is a quantitative measure of the strength of the seismic waves it creates. The earthquake’s seismic-wave energy release can be estimated using a rough semi-empirical formula due to Bath (1966): The largest earthquakes have magnitude ?9.5. One type of earthquake is caused by slippage along a fault deep in the crust. Suppose
Answer Preview: Given: Magnitude of earthquake, M = 8 5 Duration of the earthquake, T = 100 s Average wave speed, C The equation for seismic power released, P = (4 9 …

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, Chapter: 7 -Problem: 5 >> Consider sound waves propagating in an atmosphere with a horizontal wind. Assume that the sound speed C, as measured in the air’s local rest frame, is constant. Let the wind velocity u = uxex increase linearly with height z above the ground: ux = Sz, where S is the constant shearing rate. Consider only rays in the x-z plane. (a) Give an expression for the dispersion relation ? = ?(x, t ; k). (b) S
Answer Preview: (a) The dispersion relation relates the wave frequency to the wave vector k. In this problem, the wave vector is in the x-z plane and has two components, kx and kz. Assuming the sound speed C is const…

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, Chapter: 7 -Problem: 11 >> A microscope takes light rays from a point on a microscopic object, very near the optic axis, and transforms them into parallel light rays that will be focused by a human eye’s lens onto the eye’s retina (Fig. 7.8). Use matrix methods to explore the operation of such a microscope. A single lens (magnifying glass) could do the same job (rays from a point converted to parallel rays). Why does a micr
Answer Preview: ANSWER A microscope uses two lenses to magnify an object because each lens has a limit to its magnification power. The first lens, called the objectiv…

, Chapter: 12 -Problem: 8 >> Derive the junction condition [Tjz] = 0 at a horizontal discontinuity between two media by the same method as one uses in electrodynamics to show that the normal component of the magnetic field must be continuous: Integrate the equation of motion ?dv/dt = ?? · T over the volume of an infinitesimally thin pill box centered on the boundary (see Fig. 11.7), and convert the volume integral to a surfac
Answer Preview: We may take a similar approach to electrodynamics to demonstrate that the normal component of the magnetic field must be continuous in order to derive …

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, Chapter: 10 -Problem: 8 >> Derive Eqs. (10.34b) and (10.34c) for the amplitudes of waves 1 and 2 produced by three-wave mixing. Transcribed Image Text: dA(¹) dz dA (²) dz = i^/ / dijk(³) A(?) at w? = 03 — @?, k? = k3 — k?i jkAA(2)* = ikdijkA@40* (3) (1)* - at @? = @3@?, K?=k3 - k?. (10.34b) (10.34c)
Answer Preview: Three-wave mixing is a nonlinear optical process that involves the interaction of three waves: a pump wave at frequency p, a signal wave at frequency …

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, Chapter: 18 -Problem: 5 >> The density and temperature in the deep interior of the Sun are roughly 0.1 kgm?3 and 1.5 × 107 K.(a) Estimate the central gas pressure and radiation pressure and their ratio.(b) The mean free path of the radiation is determined almost equally by Thomson scattering, bound-free absorption, and free-free absorption. Estimate numerically the photon mean free path and hence estimate the photon escape
Answer Preview: (a) To estimate the central gas pressure and radiation pressure, we can use the ideal gas law and the Stefan-Boltzmann law. Gas Pressure: The gas pressure can be estimated using the ideal gas law, whi…

, Chapter: 7 -Problem: 3 >> (a) In connection with Eq. (7.35b), explain why is the tiny volume occupied by a collection of the wave’s quanta. (b) Choose for the collection of quanta those that occupy a cross sectional area A orthogonal to a chosen ray, and a longitudinal length ?s along the ray, so V = A?s. Show that d ln ?s/dt = d ln C/dt and correspondingly, (c) Given part (b), show that the conservation law (7.35b) is e
Answer Preview: To understand the connections between the given equations, let's go through each part step by step: (a) In Eq. (7 35b), the equation states that the rate of change of momentum flux density, Vg, is equ…

, Chapter: 11 -Problem: 9 >> (a) Verify Eqs. (11.46) for the sag in a horizontal beam clamped at one end and allowed to hang freely at the other end. (b) Now consider a similar beam with constant cross section and loaded with weights, so that the total weight per unit length is W(x). What is the sag of the free end, expressed as an integral over W(x), weighted by an appropriate Green’s function? Data from Eqs. (11.46)
Answer Preview: (a) The sag in a horizontal beam clamped at one end and allowed to hang freely at the other end can be found using Eqs. (11 46). First, let's conside…

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, Chapter: 6 -Problem: 22 >> (a) Wr

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