Tiple...4ª Ed - 29997852 - chapter - 3 - solutions - modern - physics - 4th - edition

Chapter 3 ! Quantization of Charge, Light, and Energy 3-1.The radius of curvature is given by Equation 3-2.

Substituting particle masses from Appendices A and D:

3-2.(a) Using mass number to compute an approximate value for B that will yield R of one meter, Equation 3-2 gives

(b) Using Equation 3-2,

(c) For doubly ionized atoms so and is unchanged because as before.

Chapter 3 ! Quantization of Charge, Light, and Energy

3-4. and

3-5. (a)

(b) period

3-6.(a)

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-6 continued) (b)

(c)

where the ni are integers. Assuming the smallest )n = 1, then )n12 = 3.0, )n23 = 2.0, )n43 =

1.0, )n45 = 4.0, and )n65 = 1.0. The assumption is valid and the fundamental charge implied is

3-9.For the rise time to equal the field-free fall time, the net upward force must equal the weight. .

3-10.(See Millikan’s Oil Drop Experiment on the home page at w.whfreeman.com/modphysics4e.) The net force in the y-direction is .

 The net force in the x-direction is At terminal speed and

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-10 continued)

 (a) At terminal speed where and Substituting gives

(b)

 (b)

3-12.. (a)

(c) 3-13.Equation 3-10: . Equation 3-12: . From Example 3-5:

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-13 continued) 3-14.Equation 3-24: ,

3-15. (a)

(b) (c) Equation 3-12:

Area of Earth . Total power =

Chapter 3 ! Quantization of Charge, Light, and Energy

3-16. (b) (c)

3-17.Equation 3-10:

3-18.(a) Equation 3-23:

(b) .

Equipartition theorem predicts The long wavelength value is very close to kT, but the short wavelength value is much smaller than the classical prediction.

3-19.(a)

(b) 3-20. (Equation 3-20)

Chapter 3 ! Quantization of Charge, Light, and Energy

 where RE = radius of Earth

3-21.Equation 3-10:

3-2.(a)

(b) Each photon has average energy E = hf and NE = 40 J/s.

(c)At 5m from the lamp N photons are distributed uniformly over an area The density of photons on that sphere is . The area of the pupil of the eye is so the number n of photons entering the eye per second is

Chapter 3 ! Quantization of Charge, Light, and Energy

 3-23.Equation 3-24: Letting

The maximum corresponds to the vanishing of the quantity in brackets. Thus, . This equation is most efficiently solved by iteration; i.e., guess at a value for B/8 in the expression, solve for a better value of B/8; substitute the new value to get an even better value, and so on. Repeat the process until the calculated value no longer changes. One succession of values is: 5, 4.966310, 4.965156, 4.965116, 4.965114, 4.965114. Further iterations repeat the same value (to seven digits), so we have

3-25.(a)

(b)It is the fraction of the total solar power with wavelengths less than 255 nm, i.e., the area under the Planck curve (Figure 3-7) up to 255 nm divided by the total area. The latter

Chapter 3 ! Quantization of Charge, Light, and Energy

 is: Approximating the former

(Problem 3-25 continued) with with :

3-26.(a)

(b) (c)

3-27.(a) Choose 8 = 550 nm for visible light.

(b)

Chapter 3 ! Quantization of Charge, Light, and Energy

3-28.(a)

(b)

Available energy'photon This is less than N.

3-29.(a)

(b) (c)

3-30.Using Equation 3-36, (1)

(2)

Subtracting (2) from (1),

Solving for h yields: h = 6.56 × 10!34 J@s Substituting h into either (1) or (2) and solving for N/e yields: N/e = 1.87 eV. Threshold frequency is given by hf/e = N/e or

Chapter 3 ! Quantization of Charge, Light, and Energy

3-31.

3-32.(a) (b) 3-3.Equation 3-31:

3-34.Equation 3-30:

3-35.(a)

(b)

(c) (d)

Chapter 3 ! Quantization of Charge, Light, and Energy

 Electron recoil energy (Conservation of energy)

. The recoil electron momentum makes an angle 2 with the direction of the initial photon.

(Conservation of momentum) 3-37.

Chapter 3 ! Quantization of Charge, Light, and Energy

3-38.(a)

(b) (c) (d)

Chapter 3 ! Quantization of Charge, Light, and Energy

3-41.(a) Compton wavelength = electron:

proton:

(b) (i) electron: (i) proton:

 3-42.(a)

Multiplying (i) by 450 nm/e and (i) by 300 nm/e, then subtracting (i) from (i) and rearranging gives

(b)

3-43. For small 2, Substituting for uy,

Chapter 3 ! Quantization of Charge, Light, and Energy

3-4.Including Earth’s magnetic field in computing y2, Equation 3-6 becomes where the second term in the brackets comes from and Thus, The first term inside the brackets is the reciprocal of , Thomson’s value for e/m. Using Thomson’s data (B = 5.5x10!4

T, õ = 1.5x104 V/m, x1 = 5 cm, y2/x2 = 8/110) and the modern value for e/m = 1.76x1011 C/kg and solving for BE:

. The minus sign means that B and BE are in opposite directions, which is why Thomson’s value underestimated the actual value.

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-45 continued) (a) The intercept on the vertical axis is the work function N. N = 2.08 eV (b)The intercept on the horizontal axis corresponds to the threshold frequency.

(c) The slope of the graph is h/e. Using the vertical intercept and the largest experimental point,

 momenta
 By conservation of momentum, the final state is an electron at rest, Conservation of

3-46.In the center of momentum reference frame, the photon and the electron have equal and opposite The total energy is: . energy requires that the final state energy is

Squaring yields,

. This can be true only if EK vanishes identically, i.e., if there is no photon at all.

Chapter 3 ! Quantization of Charge, Light, and Energy

 3-47.Bragg condition: This

67 is the minimum wavelength 8m that must be produced by the X ray tube.

3-48.(a) The momentum p absorbed is

(b)

(c)

The difference in energy has been (i) used to increase the object’s temperature and (i) radiated into space by the blackbody.

3-49.Conservation of energy: From Compton’s equation, we have:

Substituting this expression for f2 into the expression for Ek (and dropping the subscript on f1):

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-49 continued)

Ek has its maximum value when the photon energy change is maximum, i.e., when so . Then

3-50.(a)

(b) Equation 3-24: where and

Similarly,

3-51.Fraction of radiated solar energy in the visible region of the spectrum is the area under the Planck curve (Figure 3-7) between 350 nm and 700 nm divided by the total area. The latter is 6.42×107 W/m2 (see solution to Problem 3-25). Evaluating with 8 = 525 nm (midpoint of visible) and )8 = 700 nm ! 350 nm = 350 nm,

Fraction in visible =

Chapter 3 ! Quantization of Charge, Light, and Energy

f (x1014 Hz)

3-52.(a) Make a table of

The work function for Li (intercept on the vertical axis) is N = 2.40 eV. (b) The slope of the graph is h/e. Using the largest V0 and the intercept on the vertical axis,

(c) The slope is the same for all metals. Draw a line parallel to the Li graph with the work function (vertical intercept) of Pb, N = 4.14 eV. Reading from the graph, the threshold frequency for Pb is 9.8x1014 Hz; therefore, no photon wavelengths larger than will cause emission of photoelectrons from Pb.

Chapter 3 ! Quantization of Charge, Light, and Energy

3-53.(a) Equation 3-24: Letting gives

(b)

The maximum corresponds to the vanishing of the quantity in the brackets. Thus, .

(c)This equation is most efficiently solved by trial and error; i.e., guess at a value for a'8 in the expression , solve for a better value of a'8; substitute the new value to get an even better value, and so on. Repeat the process until the calculated value no longer changes. One succession of values is: 5, 4.966310, 4.965156, 4.965116, 4.965114, 4.965114. Further iterations repeat the same value (to seven digits), so we have

(d) . Therefore,

3-54.(a) (b) let the atom occupy an area of .

(c)

Chapter 3 ! Quantization of Charge, Light, and Energy

3-5.(a) The nonrelativistic expression for the kinetic energy of the recoiling nucleus is

Internal energy

(b) the nucleus must recoil with momentum equal to that of the emitted photon, about 14.98 MeV/c.

3-56.Derived in Problem 3-47, the electron’s kinetic energy at the Compton edge is

Thus,

Solving with the quadratic formula: (only the + sign is physically meaningful). Energy of the incident gamma ray hf = 708 keV.

3-57. (a)

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-57 continued)

(b) 3-58.Let in Equation 3-21:

where y = e!x. This sum is the series expansion of , i.e., Then gives A = 1!y. Writing Equation 3-2 in terms of x and y:

 Note that But , so we have

since . Multiplying this sum by hf and by A = (1!y), the average

Chapter 3 ! Quantization of Charge, Light, and Energy

(Problem 3-58 continued) energy is

Multiplying the numerator and the denominator by e!x and substituting for x, we obtain

, which is Equation 3-23.

Chapter 3 ! Quantization of Charge, Light, and Energy 74