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The size of the atom in Rutherfords, model is about 1.0 X 10-10 m. (a) Determine the speed of an electron moving about the proton using the attractive electrostatic force between an electron and a proton separated by this distance. (b) Does this speed suggest that Einsteinian relativity must be considered in studying the atom? (c) Compute the de Broglie wavelength of the electron as it moves about the proton, (d) Does this wavelength suggest that wave effects, such as diffraction and interference, must he considered in studying the atom?

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Indranil Banerjee
Indranil BanerjeeLv3
13 Nov 2020

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

3. a) Show that the phase velocity of the de Broglie waves of aparticle of mass m and de Broglie wavelength ? is given by .
b) Compare the phase and group velocities of an electron whose deBroglie wavelength is exactly 1x10-13 m.
4. To what voltages must we accelerate electrons (as in an electronmicroscope, for example) if we wish to resolve;
a) a virus of diameter 12 nm , b) an atom of 0.12 nm, c) a protonof diameter 1.2 fm?
5. a) The x-coordinate of an electron is measured with anuncertainty of 0.20 mm. What is the x- component of the electron’svelocity, vx, if the minimum percentage uncertainty in asimultaneous measurement of vx is 1.0%.
b) Repeat part (a) for a proton. 6. Suppose Fuzzy, a quantummechanical duck, lives in a world in which h = 2? J.s. Fuzzy has amass
of 2.0 kg and is initially known to be within a region 1.0 mwide.
a) What is the minimum uncertainty in his speed? b) Assuming thisuncertainty in speed to prevail for 5.0 s, determine theuncertainty in position after
this time.
7. The seeing ability, or resolution, of radiation is determined byits wavelength. If size of an atom is of the order of 0.1 nm, howfast must an electron travel to have a wavelength small enough to“see” an atom?
8. An air rifle is used to shoot 1.0 g particles at 100 m/s througha hole of diameter 2.0 mm. How far from the rifle must an observerbe to see the beam spread by 1.0 cm because of the uncertaintyprinciple? Compare this answer with the diameter of the universe (? 1026 m).

In 1924, Louis de Broglie postulated that particles such aselectrons and protons might exhibit wavelike properties. Histhinking was guided by the notion that light has both wave andparticle characteristics, so he postulated that particles such aselectrons and protons would obey the same wavelength-momentumrelation as that obeyed by light: \lambda = h/p, where lambda isthe wavelength, p the momentum, and h Planck's constant.

a. Find the de Broglie wavelength lambda for an electron moving ata speed of 1.00 \times 10^6 \; {\rm m/s}. (Note that this speed islow enough that the classical momentum formula p=mv is stillvalid.) Recall that the mass of an electron is m_{\rm e} =9.11\times 10^{-31}\; {\rm kg}, and Planck's constant is h = 6.626\times 10^{-34}\; {\rm J \cdot s}.
Express your answer in meters to three significant figures.
lambda =7.270×10-10 \rm m

b. Find the de Broglie wavelength lambda of a baseball pitched at aspeed of 43.7 m/s. Assume that the mass of the baseball is 0.143\;{\rm kg}.
Express your answer in meters to three significant figures
lambda =1.06×10-34 \rm m

c. Consider a beam of electrons in a vacuum, passing through a verynarrow slit of width 2.00 \;\mu{\rm m}. The electrons then headtoward an array of detectors a distance 1.076 m away. Thesedetectors indicate a diffraction pattern, with a broad maximum ofelectron intensity (i.e., the number of electrons received in acertain area over a certain period of time) with minima of electronintensity on either side, spaced 0.518 cm from the center of thepattern. What is the wavelength lambda of one of the electrons inthis beam? Recall that the location of the first intensity minimain a single slit diffraction pattern for light is y=L \lambda /a,where L is the distance to the screen (detector) and a is the widthof the slit. The derivation of this formula was based entirely uponthe wave nature of light, so by de Broglie's hypothesis it willalso apply to the case of electron waves.
Express your answer in meters to three significant figures.

I only need C... please answer fully, thank you

A magnetic field of 0.1T forces a proton beam of 1.5 mA to move ina circle of radius 0.1 m. The plane of the circle is perpendicularto the magnetic field.

47. Of the following, which is the best estimate of the speed of aproton in the beam as it moves in the circle?
(A) 10-2 m/s (B) 103 m/s (C) 106 m/s (D) 108 m/s (E) 1015 m/s

52. An ideal gas is initially in a state that corresponds to point1 on the graph above, where it has pres¬sure p1, volume V1, andtemperature T1. The gas undergoes an isothermal process representedby the curve shown, which takes it to a final state 3 attemperature T3. If T2 and T4 are the tempera¬tures the gas wouldhave at points 2 and 4, respec¬tively, which of the followingrelationships is true?
(A) T1 < T3 (B) T1 < T2 (C) T1 < T4 (D) T1 = T2 (E) T1 =T4

53. The absolute temperature of a sample of monatomic ideal gas isdoubled at constant volume. What effect, if any, does this have onthe pressure and density of the sample of gas?
Pressure Density
(A) Remains the same Remains the same
(B) Remains the same Doubles
(C) Doubles Remains the same
(D) Doubles Is multiplied by a factor of 4
(E) Is multiplied by a factor of 4 Doubles



62. A sample of an ideal gas is in a tank of constant volume. Thesample absorbs heat energy so that its temperature changes from 300K to 600 K. If v1 is the average speed of the gas molecules beforethe absorption of heat and v2 is their average speed after theabsorption of heat, what is the ratio v2/ v1 ?
(A) 1/2 (B) 1 (C) (D) 2 (E) 4

63. Two people of unequal mass are initially standing still on icewith negligible friction. They then simul¬taneously push each otherhorizontally. Afterward, which of the following is true?
(A) The kinetic energies of the two people are equal.
(B) The speeds of the two people are equal.
(C) The momenta of the two people are of equal magnitude.
(D) The center of mass of the two person system moves in thedirection of the less massive person.
(E) The less massive person has a smaller initial acceleration thanthe more massive person.

64. Two parallel conducting plates, separated by a distance d, areconnected to a battery of emf e. Which of the following is correctif the plate separa¬tion is doubled while the battery remainsconnected?
(A) The electric charge on the plates is doubled.
(B) The electric charge on the plates is halved.
(C) The potential difference between the plates is doubled.
(D) The potential difference between the plates is halved
(E) The capacitance is unchanged.

68. A rod on a horizontal tabletop is pivoted at one end and isfree to rotate without friction about a vertical axis, as shownabove. A force F is applied at the other end, at an angle ? to therod. If F were to be applied perpendicular to the rod, at whatdistance from the axis should it be applied in order to pro¬ducethe same torque?
(A) L sin ? (B) L cos ? (C) L (D) L tan ? (E) L

70. A 4 µF capacitor is charged to a potential differ¬ence of 100V. The electrical energy stored in the capacitor is
(A) 2 x 10-10 J (B) 2 x 10 8 J (C) 2 x 10 6 J (D) 2x 10-4 J (E) 2 x10-2 J



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