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Problem 5 Medium Difficulty

The “size” of the atom in Rutherford’s model is about $1.0 \times 10^{-10} \mathrm{m} .$ (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 be considered in studying the atom?

Answer

a) $1.6 \times 10^{6} \mathrm{m} / \mathrm{s}$
b) No
c) $4.5 \times 10^{-10} m$
d) Yes

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Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

Zachary M.

Hope College

Jared E.

University of Winnipeg

Video Transcript

in question. A. We have to calculate the orbit of speed V oven electric around a protein given that the radius. But this is between the election and the problem is 10 to the minus 10 meters. Uh, and we have to do that using the electric static force. We know that the electrostatic forces, even between an electron approach, is getting by k times the chart of the elections square time divided by the radio square. And since the electron is moving in a circular motion, this is equal to end v squared over R, where m is the mass of the electorate. And from here you got the V is equal to e times the square root of K over in our, so you can substitute the numbers. Here he is one point c six times sent to the minors 19 Coombs times the square root of K, which is nine times 10 to the ninth Newtons meters per square Brooklyn square divided by the mass of the electron, which is 9.1 times 10 to the minus 31 kilograms. Times are, which is 10 to the minus 10 meters. So this is equal to 1.6 times 10 to the sixth. Neither is per second. This is the answer to question a in question be we have to. We have to decide whether we should use Ah, this special relativity theory when dealing with this with the electric at this speed that we calculated in question, they and we know that whether we should or used a special relativity or not depends on the value of beta, which is defined as of you oversee, because if it is much smaller than one, then we shouldn't use special relativity. Okay, And let's see if, in our case view ever see, is much more than what. So let's go play. Beta beta is able to 1.6 times said to the six meters per second, which is our speed divided by this bit of life, which is three times 10 to the eight meters per second. And this is approximately five times said of the minus three, which is much smaller than one. Then we don't need to use, uh, special relativity. Okay, in questions. See, we have to calculate the really wavelength off the electoral, and we know that the Beverly Wavelength Lunda is people to h over the momentum of the particle. And since our particles known relativistic as I just explained in question be the momentum is just envy. And let's insert numbers. You're so ages 61 63 times in the minus 34 Jules Second I m is the mass of the electron, which is 9.1 time sentinel minus 31 kilograms and V is 1.6 times 10 to the sixth meters per second, which is equal to 4.5 time. Stand to the minus 10 meters. Hey, this is the wavelength literally wavelength of the election. And in question D, we have to check whether we should expect the fraction phenomenon interference phenomena, Uh, given that the wavelength of the electric is 4.5 times 10 to the minus 10 meters and what we should notice is that the radius of the atom itself, the size of the atom is actually I'm not gonna write our I'm gonna write just d. That's the side of the atom. It's about 10 to the minus 10th meters. Okay, that's the So that's the typical size of the atoms. Tend to the minister and the wavelength of the electorate is of the order is about. In this case, it's equal to 4.5 times into the minus 10 meters. Okay, so notice that the to both the size of the Adam and the wavelength of the election are about the same size. They have the same order of magnitude. And for that reason we can expect to see wave phenomena happening with the Elektrim, such as interference and diffraction. So here the question is yes to

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

Andy C.

University of Michigan - Ann Arbor

Zachary M.

Hope College

Jared E.

University of Winnipeg