Reading: Griffiths, Chapter 2, sections 3-5.

You may work together and get help from other students. Your solutions must be written in your own words, without looking at someone else’s solutions while you write them.

Don’t forget the 9 points that we are looking for in your solutions (see Moodle).

In order to make sure you get your context and meaning/make sense points, next to your answers, please put a “c” with a circle around it for context, and an “m” with a circle around it for meaning.

  1. Griffiths, Chapter 2, Problem 2.21

  2. Griffiths, Chapter 2, Problem 2.34, parts a) and b only; and Problem 2.35.

  3. At an early stage in the development of the atomic theory, J. J. Thompson proposed an atom consisting of a positive charge Ze spread uniformly throughout a sphere of radius \(R\) where Z is the number of electrons each with charge -e and e is the fundamental unit of charge (\(1.6 \times 10^{-19}\) Coulombs). The point-like electrons are embedded in the uniform positive charge like raisins in raisin pudding, hence the name “raisin pudding” model of the atom. a) Find the force acting on one electron as a function of its distance \(r\) from the center of the sphere. Assume that the charge of the remaining electrons is smeared out uniformly throughout the sphere. b) What type of motion does the electron execute in the radial direction? c) What is the frequency of this motion for a typical atomic radius of 1 angstrom (0.1 nm)? d) What is the frequency of an electron that is in a circular orbit about the center? Interesting, no? e) How does this frequency compare with that of the fundamental Lyman \(\alpha\) transition in hydrogen? The wavelength of a Lyman \(\alpha\) photon is 121.6 nm.

  4. You are going to find the charge distribution (i.e., the charge density) that (mostly) produces the Yukawa potential, i.e.,

\[V(r) = \frac{A}{r}e^{-r/a}\]

The Yukawa potential was created to model the Nuclear Force, which drops off very quickly with distance. That’s why that exponential is there.

a) Make a plot of the Yukawa potential and the Coulomb potential (A/r) on the same plot. Show that the Yukawa potential is really close to the Coulumb potential for short distances. Show that at long distances the Yukawa potential is very different than the Coulomb potential. You might need two plots to show both these things.

b) Pretend the Yukawa potential obeys Maxwell’s equations, and find the electric field given by this potential.

c) Use Poisson’s equation (or better yet, start with your answer to part b and then use the differential form of Gauss’ law) to find the charge density that would yield this potential. (Careful, \(\nabla^2\) is the divergence of the gradient, so you need to look up the divergence in spherical coordinates.)

d) (I changed the question a little because it was confusing, but you can use all the same math did in the previous version of d. ) Integrate \(\rho\) over a spherical volume of radius \(r\). Use Gauss’ law to calculate the field at a radius \(r\). It’s not the same field you found in part (b) right?
In your “meaning” description, discuss how Yukawa’s potential violates Maxwell’s equations and is therefore unphysical (not possible with E&M as we understand it.) You may wish to discuss which way the field should point given the sign of the charge density you found in part (c), and which way the field points given the Yukawa potential (you can look at your answer to part (a) and/or remember that positive charges fall down in electric potential).

e) Plot the field you found in part (b) and the field you found in part(d). Given what Yukawa was going for, which field was closer to what he wanted, i.e. which one shows a field that is better “shielded” at short distances?

Previous versions of (d) and (e) are shown below for posterity.

previous version of d) Use the integral form of Gauss’s law to show that this overal charge distribution is not neutral. My suggestion is to integrate \(\rho\) over a spherical volume of radius \(r\), and then let \(r\) tend toward infinity. So how big a point charge do you need at the origin in order to make the charge distribution neutral?

previous version of e) Make a plot of the electric field due to the charge distribution \(\rho\) both before and after you add the point charge. (Hint: You already have a formula for \(Q_{enclosed}\) from part d, so you’re only moments away from knowing what the E-field due to that charge distribution is. You can superpose the E-field of a point charge on top of that.)