Description
1) Charged Slab with width:
In the first workshop we used Gauss’ Law and symmetry considerations to calculate the electric field due
to an infinite, uniform, thin charged slab lying in the x‐y plane.
Consider a similar infinite slab, but with a spread‐out charge distribution perpendicular to the slap.
𝜌ሺ𝑟⃗ሻ ൌ 𝜌ሺ𝑧ሻ ൌ 𝜌 𝑒ିఈ|௭|
This might be used as a model for a nano‐scale metal surface where the charge distribution is related to
the “skin depth” or finite extension of the surface states of the electrons in the metal.
a) Explain why and how you can use Gauss’ Law to determine the electric field due to this slab. What is
the general form for the electric field? Justify your answer.
b) Describe carefully, and draw a picture, showing the volume and surface you will use to solve the
integrals in Gauss’ Law
c) Solve the two integrals (volume integral and surface integral) in Gauss’ Law to determine the electric
field everywhere.
d) Determine the electrostatic potential, 𝜙ሺ𝑟⃗ሻ, that corresponds to this electric field. (Hint: you can
either note that the potential is the integral of the field or guess and check a function for the potential
with the property that the negative of the gradient of the function gives the field.
e) Consider your results for the field and the potential in the limits as 𝑧→∞ and 𝑧 → 0. Do these limits
make physical sense, comparing them to the thin slab? Explain.
f) Note that your result disagrees with our symmetry arguments from Questions 1a, 1b, and 1c of
Workshop 1. Which of the symmetry arguments doesn’t work for this case and why?
2) Classical “Hydrogen Atom” Field:
If we consider the electron’s probability density in a hydrogen atom as a static, classical charge density (I
know, quite a stretch) we can make some classical predictions about the atom.
Consider a charge distribution of the form:
𝜌ሺ𝑟ሻ ൌ 𝜌 𝑒ିఈ
The charge distribution has a total charge 𝑄 and 𝛼 ൌ
ଶ
బ
where 𝑎 is the length scale in the problem. For
a hydrogen atom, 𝑄 ൌ െ𝑒 and 𝑎 is the Bohr radius.
a) Determine what 𝜌 needs to be in terms of 𝑄 and 𝑎. Be sure to explain how you determined your
answer.
Hint: There is a standard trick for doing these sorts of integrals:
න 𝑥 𝑒ିఈ௫ 𝑑𝑥 ൌ ሺെ1ሻ 𝜕
𝜕𝛼 න 𝑒ିఈ௫ 𝑑𝑥
Of course, if you’re not in a test (or qualifier) you can always pull up Mathematica, Wolfram Alpha, or
whatever python library you might use.
b) Explain how you will use Gauss’ Law to determine the electric field everywhere for the atom. Draw a
picture to illustrate your approach.
c) Calculate both the volume and the surface integral in Gauss’ Law, showing your work. Use these
results to determine the electric field everywhere.
d) As always, we need to check our results. Show that your result for the field makes physical sense in
the limit that 𝑟 gets large.
NOTE: This means more than finding what happens if 𝑟ൌ∞. Check the functional behavior in the limit
as 𝑟→∞.
d) Check that your result also is well behaved in the limit as 𝑟 → 0. Remember that this charge
distribution is NOT due to a point charge, just the charged cloud.
3) Classical Hydrogen Atom potential:
The equation 16.27 in the textbook gives the classical hydrogen atom electric potential. This is the
potential of both the electron cloud and the proton of the hydrogen atom, modeled as a point charge.
(Note, this is in Gaussian Units while we’ve been using SI units in class. You should be able to handle this
difference.)
a) Consider your results to Problem 2. What is the total electric field of the atom, including both the
electron cloud (Problem 2) and the proton? Explain your work.
b) Show that your total electric field agrees with the potential given in 16.27.
c) Determine the electrostatic potential energy of the electron cloud of the hydrogen atom. Use the
total electrostatic potential but don’t include the energy due to the proton.
4) The “Dipole Sphere”:
The electric potential and electric field of a sphere with a uniform charge on its surface is a common,
simple example done in introductory physics classes. Let’s change this up a bit for some more useful
practice.
a) As a reminder, write down the electric potential and electric field for a sphere of radius 𝑅 centered at
the origin, 𝑟⃗ ൌ 0. Include results for both 𝑟𝑅 and 𝑟൏𝑅. You don’t have to actually solve this, but it
would be good practice just to be sure you can do it.
Next, consider the same sphere of radius 𝑅 but now the surface charge is
split between the upper and lower hemispheres.
The upper hemisphere (𝑧 0 in Cartesian Coordinates, 0 𝜃 గ
ଶ
in
Spherical Coordinates) has a constant positive surface charge density,
𝜎. The lower hemisphere (𝑧 ൏ 0, గ
ଶ 𝜃𝜋) has a constant negative
surface charge density, െ𝜎. The magnitudes of the charge densities are
the same.
b) Sketch what you think the electric field will look like in the x‐z plane shown.
c) Explain why you can’t directly use Gauss’ Law to solve for the field of this charge distribution.
d) Using a direct integration, solve for the electric potential everywhere on the z‐axis. This includes 𝑧
𝑅, 𝑧 ൏ െ𝑅, and െ𝑅൏𝑧൏𝑅. Of course, show your work.
e) Calculate the electric field along the z‐axis, 𝐸ሬ⃗ሺ𝑧ሻ ൌ 𝐸ሺ𝑧ሻ 𝑒̂
௭ using your result for the potential.
f) The properties of the potential and field of this “Dipole Sphere” are quite different from those of the
uniformly charge sphere. What are some of the differences? Explain the physics of these differences.
For example, you might consider what happens for |𝑧| → ∞, |𝑧| → 𝑅, and the field and potential in the
interior of the spheres.
x
z
𝝈
െ𝝈
