PHY 712 -- Assignment #2
Assigned: 1/15/2025 Due: 1/17/2025
Continue reading Chapter 1 in Jackson.
- Jackson Problem #1.5. Be careful to take into account the behavior of Φ(r) for r-->0.
PHY 712 -- Assignment #5
Assigned: 1/27/2025 Due: 1/29/2025
Continue reading Chaps. 1-3 in Jackson.
- Consider a square of length a which has a uniform charge density ρ0. In Lecture 5
we found an "exact" expression for the electrostatic potential of this system Φ(x,y) for the case that
the potential vanishes on the 4 boundary lines Φ(x,0)=Φ(x,a)=Φ(0,y)=Φ(a,y).
- For the 5x5 grid discussed in Lecture 6 with h=a/4, evaluate the unique values of Φ(x,y) on the grid points by summing the series in the exact expressions.
- Using the finite difference method, find the approximate unique values of Φ(x,y) on the grid points and compare with the exact values.
- Using the finite element method, find the approximate unique values of Φ(x,y) on the grid points and compare with the exact values.
PHY 712 -- Assignment #6
Assigned: 1/27/2025 Due: 1/31/2025
Continue reading Chap. 2 in Jackson.
- Eq. 2.5 on page 59 of Jackson was derived as the surface change density on a sphere of radius a due to a charge q placed at a radius y > a outside the sphere. Determine the total surface charge on the sphere's outer surface.
- Now consider the same system except assume y < a representing the charge q being placed inside the sphere. What is the surface charge density and the total charge on the inner sphere surface in this case?
PHY 712 -- Assignment #8
Assigned: 2/3/2025 Due: 2/5/2025
Complete reading Chapter 3 and start Chapter 4 in Jackson .
- Consider the charge density of an electron bound to a proton in the ground state of a hydrogen atom -- ρ(r) = (1/πa03) e-2r/a0, where a0 denotes the Bohr radius. Find the electrostatic potential Φ(r) associated with ρ(r). Compare your result to the electrostatic potential given in HW#2.
PHY 712 -- Assignment #9
Assigned: 2/5/2025 Due: 2/7/2025
Continue reading Chapter 4 in Jackson .
-
Consider the localized charge given by
ρ(r,θ)=ρ0 r exp(-ar) cos(θ)
where ρ0 and a are positive constants. Find the electrostatic potential produced by this charge distribution. Examine your result in the limit r → 0 and also in the limit r → ∞ in terms of multipole moments and other interesting features.
PHY 712 -- Assignment #10
Assigned: 2/7/2025 Due: 2/10/2025
Finish reading Chapter 4 in Jackson .
- Work problem 4.9(a) in Jackson. Hint: It may be convenient to use a coordinate system with the origin at the center of the dielectric sphere. Also, you may benefit from considering the case where ε/ε0=1 to check that your expression makes sense.
PHY 712 -- Assignment #11
Assigned: 2/10/2025 Due: 2/12/2025
Start reading Chapter 5 (Sec. 5.1-5.5) in Jackson .
- Consider an infinitely long cylindrical wire with radius a,
oriented along the z axis. There
is a steady uniform current inside the wire. Specifically, in terms of
r the radial parameter of the cylindrical coordinates of the system
the current density is J(r)=J0 , where
J0 is a constant vector pointing along the z-axis,
for r ≤ a and zero otherwise.
- Find the vector potential (A) for all r.
- Find the magnetic flux field (B) for all r.
PHY 712 -- Assignment #13
Assigned: 2/14/2025 Due: 2/17/2025
Complete reading Chapter 5 in Jackson .
- In the lecture notes, we derived the H and B fields due to
a uniform sphere of radius a which has a uniform magnetization density
M0 oriented along the z-axis. For these fields, show that:
- At the surface of the sphere (r=a), the normal component of the B field is continuous.
- At the surface of the sphere (r=a), the tangential component of the H field is continuous.
PHY 712 -- Assignment #16
Assigned: 2/21/2025 Due: 2/24/2025
Start reading Chapter 7 (Sec. 7.1-7.3) in Jackson .
- Consider the reflectivity of a plane polarized electromagnetic wave
incident from air (n=1) on a material with refractive index
n'=1.5 at an angle of incidence i. Assume μ=μ'.
Plot the reflectance
R(i)=|E"0/E0|2
as a function of i for 0 ≤ i ≤ 90 deg for both cases of polarization of (E0 in the plane of incidence or perpendicular to the plane of incidence). What is the qualitative difference between the two cases?
PHY 712 -- Assignment #17
Assigned: 2/24/2025 Due: 2/26/2025
Continue reading Chapter 7, particularly Sec. 7.10 in Jackson .
- Work problem 7.22 (a) in Jackson. In addition to the analytic results, plot the real and imaginary parts of the permittivity as a function of ω for your favorite values of the constants.
PHY 712 -- Assignment #18
Assigned: 2/26/2025 Due: 2/28/2025
Continue reading Chapter 7, particularly Sec. 7.4 in Jackson .
- This problem concerns the phenomenon of total internal reflection. Imagine that plane polarized monochromatic light is relected and refracted at an interface between two media which have real refractive indices n=2 and n'=1.1. Draw a diagram of the incident, reflected, and refracted beams. What is the range of incident angles i for which total internal reflection occurs?
