PHY 712 -- Assignment #1
Read Chapters I and 1 and Appendix 1 in Jackson.
- Jackson Problem #1.5. Be careful to take into account the behavior of Φ(r) for r-->0.
PHY 712 -- Assignment #2
Continue reading Chap. 1 in Jackson.
- Calculate and numerically evaluate the electrostatic energy of the following 5 ion molecule
scaled by the factor (1/(4πε0)) (q2 / a).
Comment on the significance
of the sign of your result.
Note that x, y, and z denote unit vectors in the
three Cartesian directions.
- Charge = -4q Position = 0
- Charge = q Position = (a/2)(x+y+ z)
- Charge = q Position = (a/2)(-x-y+ z)
- Charge = q Position = (a/2)(x-y+ z)
- Charge = q Position = (a/2)(-x+y+ z)
PHY 712 -- Assignment #3
Continue reading Chap. 1 in Jackson.
- Using the Ewald summation methods developed in class, find the electrostatic interaction energy of a NaCl lattice having a cubic lattice constant a. Check that your result does not depend of the Ewald parameter η. You are welcome to copy (and modify) the maple file used in class. A FORTRAN code is also available upon request.
PHY 712 -- Assignment #5
Continue reading Chap. 1-3 in Jackson.
-
For the two-dimensional rectangular system discussed in the lecture notes,
work out the analytic form of the electrostatic potential Φ(x,y) for
the following charge density
for 0 ≤ x ≤ a and
0 ≤ y ≤ b. Assume that the potential is 0 on the boundary of the
rectangle.
ρ(x,y)= ρ0 sin(πx/a) sin(πy/b)
Here ρ0 is a given constant. (Note there is an easy way and a hard way to solve this problem.)
PHY 712 -- Assignment #6
Continue reading Chap. 1-3 in Jackson.
Consider a two-dimensional square system, with 0 ≤ x ≤ a and 0 ≤ y ≤ a where the electrostatic potential Φ(x,y) vanishes on the boundaries -- Φ(x,0)=0, Φ(x,a)=0, Φ(0,y)=0, Φ(a,y)=0. The charge density within the square is constant (ρ0); ρ(x,y)=ρ0.
- Using the results derived in Lecture 5 (and mentioned in Lecture 6), numerically evaluate Φ(x,y) at the grid points discussed in Lecture 6.
- Now, using either the finite difference or finite element method for the two grids discussed in class, find Φ(x,y) on the grid points and compare the numerical solutions with the numerical answers you determined.
PHY 712 -- Assignment #7
Continue reading Chap. 2 in Jackson.
- Eq. 2.5 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 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 surface charge in this case?
PHY 712 -- Assignment #8
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 HW#1.
PHY 712 -- Assignment #9
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
Finish reading Chapter 4 in Jackson .
- Work problem 4.9(a) in Jackson. It is probably most convenient to use a coordinate system with the origin at the center of the dielectric sphere.
PHY 712 -- Assignment #11
Start reading Chapter 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 #12
Continue reading Chapter 5 in Jackson .
- Consider the following equation and verify its
validity (or otherwise).
PHY 712 -- Assignment #13
Finish reading Chapter 5 in Jackson .
- Work through some of the details of magnetic shielding effects of the highly permeable spherical shell given in Eq. 5.121 of Jackson and/or the equivalent presentation in the lecture notes.
PHY 712 -- Assignment #15
Start reading Chapter 7 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
Start reading Chapter 9 in Jackson .
- Problem 9.10 in Jackson lists the harmonic frequency dependent charge and current densities of a radiating H atom. Instead of answering Jackson's questions, calculate the exact scalar Φ(r) and vector potential A(r) fields for r>>a0 and compare your results with the scalar and vector potential fields calculated within the dipole approximation.
PHY 712 -- Assignment #18
Finish reading Chapters 9 and 10 in Jackson .
- Work problem 9.16(a) in Jackson. Note that you can use an approach similar to that discussed in Section 9.4 of the textbook, replacing the "center-fed" antenna with the given antenna configuration.
PHY 712 -- Assignment #19
Start reading Chapters 11 in Jackson .
- In class, we examined measurements of quantities that could be measured in two different reference frames which were related by the Lorentz transformation matrix. In particular, we are interested in the velocity components such as ux and and u'x. For the geometry used in class, work out the algebraic relationships that show that the 4-components of the vector (γuc, γuux, γuuy, γuuz) are related by a Lorentz transformation to the corresponding 4-component vector measured in the moving frame -- (γu'c, γu'u'x, γu'u'y, γu'u'z).
PHY 712 -- Assignment #20
Continue reading Chapter 11 in Jackson .
- Derive the relationships between the component of the electric and magnetic field components Ex, Ey, Ez, Bx, By, and Bz as measured in the stationary frame of reference and the components E'x, E'y, E'z, B'x, B'y, and B'z measured in a moving frame of reference which is moving at a constant relative velocity v along the x axis.
PHY 712 -- Assignment #21
Start reading Chap. 14 in Jackson .
- Consider an electron moving at constant speed βc ≈ c in a circular trajectory of radius ρ. Its total energy is E= γ m c2 = 200 GeV. Estimate the value of the ratio of the energy lost during one full circle to its total energy. Assume that synchroton radius in this case is ρ=103 meters. Note if you use the expression for this process analyzed by Jackson, please explain the details.
PHY 712 -- Assignment #22
Continue reading Chap. 14 in Jackson . This problem is designed to demonstrate Parseval's theorem using the definitions given in the lecture notes and on Page 674 in Jackson. We will use the example
A(t)=K e-(t/T)2,
where K and T are positive constants.
- Find the Fourier transform of A(t).
- Evaluate the integral of the squared modulus of A(t) between -∞ ≤ t ≤ ∞.
- Evaluate the integral of the squared modulus of the Fourier transform of A(t) between -∞ ≤ ω ≤ ∞.
- Evaluate the integral of the squared modulus of A(t) between -∞ ≤ t ≤ ∞.
PHY 712 -- Assignment #23
Finish reading Chap. 14 in Jackson .
- This problem concerns the Compton scattering of a photon having an initial
momentum magnitude of p and a final momentum magnitude p'
at an angle θ due to an electron of mass m, initially
at rest,
as discussed in lecture and on page 696
of Jackson. The wavelength of the photon before the collision is
λ=h/p and after is λ'=h/p', where h is
Planck's constant. Show that
λ'-λ=(h/(mc))(1-cosθ).