PHY 712 -- Assignment #2
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 #3
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 #4
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 #6
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 5x5x5 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 #7
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
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
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. 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
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 #12
Continue reading Chapter 5 (Sec. 5.6-5.7) in Jackson .
- Consider the following equation and verify its
validity (or otherwise).
PHY 712 -- Assignment #13
Finish reading Chapter 5 (Sec. 5.8-5.12) 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 #16
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
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 functions as a function of ω for your favorite values of the constants.
PHY 712 -- Assignment #18
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?
PHY 712 -- Assignment #20
Continue reading Chapter 9 (Sec. 9.1-9.2) 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,ω0) field for r>>a0 and compare your results with the scalar potential field calculated within the dipole approximation.
PHY 712 -- Assignment #21
Continue reading Chapter 9 (Sec. 9.1-9.4) in Jackson .
- Problem 9.16(a) in Jackson . In this case, "exactly" really means following the approach discussed in Sec. 9.4 using instead the current density given in the problem.
PHY 712 -- Assignment #22
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 #23
Continue reading Chapters 11 (Especially Sec. 11.9) 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 #24
Start reading Chap. 14 (Sec. 14.1-14.5) 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 #25
Continue reading Chap. 14 (Sec. 14.1-14.6) 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 ≤ ∞.
PHY 712 -- Assignment #26
Finish reading Chap. 14 (Sec. 14.1-14.8) 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).
PHY 712 -- Assignment #27
- In the context of analyzing Cherenkov radiation , we considered
a particle of charge q moving at constant velocity
v along the x axis producing electric and magnetic fields
at time t at a position r in the x-y plane. The
fields can be evaluated using the Liénard-Wiechert analysis
given in Lecture 33 for example, evaluated at the retarded time tr.
The analysis depends on the ratio of the speed of the particle v to
the speed cn of electromagnetic waves with the medium --
βn. We showed the following relationship between the
lengths R(t), R(tr), and the angle θ with
the x axis.
- First consider the case of the particle moving in vacuum so that βn < 1 . Determine the physical solutions in terms of choice of signs and range of angles θ for this case.
- Now consider the case of the particle moving in a dielectric medium and moving fast enough so that βn > 1 . Determine the physical solutions in terms of choice of signs and range of angles θ for this case.