PHY 712 -- Assignment #1
January 12, 2015
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
January 14, 2015
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.
January 21, 2015
- Consider a one-dimensional charge distribution of the form:
where ρ0 and a are constants.ρ(x) = ⎧
⎪
⎨
⎪
⎩0 for x < −a/2 ρ0 x/a for −a/2 ≤ x ≤ a/2 0 for x > a/2, - Solve the Poisson equation for the electrostatic potential Φ(x) with the boundary conditions [(d Φ)/dx](−a/2) = 0 and [(d Φ)/dx](a/2) = 0.
- Find the corresponding electrostatic field E(x).
- Plot Φ(x) and E(x).
- Discuss your results in terms of elementary Gauss's Law arguments.
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PHY 712 -- Assignment #4
January 23, 2015
Continue reading Chap. 1 & 2 in Jackson.
- Jackson Problem #2.16. Note: as long as you show that your result is equivalent to the result given in the text, it is not necessary to put your result in the identical form.
PHY 712 -- Assignment #5
January 26, 2015
Review last section of Chap 1 in Jackson .
- Work Problem #1.24 in Jackson. Note that you can set this up as a linear algebra problem as we did in the lecture notes and can be solved directly for the three unknown values in Maple. It is not then necessary to use iteration methods. Also note that it is convenient to multiply the entire equation by 4πε0 so that the values of 4πε0 Φ are calculated directly. Also note that in these units, ρ = 1. These can be compared to the exact results in part (c) and to the series solution of the same system in Jackson problem 2.16.
PHY 712 -- Assignment #6
January 28, 2015
Finish reading Chapters 1-2 in Jackson .
- Work Problem #2.30 in Jackson after correcting the equation for SI units. Choose ρ=1 in these units and compare your results with those from previous homework sets involving Jackson's problems 2.16 and 1.24.
PHY 712 -- Assignment #7
January 30, 2015
Continue reading Chapter 3 in Jackson .
- Work Problem #3.9 in Jackson.
Work out a general expression for the potential Φ(ρ,φ,z); then
evaluate the unknown constants for the particular boundary potential
Φ(ρ=b,φ,z)=V(φ,z)= V0 sinh(z/L) sinh(1-z/L), where V0 and L are given potential and length constants, respectively and "b" is the cylinder radius given in the problem.
PHY 712 -- Assignment #8
February 2, 2015
Complete reading Chapter 3 and start Chapter 4 in Jackson .
- Consider the charge density of an electron bound to a proton in 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
February 4, 2015
Continue reading Chapter 4 in Jackson .
- Work problem #4.1 (parts a and b), in Jackson. For each case, find the lowest order multipole moment qlm and its cartesian equivalent. Comment on other non-vanishing higher multipole moments qlm.
PHY 712 -- Assignment #10
February 6, 2015
Finish reading Chapter 4 in Jackson .
- Work problem #4.9 in Jackson. In order to slightly simplify the analysis, you can assume that the point charge is in the z direction so that you can use the expansion given in equation 3.33 instead of a full spherical harmonic expansion.
PHY 712 -- Assignment #11
February 9, 2015
Start reading Chapter 5 in Jackson .
- Consider an infinitely long wire with radius a,
oriented along the z axis. There
is a steady uniform current inside the wire. Specifically the current
is along the z-axis with the magnitude of J0 for ρ ≤ a and
zero for ρ > a, where ρ denotes the radial parameter of the natural
cylindrical coordinates of the system.
- Find the vector potential (A) for all ρ.
- Find the magnetic flux field (B) for all ρ.
PHY 712 -- Assignment #12
February 11, 2015
Continue reading Chapter 5 in Jackson .
- Work problem #5.13.
PHY 712 -- Assignment #13
February 13, 2015
Finish reading Chapter 5 in Jackson .
- Work through the details of the magnetic shielding example given in Section 5.12 of your textbook. Verify Eq. 5.121 and 5.122.
February 16, 2015
- This problem relates to the evaluation of the retarded time Green's
function for a charged particle as given in Eq. 6.44 of Jackson and
in the lecture notes. Suppose that the particle trajectory is given by
where R0 and v0 are fixed constant position and velocity vectors respectively. Write an expression for the integralRq(t′) = R0+v0 t′,
expressing your answer in terms of the arbitrary function f and the field time t and position r.⌠
⌡∞
−∞f(t′) δ(t′−(t−|r− Rq(t′)|/c) ),
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February 18, 2015
- Suppose that an electromagnetic wave of pure (real) frequency ω is
traveling along the z-axis of
a wave guide having a square cross section with side dimension a
composed of
a medium having a real
permittivity constant ϵ and a real
permeability constant μ. Suppose that the wave is known to have the form:
E(r,t) = ℜ ⎧
⎨
⎩H0 ei k z − i ωt (i μω) a πsin ⎛
⎝πx a⎞
⎠^y⎫
⎬
⎭
Here H0 denotes a real amplitude, and the parameter k is assumed to be real and equal toH(r,t) = ℜ ⎧
⎨
⎩H0 ei k z − i ωt ⎡
⎣−ik a πsin ⎛
⎝πx a⎞
⎠^x+ cos ⎛
⎝πx a⎞
⎠^z⎤
⎦⎫
⎬
⎭.
for μϵω2 > ([(π)/a] )2.k ≡ ⎛
√μϵω2 − ⎛
⎝π a⎞
⎠2
, - Show that this wave satisfies the sourceless Maxwell's equations.
-
Find the form of the time-averaged Poynting vector
for this electromagnetic wave.〈S 〉avg ≡ 1 2ℜ{ E(r,t)×H*(r,t) }
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PHY 712 -- Assignment #16
February 20, 2015
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, Plot the reflectance
R(i)=|E"0/E0|2
as a function of i for both cases of polarization (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
February 23, 2015
Continue reading Chapter 7 in Jackson .
- Work problem 7.4 in Jackson .
PHY 712 -- Assignment #18
February 25, 2015
Continue reading Chapter 7 in Jackson .
- Work problem 7.22(a) in Jackson .
March 16, 2015
-
The figure above shows the cross section of a
magnetostatic solenoid which is uniform in the ∧z
direction (perpendicular to the page). The current flows in the azimuthal
∧ϕ direction; specifically the
current density is given in cylindrical coordinates by:
Here J0 is a constant, a and b denote the inner and outer diameters of the cylinder, respectively, and ∧ϕ = −sin(ϕ) ∧x+ cos(ϕ) ∧y.J = ⎧
⎪
⎨
⎪
⎩J0 ^ϕa ≤ ρ ≤ b 0 otherwise. (1) - Show that the vector potential A for this system
can be written as
where the scalar function f(ρ) satisfies the equationA = f(ρ) ^ϕ, (2) ⎡
⎣d2 d ρ2+ 1 ρd d ρ− 1 ρ2⎤
⎦f(ρ) = ⎧
⎪
⎨
⎪
⎩−μ0 J0 a ≤ ρ ≤ b 0 otherwise. (3) - Find the function f(ρ) in the three regions: 0 ≤ ρ ≤ a, a ≤ ρ ≤ b, and ρ ≥ b.
- Find the B field in the three regions. Check to make sure that your answer is consistent with what you know about solenoids. (Hint: B ≡ 0 outside the solenoid.)
- Show that the vector potential A for this system
can be written as
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PHY 712 -- Assignment #20
March 18, 2015
Continue reading Chapter 8 in Jackson .
- Following section 8.4 of Jackson which works out the form of the electromagnetic fields for the TE1,0 mode. Find the corresponding fields for the TM mode with the lowest cutoff frequency.
PHY 712 -- Assignment #21
March 20, 2015
Start reading Chapter 9 in Jackson .
- Work problem # 9.10(b) in Jackson.
PHY 712 -- Assignment #22
March 23, 2015
Continue reading Chapter 9 & 10 in Jackson .
- Work problem # 9.16(a) in Jackson.
PHY 712 -- Assignment #23
March 27, 2015
Start reading Chapter 11 in Jackson .
- Work out the details of the derivation of the velocity transformation equations 11.31 in Jackson.
PHY 712 -- Assignment #24
March 30, 2015
Continue reading Chap. 11 in Jackson .
- Work problem 11.5 at the end of Chapter 11 in Jackson.
PHY 712 -- Assignment #25
April 1, 2015
Continue reading Chap. 11 in Jackson .
- Verify Eq. 11.148 in Jackson by evaluating the transformation equations.
PHY 712 -- Assignment #26
April 6, 2015
Continue reading Chap. 14 in Jackson .
- "Prove" equation 14.66 in Jackson.
PHY 712 -- Assignment #27
April 8, 2015
Continue reading Chap. 14 in Jackson .
- Consider an electron moving at constant velocity βc ≈ c in a circular trajectory of radius ρ. Its total energy is E= γ m c2. Determine the ratio of the energy lost during one full cycle to the total energy. Evaluate the expression for an electron with total energy 200 GeV in a synchroton of radius ρ=103 m.
PHY 712 -- Assignment #28
April 10, 2015
Continue reading Chap. 14 in Jackson .
- Supply some of the intermediate steps to derive the Thompson formula for scattering of radiation by a free electron in Eq. 14.125 in Jackson.
PHY 712 -- Assignment #29
April 13, 2015
Start reading Chap. 15 in Jackson .
- Supply the steps to show that Eq. 15.10 follows from 15.9 in Jackson. Perform the integrals over solid angle to verify the form of total intensity per unit frequency given in the equation at the bottome of page 712.
PHY 712 -- Assignment #30
April 15, 2015
Continue reading Chapters 13 and 15 in Jackson .
- Consider Cherenkov radiation in water where ε/ε0 = 1.33. Estimate the critical angles θC for various velocities of a proton producing Cherenkov radiation.