PHY 712 Graduate Level Electrodynamics

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.

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January 23, 2013

PHY 712 - Problem Set #3

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Continue reading Chaper 1 & 2 in Jackson; homework is due Friday Jan. 25, 2013.

Consider a one-dimensional charge distribution of the form:

ρ(x) = ⎧
⎪
⎨
⎪
⎩

0
for
x < −a/2

ρ₀ x/a
for
−a/2 ≤ x ≤ a/2

0
for
x > a/2,

where ρ₀ and a are constants.
1. 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.
2. Find the corresponding electrostatic field E(x).
3. Plot Φ(x) and E(x).
4. Discuss your results in terms of elementary Gauss's Law arguments.

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PHY 712 -- Assignment #4

January 25, 2013

Continue reading Chap. 1 & 2 in Jackson. The following problem will be due Mon, Jan. 28, 2013.

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 28, 2013

Review last section of Chap 1 in Jackson . This problem is due January 30, 2013,

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 30, 2013

Finish reading Chapters 1-2 in Jackson . This problem is due February 1, 2013.

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

February 1, 2013

Read Chapter 3 in Jackson . This problem is due February 4, 2013.

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).

PHY 712 -- Assignment #8

February 4, 2013

Start reading Chapter 4 in Jackson . This problem is due February 6, 2013.

Work problem #4.7 in Jackson. In order that the units come out correctly, multiply the given expression for ρ(r) by q/a₀³, where q is the elementary charge and a₀ denotes the Bohr radius. Also replace r in the expression with r/a₀.

PHY 712 -- Assignment #9

February 6, 2013

Continue reading Chapter 4 in Jackson . This problem is due February 8, 2013.

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 and that you can use the expansion given in equation 3.33 instead of a full spherical harmonic expansion.

PHY 712 -- Assignment #10

February 11, 2013

Start reading Chapter 5 in Jackson . This problem is due February 13, 2013.

Consider an infinitely long wire 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.
1. Find the vector potential for all ρ.
2. Find the magnetic flux field for all ρ.

PHY 712 -- Assignment #11

February 25, 2013

Finish reading Chapter 6 in Jackson . This problem is due February 27, 2013.

Supply the intermediate steps to deriv Eq. 6.105.

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Feb 27, 2013

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PHY 712 - Problem Set # 12

Finish reading Chapter 6 and start reading Chapter 7 of Jackson. The problem will be due Fri. Mar. 1, 2013.

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 e^{i k z − i ωt} (i μω) a
π
sin ⎛
⎝ πx
a
⎞
⎠
^

y

⎫
⎬
⎭

H(r,t) = ℜ ⎧
⎨
⎩ H₀ e^{i k z − i ωt} ⎡
⎣ −ik a
π
sin ⎛
⎝ πx
a
⎞
⎠
^

x

+ cos ⎛
⎝ πx
a
⎞
⎠
^

z

⎤
⎦ ⎫
⎬
⎭ .

Here H₀ denotes a real amplitude, and the parameter k is assumed to be real and equal to

k ≡ ⎛
√

μϵω² − ⎛
⎝ π
a
⎞
⎠ 2

,

for μϵω² > ([(π)/a] )².
1. Show that this wave satisfies the sourceless Maxwell's equations.
2. Find the form of the time-averaged Poynting vector
  
  〈S 〉_avg ≡ 1
  2
  ℜ{ E(r,t)×H^*(r,t) }
  
  for this electromagnetic wave.

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PHY 712 -- Assignment #13

February 28, 2013

Continue reading Chapter 7 in Jackson . This problem is due March 1, 2013.

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 reflectivity R(i) as a function of i for both cases of polarization (E in the plane of incidence or perpendicular to the plane of incidence). What is the qualitative difference between the two cases?

PHY 712 -- Assignment #14

March 01, 2013

Continue reading Chapter 7 in Jackson . This problem is due March 4, 2013.

Work problem 7.22(a) in Jackson .

PHY 712 -- Assignment #15

March 04, 2013

Finish reading Chapter 7 and start reading Chapter 8 in Jackson . This problem is due March 6, 2013.

Consider a material having a complex dielectric function of the form:
ε=ε_b + i σ/ω
where ε_b and σ are real valued constants. Assume that the permeability of the material has a real value μ. Find the corresponding form of the real and imginary refractive index:
(n_R +i n_I)²= (ε μ)/(ε₀ μ₀).

PHY 712 -- Assignment #16

March 06, 2013

Continue reading Chapter 8 in Jackson . This problem is due March 8, 2013.

Work out the details of the analysis of the electromagnetic fields within a coaxial cable discussed in Lecturenotes 22. (Similar to problem 8.2 (a) in Jackson).

PHY 712 -- Assignment #17

March 25, 2013

Start reading Chap. 11 in Jackson . This problem is due March 27, 2013.

Work out the details of the derivation of the velocity transformation equations 11.31 in Jackson.

PHY 712 -- Assignment #18

March 27, 2013

Continue reading Chap. 11 in Jackson . This problem is due April 1, 2013.

Work problem 11.5 at the end of Chapter 11 in Jackson.

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Mar 31, 2013

PHY 712 - Problem Set # 19

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Finish reading Chapter 11 and start reading Chapter 14 of Jackson. The problem will be due Wed. Apr. 3, 2013. Note: we are using cgs (Gaussian) units.

Consider the electromagnetic field transformations between a stationary frame S relative to a frame S′ which is moving at constant velocity v along the x axis. In the S′ frame, there is a plane wave in vacuum propagating along the z axis with electric field amplitude E₀

E′ = E₀
^

x

e^{i k^′·r^′− i ω′t′}.

B′ = E₀
^

y

e^{i k^′·r^′− i ω′t′}.
1. Determine the E and B fields in the stationary frame.
2. Check whether the fields in the stationary frame behave like normal plane waves. That is that in cgs (Gaussian) units the amplitudes of the E and B fields have the same magnitude and the wavevector k is perpendicular to both E and B .

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On 31 Mar 2013, 19:07.

PHY 712 -- Assignment #20

April 3, 2013

Continue reading Chap. 14 in Jackson . This problem is due April 5, 2013.

Starting with Eq. 14.38 in Jackson, derive Eq. 14.44, the radiation distribution for a charged particle in circular motion.

PHY 712 -- Assignment #21

April 5, 2013

Continue reading Chap. 14 in Jackson . This problem is due April 8, 2013.

Derive Eq. 14.66 in Jackson.

PHY 712 -- Assignment #22

April 8, 2013

Continue reading Chap. 14 in Jackson . This problem is due April 10, 2013.

Derive Eq. 14.79 in Jackson or the equivalent presented in the lecture notes. Note that there is some information about the derivation in the revised version of the notes for Lecture 30.

PHY 712 -- Assignment #23

April 12, 2013

Start reading Chap. 15 in Jackson . This problem is due April 22, 2013.

Problem 15.2 in Jackson.

PHY 712 -- Assignment #24

April 22, 2013

Continue reading Chap. 15 in Jackson . This problem is due April 24, 2013.

Problem 15.6 in Jackson.

PHY 712 -- Assignment #25

April 24, 2013

Review topics in Chap. 9 in Jackson . This problem is due April 26, 2013.

Problem 9.16 in Jackson.

Last modfied: Monday, 22-Apr-2013 09:54:20 EDT