PHY 712 Electrodynamics

MWF 10-10:50 PM

OPL 107

http://www.wfu.edu/~natalie/s09phy712/

Instructor: Natalie Holzwarth

Phone:758-5510

Office:300 OPL

e-mail:natalie@wfu.edu

Homework Assignments

Problem Set #1 (1/14/2009)

Problem Set #2 (1/16/2009)

Problem Set #3 (1/23/2009)

Problem Set #4 (1/26/2009)

Problem Set #5 (1/28/2009)

Problem Set #6 (1/30/2009)

Problem Set #7 (2/02/2009)

Problem Set #8 (2/04/2009)

Problem Set #9 (2/06/2009)

Problem Set #10 (2/09/2009)

Problem Set #11 (2/18/2009)

Problem Set #12 (2/20/2009)

Problem Set #13 (2/23/2009)

Problem Set #14 (2/27/2009)

Problem Set #15 (3/02/2009)

Problem Set #16 (3/23/2009)

Problem Set #17 (3/25/2009)

Problem Set #18 (3/27/2009)

Problem Set #19 (3/30/2009)

Problem Set #20 (4/01/2009)

Problem Set #21 (4/03/2009)

Problem Set #22 (4/06/2009)

Problem Set #23 (4/13/2009)

Problem Set #24 (4/15/2009)

Problem Set #25 (4/17/2009)

Problem Set #26 (4/20/2009)

Problem Set #27 (4/22/2009)

PHY 712 -- Assignment #1

January 14, 2009

Read Chapters I and 1 in Jackson. The following problem will be due Fri, Jan. 16, 2009.

1. Jackson Problem #1.5

PHY 712 -- Assignment #2

January 16, 2009

Continue reading Chapters 1 in Jackson. Homework due Fri. Jan. 23, 2009.

1. Read the lecturenotes for the Ewald summation " Notes for Lectures 2 & 3" (pdf file) . Check the result for the CsCl structure, using the maple script as a guide. For example, does your result depend on the summation limits or on the value of η?
2. Evaluate the interaction energy for a different crystal form such as the NaCl structure described in the lecture notes, or for a different structure of your choosing.
3. Extra credit or computational assignment topic -- If you are interested in further study in this area, you might want to use a fortran90 code which is also available -- ewaldsum.f90. sample run.

January 23, 2009

PHY 712 - Problem Set #3

Continue reading Chaper 1 & 2 in Jackson; homework is due Monday, Jan. 26, 2009.

1. Consider a one-dimensional charge distribution of the form:
   
   $$\rho(x) = \left\{ \begin{array}{lll} 0 \;\; & {\rm {for}}\... ...0 \;\;\; & {\rm {for}} \; & x \geq a/2, \end{array} \right.$$
   
   
   where $\rho_0$ and $a$ are constants.
   1. Solve the Poisson equation for the electrostatic potential $\Phi(x)$ with the boundary conditions $\Phi(-a/2) = 0$ and $\frac{d \Phi}{dx}(-a/2) = 0$.
   2. Find the corresponding electrostatic field $E(x)$.
   3. Plot $\Phi(x)$ and $E(x)$.
   4. Discuss your results in terms of elementary Gauss's Law arguments.

PDF version

January 25, 2009

PHY 712 - Problem Set #4

Continue reading Chaper 1 & 2 in Jackson; homework is due Wednesday, Jan. 28, 2009.

1. Consider a one-dimensional charge distribution of the form:
   
   $$\rho(x) = \left\{ \begin{array}{lll} 0 \;\; & {\rm {for}}\; ... ...\\ 0 \;\;\; & {\rm {for}} \; & x \geq a, \end{array} \right.$$
   
   
   where $\rho_0$ and $a$ are constants.
   1. Solve the Poisson equation for the electrostatic potential $\Phi(x)$ with the boundary conditions $\Phi(0) = 0$ and $\frac{d \Phi}{dx}(0) = 0$.
      1. Use the Green's function discussed in Lecture Notes #4:

         
         

         $$G(x,x') = 4 \pi x_<.$$
         
         

      2. Use the Green's function discussed in Lecture Notes #5:

         
         

         $$G(x,x') = \frac{8 \pi}{a} \sum_n \frac{\sin(n\pi x/a)\sin(n\pi x'/a)}{\left(\frac{n\pi}{a}\right)^2}.$$
         
         

   2. In both cases, check whether the Green's function-derived solutions satisfy the boundary conditions. If they do not, you will need to add contributions from solutions to the homogeneous equations as discussed in Lecture Notes #5. Obviously, you should obtain the same answer for both methods.

PDF version

PHY 712 -- Assignment #5

January 28, 2009

Continue reading Chaps. 1-3 in Jackson. The following problem will be due Fri, Jan. 30, 2009.

1. Jackson Problem #2.16.

PHY 712 -- Assignment #6

January 30, 2009

Continue reading Chaps. 2 in Jackson. The following problem will be due Mon, Feb. 2, 2009.

1. Jackson Problem #2.2.

PHY 712 -- Assignment #7

February 2, 2009

Review last section of Chap 1 in Jackson . This problem is due Feb. 4, 2009

1. Work Problem #1.24 in Jackson. Note that you can set this up as a linear algebra problem as we did in Lecture Notes #8 & 9 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 #8

February 4, 2008

Finish reading Chapters 1-2 in Jackson . This problem is due Feb. 6, 2009.

1. 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 solve Jackson's problems 2.16 and 1.24.

PHY 712 -- Assignment #9

February 6, 2008

Start reading Chapter 3 in Jackson . This problem is due Feb. 9, 2009.

1. Work Problem #3.9 in Jackson.

PHY 712 -- Assignment #10

February 9, 2008

Finish reading reading Chapter 3 in Jackson . This problem is due Feb. 11, 2009.

1. Evaluate the left and right hand sides of Eq. 3.62 for l=0,1, and (optionally) 2, convincing yourself the of their equality. Note that cos γ = cos θ cos θ' + sin θ sin θ' cos( φ - φ').

PHY 712 -- Assignment #11

February 18, 2008

Finish reading reading Chapter 4 in Jackson . This problem is due Feb. 20, 2009.

1. Work problem 4.11 in Jackson .

February 19, 2009

PHY 712 - Problem Set # 12

Start reading Chapter 5 of Jackson. This problem is due Monday Feb. 23, 2009.

1. The figure above shows the cross section of a magnetostatic solenoid which is uniform in the ${\bf {\hat{z}}}$ direction (perpendicular to the page). The current flows in the azimuthal $\bf {\hat{\phi}}$ direction; specifically the current density is given in cylindrical coordinates by:
   

   $${\bf {J}} = \left\{ \begin{array}{ll} J_0 \bf {\hat{\phi}} ... ...e \rho \le b \\ 0 & {\rm {otherwise.}} \end{array} \right.$$ (1)

   
   
   Here $J_0$ is a constant, $a$ and $b$ denote the inner and outer diameters of the cylinder, respectively, and ${\bf {\hat{\phi}}} = -\sin(\phi) {\bf {\hat{x}}}+ \cos(\phi) {\bf {\hat{y}}}.$
   1. Show that the vector potential ${\bf {A}}$ for this system can be written as
      

      $${\bf {A}} = f(\rho) {\bf {\hat{\phi}}},$$ (2)

      
      
      where the scalar function $f(\rho)$ satisfies the equation
      

      $$\left[ \frac{d^2}{d \rho^2} + \frac{1}{\rho} \frac{d}{d \rho}... ...e \rho \le b \\ 0 & {\rm {otherwise.}} \end{array} \right.$$ (3)

      
      

   2. Find the function $f(\rho)$ in the three regions: $0 \le \rho \le a$, $a \le \rho \le b$, and $\rho \ge b$.
   3. Find the $\bf {B}$ field in the three regions. Check to make sure that your answer is consistent with what you know about solenoids. (Hint: $\bf {B} \equiv 0$ outside the solenoid.)

PDF version

PHY 712 -- Assignment #13

February 23, 2009

Continue reading Chap. 5 in Jackson. The following problem will be due Wed. Feb. 25, 2009.

1. Jackson Problem #5.13

PHY 712 -- Assignment #14

February 25, 2009

Finish reading Chap. 5 in Jackson. The following problem will be due Mon. Mar. 2, 2009.

1. Jackson Problem #5.14.

February 27, 2009

PHY 712 - Problem Set # 15

Start reading Chapter 6 of Jackson. The problem will be due Wed. Mar. 4, 2009.

1. In the Lorentz gauge and in the absence of sources, we found that the vector ${\bf {A}}({\bf {r}},t)$ and scalar $\Phi({\bf {r}},t)$ potentials must satisfy the following equations:
   

   $$\nabla^2 \Phi({\bf {r}},t) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \Phi({\bf {r}},t) = 0$$ (1)

   
   
   and
   

   $$\nabla^2 {\bf {A}}({\bf {r}},t) - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} {\bf {A}}({\bf {r}},t) = 0.$$ (2)

   
   

   Assuming solutions of the form:
   

   $$\Phi({\bf {r}},t) \equiv \Phi_0 {\rm {e}}^{i ({\bf {k}}\cdot ... ...bf {A}}_0 {\rm {e}}^{i ({\bf {k}}\cdot {\bf {r}}- \omega t)} ,$$ (3)

   
   
   where $\Phi_0$ and $\omega$ are scalar constants and ${\bf {A}}_0$ and ${\bf {k}}$ are vector constants, find relationships between these constants that must be satisfied in order to satisfy the Maxwell's equations and the Lorentz gauge conditions. Also, determine the corresponding forms of the $\bf {E}$ and $\bf {B}$ fields.

PDF version

March 21, 2009

PHY 712 - Problem Set # 16

Continue reading Chapter 6 of Jackson. This problem will be due Wed. March 25, 2009.

1. Suppose that an electromagnetic wave of pure (real) frequency $\omega$ 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 $\epsilon$ and a real permeability constant $\mu$. Suppose that the wave is known to have the form:
   
   $${\bf {E}}({\bf {r}},t) = \Re \left \{ H_0 {\rm {e}}^{i k z - ... ...} \sin \left(\frac{\pi x}{a} \right) {\bf {\hat{y}}}\right \}$$
   
   
   
   
   $${\bf {H}}({\bf {r}},t) = \Re \left \{ H_0 {\rm {e}}^{i k z - ... ...left(\frac{\pi x}{a} \right) {\bf {\hat{z}}}\right] \right \}.$$
   
   
   Here $H_0$ denotes a real amplitude, and the parameter $k$ is assumed to be real and equal to
   
   $$k \equiv \sqrt{ \omega^2 - \left(\frac{\pi}{a} \right)^2},$$
   
   
   where $\omega > \frac{\pi}{a}$. Find the form of the time-averaged Poynting vector
   
   $$\langle {\bf {S}} \rangle_{avg} \equiv \frac{1}{2} \Re \left ... ...bf {E}}({\bf {r}},t) \times {\bf {H}}^*({\bf {r}},t) \right \}$$
   
   
   for this electromagnetic wave.

PDF version

PHY 712 -- Assignment #17

March 25, 2009

Start reading Chapter 7 in Jackson . This problem is due Friday, March 27, 2009.

1. Use the results in section 7.3 to calculate the reflectance as a function of incident angle i. Assuming μ=μ'=&mu₀ choose typical values of the refractive indicies n and n' and plot the reflectances as a function of i for the two polarizations showing the effects of Brewster's angle.

PHY 712 -- Assignment #18

March 27, 2009

Continue reading Chapter 7 in Jackson . This problem is due Monday, March 30, 2009.

1. Problem 7.2a in Jackson . For the plots, assume a reasonable value of d.

PHY 712 -- Assignment #19

March 30, 2009

Continue reading Chapter 7 in Jackson . This problem is due Wednesday, April 1, 2009.

1. Consider the relection and refraction of plane electromagnetic waves from an anisotropic crystal as discussed in Lecture notes 22 ( http://www.wfu.edu/~natalie/s09phy712/lecturenote/lecture22.pdf). For both s-polarization and p-polarization:
   1. Find the Poynting vector S' corresponding to the transmitted power of the wave within the crystal.
   2. Check that the "normal" (y) component of the Poynting vectors for the incident beam (S), reflected beam (S") and transmitted beam (S') are consistent: S • y + S" • y = S' • y .

PHY 712 -- Assignment #20

April 1, 2009

Continue reading Chapter 7 in Jackson . This problem is due Friday, April 3, 2009.

1. Work Problem 7.22(a) in Jackson .

PHY 712 -- Assignment #21

April 3, 2009

Start reading Chapter 11 in Jackson . This problem is due Monday, April 6, 2009.

1. Consider a light source which is generating a plane wave of wavelength λ = 400 nm as measured in its own frame of reference.
   1. Suppose this light is detected in a frame of reference which views the source moving with respect the detector in the same direction as the propagation direction and finds the wavelength to be λ' = 600 nm. What is the relative speed β of the source and detector?
   2. Now suppose that light source is propagating at an angle of 90^o with respect to the direction which it is moving relative to the detector. What is the dectected wavelength assuming β remains the same?

PHY 712 -- Assignment #22

April 6, 2009

Continue reading Chapter 11 and start reading Chapter 14 in Jackson . This problem is due Monday, April 13, 2009.

1. Using the lecturenotes from Lecture 27 as a guide, derive the electromagnetic fields for a charge moving at constant velocity (Eq. 19,20) using the Liénard-Wiechert equations, supplying some of the extra steps not given in the notes.

PHY 712 -- Assignment #23

April 13, 2009

Continue reading Chapter 14 in Jackson . This problem is due Wednesday, April 15, 2009.

1. Consider a particle of charge q moving in a circular trajectory of radius a in the x-y plane: R_q(t')= a ( cos(ωt') x + sin(ωt') y ). Radiation from this moving particle is observed at a point r = r ( sin θ cos φ x + sin θ sin φ y + cos θ z), where r >> a. Find an expression that represents the radiated power per unit solid angle evaluated at the retarded time. Compare the behaviors of the power distribution in the non-relativistic and highly relativistic limits.

PHY 712 -- Assignment #24

April 15, 2009

Continue reading Chapter 14 in Jackson . This problem is due Friday, April 17, 2009.

1. The following paper Young-Sea Huang and Kang Hao Lu, Foundations of Physics 38 151-159 (2007) claims that Jackson's Eq. 14.38 is incorrect and gives an alternative derivation and result. Read through the paper with enough detail to form your own opinion of which is the correct expression for the power distribution.

PHY 712 -- Assignment #25

April 17, 2009

Finish reading Chapter 14 and start Chapter 9 in Jackson

1. Consider a synchrotron light source such as the "Advanced Light Source" in Berkeley CA which has a ring radius of ρ=100m and the radiating electrons have energy E=2GeV. Determine the critical frequency ω_c. With the help of Maple, plot the spectral intensity as a function of ω for θ=0 and for at least one other angle. A webpage with links to several synchrotron facilities is http://lightsources.org/cms/.

April 20, 2009

PHY 712 - Problem Set # 26

Continue reading Chapter 9 of Jackson. This homework is due Wed. Apr. 22, 2009.

1. Suppose that you have a source with the following charge and current density distributions:
   
   $$\rho({\bf {r}},t) = \frac{2 D z {\rm {e}}^{-r^2/R^2}}{R^5 \pi^{3/2}} {\rm {e}}^{-i\omega t}.$$
   
   
   
   
   $${\bf {J}}({\bf {r}},t) = {\bf {\hat{z}}}\frac{-i \omega D {\rm {e}}^{-r^2/R^2}}{R^3 \pi^{3/2}} {\rm {e}}^{-i\omega t}.$$
   
   
   In this expression, the constant $D$ denotes the dipole moment, $R$ is a length parameter, and $\omega$ is the (constant) harmonic frequency.
   1. Show that this source is consistent with the continuity equation.
   2. Write an expression for the scalar and vector potentials $\Phi({\bf {r}},t)$ and ${\bf {A}}({\bf {r}},t)$, evaluating as many of the integrals as is feasible.
   3. Write the forms of $\Phi({\bf {r}},t)$ and ${\bf {A}}({\bf {r}},t)$ for distances $r>>R$.
   4. Find the electric and magnetic fields ${\bf {E}}({\bf {r}},t)$ and ${\bf {B}}({\bf {r}},t)$ for distances $r>>R$.
   5. Find the time averaged Poynting vector for this source for distances $r>>R$.

Link to PDF version.

PHY 712 -- Assignment #27

April 22, 2009

Finish reading Chapter 9 and start Chapter 10 in Jackson. This problem is due Friday, Apr. 24, 2009.

1. Derive Eqs. 9.55 and 9.56 in Jackson.
2. Plot the power distribution as a function of θ for several choices of kd.

[an error occurred while processing this directive] Last modfied: Tuesday, 21-Apr-2009 18:16:29 EDT