PHY 712 Electrodynamics

MWF 11-11:50 AM OPL 107 http://www.wfu.edu/~natalie/s02phy712/

Instructor: Natalie Holzwarth Phone:758-5510Office:300 OPL e-mail:natalie@wfu.edu

Homework Assignments

Information about Computational Project

PHY 712 -- Assignment #1

January 16, 2002

Read Chapters I and 1 in Jackson.

  1. Jackson Problem #1.5
Jan 22, 2002

PHY 712 -- Problem Set PHY 712 - Problem Set #2



  1. Consider a one-dimensional charge distribution of the form:
    r(x) = ì
    ï
    í
    ï
    î
    0     
    for  
    x £ -a/2
    r0 x/a     
    for   
    -a/2 £ x £ a/2
    0       
    for   
    x ³ a/2,
    where r0 and a are constants.

    1. Solve the Poisson equation for the electrostatic potential F(x) with the boundary conditions F(x ® -¥) = 0 and d F(x ® ¥)/dx = 0.
    2. Find the corresponding electrostatic field E(x).
    3. Plot F(x) and E(x).
    4. Discuss your results in terms of elementary Gauss's Law arguments.

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On 22 Jan 2002, 14:59.

PHY 712 -- Assignment #3

January 22, 2002

Continue Reading Chap 1 in Jackson

  1. Complete the proof of the "mean value theorm" for a solution of the Laplace equation, following "Lecture Notes #3". In particular, carry out several of the angular integrals that result in Eqs. 9-13 of the notes.
  2. Extra credit given for checking more non-trivial terms.

PHY 712 -- Assignment #4

January 25, 2002

Continue reading Chap 1 in Jackson

  1. Read the lecturenotes for the Ewald summation "Lecture Notes #4" or (preferably) the pdf file . Check the result for the CsCl structure, using the maple script as a guide.
  2. Extra credit -- Evaluate the interaction energy for another crystal form such as the NaCl structure, for example. If you are interested in further study in this area, you might want to use a fortran90 code which is also available -- , ewaldsum.f90
Jan 31, 2002

PHY 712 -- Problem Set PHY 712 - Problem Set #5



  1. Consider a three-dimensional charge distribution of the form:
    r(r) = q
    p3/2 a3
    		 e-(r/a)2
    where q and a are constants. In the following, you may which to use the result that for this particular charge density,
    1
    4pe0
    		 ó
    õ     		d3r¢ r(r¢)
    |r - r¢|
    		 = q
    4pe0
    		 erf(r/a)
    r
    		 .

    1. Find the electrostatic potential F(r) , as a function of the distance r from the center of the charge distribution r(r).
    2. Now suppose that a grounded metal plate is placed at a distance -d[^(z)] from the center of the charge distribution. Find the electrostatic potential due to r(r) and the boundary condition F(x,y,z = -d) = 0.
    3. How would your result change if you maintained a potential V0 on the grounded sheet? (That is that the charge distribution is still r(r) and the boundary condition F(x,y,z = -d) = V0).

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On 31 Jan 2002, 18:23.

PHY 712 -- Assignment #6

January 30, 2002

Continue reading Chap.2 in Jackson

  1. Show that the series given in Eq. (19) in the lecture notes for Lecture #6 is a valid representation of the solution to the given Laplace problem.
  2. Using the two-dimensional analogue of Eq. (8) appropriate for this problem, find another series expansion for this solution.
  3. Extra-credit Plot your results for F(x,y) using Maple.

PHY 712 -- Assignment #7

February 1, 2002

Complete Chap 1 in Jackson

  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 #7 and solve 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 (for example, equation 7 in Lecture Notes #7) by 4pe0 so that the values of 4pe0F are calculated directly. Also note that in these units, r = 1. These can be compared to the exact results in part (c).

PHY 712 -- Assignment #8

February 4, 2001

Finish reading Chap 2 in Jackson

  1. Work Problem #2.30 in Jackson after correcting the equation for SI units. Choose r=1 in these units and compare your results with those of 2.16 and 1.24.
  2. Extra credit: Work Problem #2.16 in Jackson so that you can make a quantitative comparison with the finite element results.
Feb 5, 2002

PHY 712 -- Problem Set PHY 712 - Problem Set # 9

Read Chapter 3 of Jackson.

  1. Convince yourself that Eqs. 3.62 and 3.70 are correct by expanding the expressions to second order. That is, verify the following:
    Pl( ^
    r
     
    		 · ^
    r
     
    		 ¢
     
    		 ) = 4 p
    2 l + 1
    		 l
    å
     m = -l 
    		 Y*lm( ^
    r
     
    		 )Ylm( ^
    r
     
    		 ¢
     
    		 )
    		(1)
    and
    1
    |r - r¢|
    		 = ¥
    å
     l = 0 
    		 l
    å
     m = -l 
    		 4 p
    2l+1
    		 rl <
    rl+1 >
    		  Y*lm( ^
    r
     
    		 )Ylm( ^
    r
     
    		 ¢
     
    		 ).
    		(2)

  2. Using these results consider the equation we had from Problem Set #5:
    1
    4pe0
    		 ó
    õ     		d3r¢ r(r¢)
    |r - r¢|
    		 = q
    4pe0
    		 erf(r/a)
    r
    		 .
    		(3)
    Show how you can obtain this result.

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On 5 Feb 2002, 14:42.

PHY 712 -- Assignment #10

February 8, 2002

Finish reading Chap 3 in Jackson

  1. Work Problem #3.9 in Jackson.

PHY 712 -- Assignment #11

February 11, 2002

Start reading Chap 4 in Jackson

  1. Work Problem #4.1 b,c,d in Jackson.

PHY 712 -- Assignment #12

February 13, 2002

Continue reading Chap 4 in Jackson

  1. Work Problem #4.7 in Jackson.

PHY 712 -- Assignment #13

February 15, 2002

Continue reading Chap 4 in Jackson

  1. Work Problem #4.9 in Jackson.

PHY 712 -- Assignment #14

February 18, 2002

Finish reading Chap 4 in Jackson

  1. Work Problem #4.11 in Jackson. For this purpose, you can assume that the density of air may be approximated by the ideal gas formula.

PHY 712 -- Assignment #15

February 20, 2002

Start reading Chap 5 in Jackson

  1. Work Problem #5.13 in Jackson.
Mar 1, 2002

PHY 712 -- Problem Set PHY 712 - Problem Set # 16

Continue reading Chapter 5 of Jackson.

  1. Consider a uniform cylindrical current expressed in cylindrical equations in the form:
    J(r) º j0 Q(a-r) ^
    z
     
    		 ,
    		(1)
    where j0 is a constant current density, a is the radius of the cylinder, and Q(a-r) denotes the Heaviside step function (Q(x) = 1 if x ³ 0 and Q(x) = 0 if x < 0).

    1. Find the vector potential A in the Coulomb gauge (Ñ·A = 0). Assuming the appropriate boundary conditions at r = a, find the form of A for both r < a and for r > a up to an arbitrary constant.
    2. Find the magnetic field B for both r < a and for r > a.
    3. Sketch A(r) and B(r) as a function of r.

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On 1 Mar 2002, 08:09.

PHY 712 -- Assignment #17

March 4, 2002

Finish reading Chap 5 in Jackson

  1. Work through the example in Section 5.12 of your text and verify the results, 5.121 and 5.122.

PHY 712 -- Assignment #18

March 25, 2002

Finish reading Chap 6 in Jackson

  1. Work either Problem 6.2 or 6.20 in Jackson (extra credit for working both).

PHY 712 -- Assignment #19

March 27, 2002

Start reading Chap 7 in Jackson

  1. Work Problem 7.2 in Jackson.

PHY 712 -- Assignment #20

April 1, 2002

Continue reading Chap 7 in Jackson

  1. Work Problem 7.6 in Jackson.

PHY 712 -- Assignment #21

April 3, 2002

Continue reading Chap 7 in Jackson

  1. Work Problem 7.22(a) in Jackson.
Apr 8, 2002

PHY 712 -- Problem Set PHY 712 - Problem Set # 22

Read Chapter 8 in Jackson.

Consider a TM wave propagating within a medium having, real dielectric constant e and permeability m, along the z axis within an ideal retangular waveguide with a cross section as shown in Fig. 8.5 and with z- component of electric field given by:

Ez(x,y,z,t) = E0 sin æ
ç
è 		m px
a
 ö
÷
ø 		sin æ
ç
è 		n py
b
 ö
÷
ø 		ei k z -i wt.

Here m and n are integers.

  1. Determine the value of k.
  2. Determine the other 5 components of electric and magnetic fields:

    1. Ex.
    2. Ey.
    3. Hx.
    4. Hy.
    5. Hz.

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On 8 Apr 2002, 12:37.

PHY 712 -- Assignment #23

April 10, 2002

Continue reading Chap 8 in Jackson

  1. Work Problem 8.4 in Jackson.

PHY 712 -- Assignment #24

April 12, 2002

Start reading Chap 9 in Jackson

  1. Work Problem 9.16 in Jackson.

PHY 712 -- Assignment #25

April 15, 2002

Continue reading Chap 9 in Jackson

  1. Work Problem 9.10(b) (not necessary to work entire problem) in Jackson.

PHY 712 -- Assignment #26

April 17, 2002

Start reading Chap 11 in Jackson

  1. Work Problem 11.5 in Jackson.

PHY 712 -- Assignment #27

April 19, 2002

Continue reading Chap 11 in Jackson

  1. Using the derivations in the Lecture Notes for Chapter 11 as a guide, show that Eq. (1) and (3) and (2) and (4) are consistent for a point charge moving at constant velocity v along the x-axis.

PHY 712 -- Assignment #28

April 22, 2002

Start reading Chap 14 in Jackson

  1. Work problem 14.4a in Jackson

PHY 712 -- Assignment #29

April 24, 2002

Continue reading Chap 14 in Jackson

  1. Work problem 14.12 in Jackson

PHY 712 -- Assignment #30

April 26, 2002

Continue reading Chap 14 in Jackson

  1. Consider Eq. 14.79 in Jackson. Plot the spectrum as a function of frequency and angle for various values of the parameters or in suitably parameterized form in order to appreciate the dependence on the critical frequency and on the angle of observation.
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