PHY 712 Electrodynamics

MWF 11-11:50 AM OPL 107 http://www.wfu.edu/~natalie/s00phy712/

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

Homework Assignments

Information about Computational Project for week of March 20-24.

PHY 712 -- Assignment #1

January 14, 2000

Read Chapters I and 1 in Jackson.

  1. Jackson Problem #1.5
Jan 18, 2000

PHY 712 -- Problem Set PHY 712 - Problem Set #2 PDF version



  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 condition F(x ® -¥) = 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

File translated from TEX by TTH, version 2.20.
On 18 Jan 2000, 19:14.

PHY 712 -- Assignment #3

January 21, 2000

Complete 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 the angular integrals that result in Eqs. 4-8 of the notes.
  2. Extra credit -- Check at least one more non-trivial term.

PHY 712 -- Assignment #4

January 25, 2000

Complete 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 interation energy for another crystal form such as the NaCl structure, for example.

PHY 712 -- Assignment #5

January 28, 2000

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 #5 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 #5) 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 #6

January 31, 2000

Start reading Chap 2 in Jackson

  1. Work Problem #2.16 in Jackson.

PHY 712 -- Assignment #7

February 4, 2000

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.
Feb 5, 2000

PHY 712 -- Problem Set PHY 712 - Problem Set # 8 PDF version

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. Consider a charge distribution of the form:
    r(r, q, f) = r0 r2 e-ar cos2(q),
    		(3)
    where r0 and a are constants.

    1. Express r(r, q, f) as a sum of radial functions time spherical harmonic functions in the form:
      r(r, q, f) =
      å
       lm 
      		 rlm(r)   Ylm(q, f).
      		(4)
    2. Using the results of the previous problem, find the corresponding electrostatic potential which vanishes at ¥.

File translated from TEX by TTH, version 2.20.
On 5 Feb 2000, 15:30.

PHY 712 -- Assignment #9

February 9, 2000

Finish reading Chap 3 in Jackson

  1. Work Problem #3.9 in Jackson.

PHY 712 -- Assignment #10

February 11, 2000

Start reading Chap 4 in Jackson

  1. Work Problem #4.1(a) in Jackson.
  2. Work Problem #4.7(a) in Jackson; extra credit for working complete problem.

PHY 712 -- Assignment #11

February 14, 2000

Continue reading Chap 4 in Jackson

  1. Work Problem #4.7(c) in Jackson, using the result quoted in part (b). Extra credit for deriving (b) from your results in part(a). Note: If you have already worked this problem as extra credit for Problem Set #10, it is not necessary to redo your work.

PHY 712 -- Assignment #12

February 16, 2000

Continue reading Chap 4 in Jackson

  1. Work Problem #4.8(a) in Jackson.

PHY 712 -- Assignment #13

February 18, 2000

Continue reading Chap 4 in Jackson

  1. Use Eq. 4.64 to investigate the behavior of Enear. For example, find the field at the center of the cube shown in Fig. 4.9. The maple file: dipole.mws* shows an example in for a different geometry.

PHY 712 -- Assignment #14

February 21, 2000

Complete reading Chap 4 in Jackson

  1. Use the data given in Problem #4.11 of Jackson for the dielectric constant of pentane to estimate the molecular susceptibility, gmol, of pentane.
Mar 9, 2000

PHY 712 -- Problem Set PHY 712 - Problem Set # 15 PDF version

Complete reading of Chapter 5 of Jackson.

You may choose one of the following problems. (Extra credit for working both problems.)

  1. Work Problem #5.13 in Jackson.
  2. 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 function.

    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.

File translated from TEX by TTH, version 2.20.
On 9 Mar 2000, 17:40.
Mar 9, 2000

PHY 712 -- Problem Set PHY 712 - Problem Set # 16 PDF version

Read Chapter 6 of Jackson.

Consider the electric field produced by a point charge q moving on a trajectory described by r0(t) with r(r,t) º q d3 (r - r0(t)). Assume that v0(t) º r0(t)/t and 2 r0(t)/t2 = 0. Show that the electric field can be written in the form:

E(r,t) = q
4pe0
 (1 - v02/c2 )( R -v0R/c)
(R - v0 ·R/c)3
 ,
(1)
where R º |R(tr)|, R(tr) º r - r0(tr), and where all quantities which depend on time on the right hand side of the equation are evaluated at the retarded time tr º t - R(tr)/c. This is the result which we will obtain from the Lienard-Wiechert potentials in Chapter 14. (See equation 14.14.) You may wish to consult Problem #6.2 and section 6.5 of your text.

If you can prove them, you may wish to use some of the following results:

ó
õ 		d3 r¢ dt¢ d3(r¢ - r0(t¢) d æ
ç
è 		t¢ - t + |r-r¢|
c
 ö
÷
ø 		= ó
õ 		dt¢ d(t¢ - t + R(t¢)/c) = 1
1-v0 ·R/cR
 .
(2)

tr
t
 = R
R-v0 ·R/c
(3)

R(tr)
t
 = - v0 ·R
R-v0 ·R/c
 .
(4)

File translated from TEX by TTH, version 2.20.
On 9 Mar 2000, 18:04.

PHY 712 -- Assignment #17

March 17, 2000

Complete reading Chap 6 in Jackson

  1. Problem #6.20 in Jackson

PHY 712 -- Assignment #18

March 24, 2000

Start reading Chap 7 in Jackson

  1. Problem #7.2 in Jackson

PHY 712 -- Assignment #19

March 27, 2000

Continue reading Chap 7 in Jackson

  1. Write expressions for the transmittance and reflectance of plane polarized light for E polarized perpendicular and E polarized parallel to the plane of incidence corresponding to equations 7.39 and 7.41 respectively.
  2. Choosing your favorite values of n and n', plot R(i), the reflectance as a function of angle of incidence, for these two cases and find the Brewster's angle. For this purpose, assume that m=m'=1.

PHY 712 -- Assignment #20

March 29, 2000

Continue reading Chap 7 in Jackson

  1. Problem #7.6 in Jackson

PHY 712 -- Assignment #21

March 31, 2000

Complete reading Chap 7 in Jackson

  1. Problem #7.22(a) in Jackson
Apr 4, 2000

PHY 712 -- Problem Set PHY 712 - Problem Set # 22 PDF Version

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.

File translated from TEX by TTH, version 2.20.
On 4 Apr 2000, 16:38.

PHY 712 -- Assignment #23

April 7, 2000

Read Chapter 9 in Jackson

Choose one of the following problems. (Extra credit for working both.)

  1. Problem #9.10 in Jackson
  2. Problem #9.16 in Jackson

PHY 712 -- Assignment #24

April 10, 2000

Read Chapter 11 in Jackson

  1. Problem #11.5 in Jackson

PHY 712 -- Assignment #25

April 12, 2000

Read Chapter 11 in Jackson

  1. Verify Eq. 11.148 in Jackson, showing the relationship between the electric and magnetic fields in two frames of reference moving at a velocity bc relative to each other, by evaluating the matrix multiplication of Eq. 11.147.

PHY 712 -- Assignment #26

April 17, 2000

Read Chapter 14 in Jackson

  1. Read the lecture notes for Lecture #34 and convince yourself that the electric field derived in the two different methods are equivalent.

PHY 712 -- Assignment #27

April 19, 2000

Read Chapter 14 in Jackson

  1. Problem # 14.12 in Jackson.
  2. Extra credit Problem # 14.15 in Jackson
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