PHY 712 Electrodynamics

MWF 9-9:50 AM OPL 107 http://www.wfu.edu/~natalie/s99phy712/

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

Homework Assignments

Information about Computational Project for week of March 22-26.

PHY 712 -- Assignment #1

January 13, 1999

Read Chapters I and 1 in Jackson.

  1. Jackson Problem #1.5
Jan 15, 1999

PHY 712 - Problem Set #2



  1. Consider a one-dimensional charge distribution of the form:

    r(x) = ì
    ï
    ï
    í
    ï
    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

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On 15 Jan 1999, 07:47.
Jan 19, 1999

PHY 712 - Problem Set # 3

  1. Evaluate the ``self-energy" of an electron bound in a hydrogen atom using equations 1.52, 1.53, or 1.54, with the electron density for the ground state of the hydrogen atom:
    r(r) = - e
    pa03
    		 e-2r/a0,
    		(1)
    where e is the elementary charge and a0 is the Bohr radius. Hint: Note that in Problem Set #1, you used the electrostatic potential corresponding to the ground state electronic density plus the electostatic potential of the proton.

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On 19 Jan 1999, 10:24.

PHY 712 -- Assignment #4

January 22, 1999

Complete Chapter 1 of Jackson.

  1. Jackson Problem #1.23 Note: Quoted results were apparently obtained using a very small grid.
  2. Jackson Problem #1.24. Note: Quoted results are multiplied by 4pe0.

PHY 712 -- Assignment #5

January 29, 1999

Complete Chapter 2 of Jackson.

  1. Jackson Problem #2.16.
  2. Jackson Problem #2.30, choosing r = 1 (and correcting equations for S.I. units). Compare your results for the two problems and also compare your your results for problem #1.24.
Jan 31, 1999

PHY 712 - Problem Set # 6

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 ¥.

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On 31 Jan 1999, 19:31.

PHY 712 -- Assignment #7

February 3, 1999

Complete Chapter 3 of Jackson.

  1. Jackson Problem #3.9.

PHY 712 -- Assignment #8

February 4, 1999

Read Chapter 4 of Jackson.

  1. Jackson Problem #4.1(a).
  2. Jackson Problem #4.7(a).

PHY 712 -- Assignment #9

February 8, 1999

Read Chapter 4 of Jackson.

  1. Jackson Problem #4.8.

PHY 712 -- Assignment #10

February 12, 1999

Complete Chapter 4 of 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.

PHY 712 -- Assignment #11

February 15, 1999

Read Chapter 5 in Jackson.

  1. Problem #5.11 in JDJ.
  2. Supply the steps which are necessary to derive the delta-function term in the expression for the magnetic field due to a magnetic dipole moment shown in Eq. 5.64. Compare the magnetic and electric dipole cases.
Mar 30, 1999

PHY 712 - Problem Set # 12

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)

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On 30 Mar 1999, 09:15.

PHY 712 -- Assignment #13

March 1 , 1999

Read Chapter 7 in Jackson.

  1. Problem #7.22(a) in JDJ.
  2. Problem #7.23 in JDJ.

PHY 712 -- Assignment #14

March 15, 1999

Complete Chapter 7 in Jackson.

  1. Problem #7.6 in JDJ.
  2. Problem #7.26 in JDJ.
Mar 14, 1999

PHY 712 -- Problem SetPHY 712 - Problem Set # 15

Read Chapter 8 in JDJ.

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 14 Mar 1999, 15:25.

PHY 712 -- Assignment #16

March 29, 1999

Read Chapter 9 in Jackson.

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

PHY 712 -- Assignment #17

April 9, 1999

Read Chapter 11 in Jackson.

  1. Problem #11.5 in JDJ.

PHY 712 -- Assignment #18

April 12, 1999

  1. Verify equations 11.148 for the electromagnetic field transformations by evaluating the transformation matrix product relating F and F'.

PHY 712 -- Assignment #19

April 16, 1999

  1. Use the Lienard-Wiechert potential results derived in problem #12 to verify equations 11.152 for the electric and magnetic field due to a particle have charge q moving at constant velocity v along the x axis, measured at the field point r= 0 x + b y. (Hint: note that the Lienard-Wiechert potentials are evaluated at the retarded time t - R/c.)
You might find the following image helpful

PHY 712 -- Assignment #20

April 19, 1999

  1. Problem #14.12 in JDJ
  2. Problem #14.15 in JDJ . (Note this problem is more challenging than the other one, and consequently worth more points.)
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