PHY 712 Graduate Level Electrodynamics

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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Jan 18, 2011

PDF version PHY 712 - Problem Set #3

Continue reading Chaper 1 - 3 in Jackson; homework is due Friday Jan. 21, 2011.

Consider a one-dimensional charge distribution of the same form considered in HW2:

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 using the appropriate Green's function derived from an orthogonal function expansion as discussed in Lecture Notes #3.
2. Compare your results for the potential with the results obtained using the Green's function G(x,x^′) = 4 πx_<, also considering the convergence with increasing numbers of expansion terms.

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PHY 712 -- Assignment #4

January 21, 2011

Continue reading Chap. 2 in Jackson. The following problem will be due Mon, Jan. 24, 2011.

Jackson Problem #2.16. In order to complete the problem, as long as you show that your result is equivalent to the result given in the problem, it is not necessary to put your result in the identical form.

PHY 712 -- Assignment #5

January 24, 2011

Complete reading Chap. 1 in Jackson. The following problem will be due Wed, Jan. 26, 2011.

Using the information in Lecture 5, 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.

PHY 712 -- Assignment #6

January 26, 2011

Review last section of Chap 1 in Jackson . This problem is due January 31, 2011,

Work Problem #1.24 in Jackson. Note that you can set this up as a linear algebra problem as we did in Lecture Notes 6 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 #7

January 28, 2011

Finish reading Chapters 1-2 in Jackson . This problem is due January 31, 2011.

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 #8

January 31, 2011

Continue reading Chapter 3 in Jackson . This problem is due February 2, 2011.

Work Problem #3.9 in Jackson. Work out a general expression for the potential Φ(ρ,φ,z); then evaluate the unknown constants for the particular boundary potential

V(&phi,z)= V₀ sin²(πz/L),
where V₀ and L are given potential and length constants, respectively.

PHY 712 -- Assignment #9

February 2, 2011

Finish reading Chapter 3 in Jackson . This problem is due February 4, 2011.

Find the potential Φ(r) inside a sphere of radius a due to a uniform charge ρ(r)=&rho₀ for 0 ≤ r ≤ a and 0 otherwise. Assume that the potential is fixed at V₀ at r=a. Note: You may solve this problem using any of the methods you have learned. However, one of these solutions should use the appropriate Green's function and Green's theorem. For that purpose, note that Eq. 3.114 is the correct Green's function for finding a potential outside a sphere. A modified version of Eq. 3.125 can be used for this purpose.

PHY 712 -- Assignment #10

February 4, 2011

Start reading Chapter 4 in Jackson . This problem is due February 7, 2011.

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 #11

February 7, 2011

Finish reading Chapter 4 in Jackson . This problem is due February 9, 2011.

Work problem #4.9 in Jackson.

PHY 712 -- Assignment #12

February 18, 2011

Finish reading Chapter 5 in Jackson . This problem is due February 21, 2011.

Work problem #5.13 in Jackson. Find the magnetic dipole moment of this system and compare your results with that of the rotating uniform charged sphere discussed in Lecture Notes 13 .

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Feb 20, 2011

PHY 712 - Problem Set #13 PDF version

Finish reading Chaper 5 in Jackson; homework is due Wednesday Feb. 23, 2011.

The figure above shows the cross section of a magnetostatic solenoid which is uniform in the ∧z direction (perpendicular to the page). The current flows in the azimuthal ∧ϕ direction; specifically the current density is given in cylindrical coordinates by:

J = ⎧
⎪
⎨
⎪
⎩

J₀
^

ϕ

a ≤ ρ ≤ b

0
otherwise.

(1)
Here J₀ is a constant, a and b denote the inner and outer diameters of the cylinder, respectively, and ∧ϕ = −sin(ϕ) ∧x+ cos(ϕ) ∧y.
1. Show that the vector potential A for this system can be written as
  
  A = f(ρ)
  ^
  
  ϕ
  
  ,
  (2)
  where the scalar function f(ρ) satisfies the equation
  
  ⎡
  ⎣ d²
  d ρ²
  + 1
  ρ
  d
  d ρ
  − 1
  ρ²
  ⎤
  ⎦ f(ρ) = ⎧
  ⎪
  ⎨
  ⎪
  ⎩
  
  −μ₀ J₀
  a ≤ ρ ≤ b
  
  0
  otherwise.
  
  (3)
2. Find the function f(ρ) in the three regions: 0 ≤ ρ ≤ a, a ≤ ρ ≤ b, and ρ ≥ b.
3. Find the B field in the three regions. Check to make sure that your answer is consistent with what you know about solenoids. (Hint: B ≡ 0 outside the solenoid.)

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Feb 22, 2011

PHY 712 - Problem Set # 14 PDF Version

Start reading Chapter 6 of Jackson. The problem will be due Fri. Feb. 25, 2011.

In the Lorentz gauge and in the absence of sources, we found that the vector A(r,t) and scalar Φ(r,t) potentials must satisfy the following equations:

∇² Φ(r,t) − 1
c²
∂²
∂t²
Φ(r,t) = 0
(1)
and

∇² A(r,t) − 1
c²
∂²
∂t²
A(r,t) = 0.
(2)

Assuming solutions of the form:

Φ(r,t) ≡ Φ₀ expi (k·r− ωt) A(r,t) ≡ A₀ expi (k·r− ωt) ,
(3)
where Φ₀ and ω are scalar constants and A₀ and 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 E and B fields.

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Feb 24, 2011

PHY 712 - Problem Set # 15 PDF Version

Continue reading Chapter 6 of Jackson. The problem will be due Mon. Feb. 28, 2011.

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 #16

March 2, 2011

Continue reading Chapter 7 in Jackson . This problem is due Monday March 14, 2011.

Consider the relection and refraction of plane electromagnetic waves from an anisotropic crystal as discussed in Lecture notes 19 ( http://www.wfu.edu/~natalie/s09phy712/lecturenote/lecture19.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 #17

March 14, 2011

Continue reading Chapter 7 in Jackson. This problem is due Wednesday March 16, 2011.

Work Problem 7.22a in Jackson. Check that you get the same answer if you use Eq. 7.119.

PHY 712 -- Assignment #18

March 16, 2011

Finish reading Chapter 7 in Jackson. This problem is due Wednesday March 18, 2011.

The reflectivity of light from a surface is defined as the ratio of the magnitudes of the reflected and incident Poynting vectors. From the information given in Chapter 7, evaluate the reflectivity of light incident from air (n=1) and reflected from water (n'=1.54). Plot the reflectivity as a function of angle of incidence for both s polarization and for p polarization. Assume μ'=μ. Find the Brewster's angle both analytically and from the form of your graph.

PHY 712 -- Assignment #19

March 28, 2011

Start reading Chapter 9 in Jackson. This problem is due Wednesday March 30, 2011.

Find the scalar potential Φ(r) analogous to the vector potential A(r) given in Eq. 9.16 for an oscillating dipole.
Derive Eq. 9.18 of your text from the forms of Φ(r) and A(r) for the oscillating dipole.

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Mar 27, 2011

PHY 712 - Problem Set # 20 PDF VERSION

Continue reading Chapter 9 of Jackson. This problem is due Fri. Apr. 1, 2011.

Suppose that you have a source with the following charge and current density distributions:

ρ(r,t) = Re ⎛
⎝ 2 D z e^−r²/R²
R⁵ π^3/2
e^−iωt ⎞
⎠ .

J(r,t) = Re ⎛
⎝
^

z

−i ωD e^−r²/R²
R³ π^3/2
e^−iωt ⎞
⎠ .

In this expression, the constant D denotes the dipole moment, R is a length parameter, and ω 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 Φ(r,t) and A(r,t), evaluating as many of the integrals as is feasible.
3. Write the forms of Φ(r,t) and A(r,t) for distances r >> R.
4. Find the electric and magnetic fields E(r,t) and B(r,t) for distances r >> R.
5. Find the time averaged Poynting vector for this source for distances r >> R.

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PHY 712 -- Assignment #21

April 1, 2011

Finish reading Chapter 9 and start Chapter 11 in Jackson. This problem is due Monday April 4, 2011.

Work problem 9.16a in Jackson.

PHY 712 -- Assignment #22

April 4, 2011

Continue reading Chapter 11 in Jackson. This problem is due Wednesday April 6, 2011.

Carry out the 4x4 matrix multiplication indicated in Eq. 11.147 in Jackson to verify equations 11.148 -- the Electromagnetic Field transformations between inertial reference frames with a relative velocity βc along the x-axis.

PHY 712 -- Assignment #23

April 6, 2011

Continue reading Chapter 11 in Jackson. This problem is due Friday April 8, 2011.

Verify the details of Lecture Notes #27 to show that the Lorentz transformation and the Lienard Wiechert potential results give the same electric and magnetic fields for a particle of charge q moving at velocity v along the x-axis.

PHY 712 -- Assignment #24

April 18, 2011

Continue reading Chapter 14 in Jackson. This problem is due Wednesday April 20, 2011.

In Section 10.6, your textbook defines unit vectors ε_|| and ε_⊥ and refers to them as "polarization" vectors. In fact, the usual definition of polarization vector refers to the direction of the electric field. Eq. 14.71 of your text shows the "polarization" decomposition of the vector part of the intensity distribution function. Find the analogous decomposition for the vector part of the radiation electric field (given for example in Eq. 14.61).

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Last modfied: Sunday, 17-Apr-2011 18:13:02 EDT