PHY 712 -- Assignment #1

Read Chapters I and 1 and Appendix 1 in Jackson.

1. Jackson Problem #1.5. Be careful to take into account the behavior of Φ(r) for r-->0.

PHY 712 -- Assignment #2

Continue reading Chap. 1 in Jackson.

1. 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 Φ(−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.

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

   ρ(x) = ρ0 sin ⎛ ⎝ πx a ⎞ ⎠ sin ⎛ ⎝ 2 πy a ⎞ ⎠ ,

   where ρ₀ represents a density constant and a represents a length constant. In the problem, you are asked to determine the electrostatic potential Φ(x,y) for 0 ≤ x ≤ a and 0 ≤ y ≤ a, which satisfies the Poisson equation for the charge density ρ(x,y). and statisfies the boundary conditions Φ(0,y)=Φ(a,y) = Φ(x,0)=Φ(x,a)=0.
   1. Find the analytic form of the electrostatic potential Φ(x,y) for 0 ≤ x ≤ a and 0 ≤ y ≤ a.
   2. Using the finite difference method for the two grids discussed in class, find Φ(x,y) on the grid points.
   3. Using the finite element method for the two grids discussed in class, find Φ(x,y) on the grid points.
   4. Compare the accuracy of the numerical solutions for this example.

PHY 712 -- Assignment #5

Continue reading Chap. 2 in Jackson.

1. Eq. 2.5 was derived as the surface change density on a sphere of radius a due to a charge q placed at a radius y > a outside the sphere. Determine the total surface charge on the sphere surface.
2. Now consider the same system except assume y < a representing the charge q being placed inside the sphere. What is the surface charge density and the total surface charge in this case?

PHY 712 -- Assignment #6

Complete reading Chapter 3 and start Chapter 4 in Jackson .

1. Consider the charge density of an electron bound to a proton in a hydrogen atom -- ρ(r) = (1/πa₀³) e^-2r/a₀, where a₀ denotes the Bohr radius. Find the electrostatic potential Φ(r) associated with ρ(r). Compare your result to HW#1.

PHY 712 -- Assignment #7

Complete reading Chapter 4 in Jackson .

1. Work problem 4.9(a) in Jackson, evaluating the coefficients of the spherical harmonic terms.

PHY 712 -- Assignment #8

Start reading Chapter 5 in Jackson .

1. Consider an infinitely long cylindrical wire with radius a, oriented along the z axis. There is a steady uniform current inside the wire. Specifically, in terms of r the radial parameter of the cylindrical coordinates of the system the current density is J(r)=J₀ , where J₀ is a constant vector along the z-axis, for r ≤ a and zero otherwise.
   1. Find the vector potential (A) for all r.
   2. Find the magnetic flux field (B) for all r.

PHY 712 -- Assignment #9

Continue reading Chapter 5 in Jackson .

1. Work problem #5.13.

PHY 712 -- Assignment #10

Finish reading Chapter 5 in Jackson .

1. Work through the details of the magnetic shielding example given in Section 5.12 of your textbook. Verify Eq. 5.121 and 5.122.

1. In deriving the Liénard Wiechert potentials, the retarded time
   

   tr=t− |r−Rq(tr)| c

   was introduced. Here r,t represent the position and time at which the field is measured, and R_q(t′) represents the trajectory of the charged particle source.
   

   v ≡ v(t′) ≡ d Rq(t′) dt′ .

   Demonstrate the following identities:
   1. 
      

      ∂tr ∂t = 1 1− v(tr) ·(r−Rq(tr)) c|r−Rq(tr)|
   2. 
      

      −c∇tr = r−Rq(tr) |r−Rq(tr)| 1− v(tr) ·(r−Rq(tr)) c|r−Rq(tr)|

PHY 712 -- Assignment #12

Finish reading Chapter 6 in Jackson .

1. Supply the details of the derivation of Eq. 6.105 in Jackson from Eq. 6.104.
2. Supply the details of the derivation of Eq. 6.121 in Jackson from Eq. 6.114.

1. 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 ei k z − i ωt (i μω) a π sin ⎛ ⎝ πx a ⎞ ⎠ ^ y   ⎫ ⎬ ⎭

   
   

   H(r,t) = ℜ ⎧ ⎨ ⎩ H0 ei 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.

PHY 712 -- Assignment #14

Continue reading Chapter 7 in Jackson .

1. Consider the reflectivity of a plane polarized electromagnetic wave incident from air (n=1) on a material with refractive index n'=1.5 at an angle of incidence i, Plot the reflectance

   R(i)=|E"₀/E₀|²

   as a function of i for both cases of polarization (E₀ in the plane of incidence or perpendicular to the plane of incidence). What is the qualitative difference between the two cases?

PHY 712 -- Assignment #15

Finish reading Chapter 8 and start reading Chapter 9 in Jackson .

1. In Lecture 19, we discussed the form of TE modes in a a x b rectangular waveguide propagating along the z-axis with mode numbers m and n, finding the field amplitudes B_x(x,y), B_y(x,y), B_z(x,y),E_x(x,y), E_y(x,y), and E_z(x,y). Find the corresponding amplitudes for TM modes in the same waveguide. (Hint: Note the footnote on page 362 or your text.)

PHY 712 -- Assignment #16

Continue reading Chapters 9 and 10 in Jackson .

1. Work problem 9.10(b) in Jackson.

PHY 712 -- Assignment #17

Finish reading Chapters 9 and 10 in Jackson .

1. Work problem 9.16(a) in Jackson.

PHY 712 -- Assignment #18

Continue reading Chapter 11 in Jackson .

1. Derive the relationships between the component of the electric and magnetic field components E₁, E₂, E₃, B₁, B₂, and B₃ as measured in the stationary frame of reference and the components E'₁, E'₂, E'₃, B'₁, B'₂, and B'₃ measured in the moving frame of reference. Note that the reverse relationships are given in Eq. 11.148.

PHY 712 -- Assignment #19

Continue reading Chapter 11 in Jackson .

1. Supply some of the intermediate steps for deriving the E and B fields resulting from a particle of charge q moving along the x-axis at constant speed v, measured at a point a distance b along the y-axis.

PHY 712 -- Assignment #20

Continue reading Chap. 14 in Jackson .

1. Consider an electron moving at constant speed βc ≈ c in a circular trajectory of radius ρ. Its total energy is E= γ m c². Determine the ratio of the energy lost during one full cycle to its total energy. Evaluate the expression for an electron with total energy 200 GeV in a synchroton of radius ρ=10³ m.

PHY 712 -- Assignment #21

Finish reading Chap. 14 in Jackson .

1. Compare Eq. 14.79 in Jackson with the corresponding expression given in the lecture notes. Are they equivalent?

PHY 712 -- Assignment #22

Start reading Chap. 15 in Jackson .

1. Show how Eq. 15.9 follows from Eq. 15.2 in the appropriate limit.