Electrodynamics I Final Exam - Part A - Closed Book KSU 2014/12/18

Name

Instructions: Use SI units. Please make your answers brief and clear.

1. (2) Give an expression for the force dF~ on a current element i d~l in a magnetic induction B.
~

~ exists in a region of magnetic induction B(r).

2. (2) A distribution of current J(r) ~ Write an expression
for the total torque on the current distribution.

~
3. (2) For a current distribution J(r), ~
how does one express the vector potential A(r) that it produces?

4. (2) A flat coil (loops confined in some plane) has N turns of some arbitrary two-dimensional shape
(along some path r(θ), for example). If it carries a current i, how do you write an expression for
its magnetic dipole moment m? ~

5. (2) A particle of charge q and mass M makes a periodic orbital motion. Write an expression
relating its orbital magnetic moment m ~
~ and its orbital angular momentum L.

~ (r)?
6. (2) What is the physical definition of magnetization M

7. (2) Give the constitutive relation between magnetic induction, magnetization, and magnetic field.

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 ~ for the following
8. (4) Use delta-functions in spherical coordinates to express the current density J(r)
situation: A spherical shell of radius a, with surface charge density σ = σ0 sin θ, that is rotating
around the z-axis at angular frequency ω.

9. (2) A square a × a coil has N turns. A uniform magnetic induction B = B0 sin(ωt) passes
perpendicularly through the coil. Calculate the time-dependent emf produced in the coil.

10. (2) Give an expression for the time-dependent electric field in terms of scalar and vector potentials.

11. (4) In the presence of any time-dependent sources, what equation is obeyed by the vector potential,
when using the Lorentz gauge?

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12. (4) Consider a wave equation, ∇2 Ψ − c12 ∂t
∂
2 Ψ = −4πf (r, t), where f (r, t) is the source that drives
some waves Ψ(r, t). Write out the space- and time-dependent Green’s function for this equation
that applies to a problem where the source turns on at time t = 0.

13. (4) Write out an equation for Poynting’s theorem in differential form. Explain in words what each
term means physically.

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14. (4) From consideration of Maxwell’s equations, what are the symmetry properties (odd or even)
of the electric polarization P~ under space inversion? What about time inversion?

15. (4) A plane EM wave is traveling in the z-direction in a medium with µ = µ0 and  = 40 . With
~ = E0 x̂ exp[i(kz − ωt)] write an expression for B(z,
linearly polarized E ~ t) in this wave.

~ t) = E0 (ŷ − iẑ) exp[i(kx − ωt)]. Looking into the

16. (4) A plane wave travels in the x-direction: E(r,
wave at a fixed point in space, in which direction does the electric field vector rotate (clockwise or
couterclockwise)? Which circular polarization is this (right or left)?

17. (2) Write an expression for the dielectric function (ω) in a plasma.

18. (2) What does (ω) imply for EM waves of low frequency traveling in a plasma?

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Electrodynamics I Final Exam - Part B - Open Book KSU 2014/12/18
Name

Instructions: Use SI units. Please write your derivations and final answers on these pages. Explain your
reasoning for full credit. Discuss the physics if the math is impossible.

1. (18) Consider a straight wire of radius a and length l in direction z, with a current density J~ = σ E
~
~ outside the wire is assumed to be negligible here.
that is uniform over its cross section. E

~ both inside and outside

a) (6) For a DC current through the wire, find the magnetic induction B
the wire.
~ inside
b) (6) Based on your result for the DC magnetic field, determine the Poynting vector S
the wire, as a function of the radial coordinate ρ from the axis of the wire.
~ satisfies Poynting’s theorem applied to the whole volume of the
c) (6) Show that the result for S
wire segment. Comment on the physical significance of the terms in the equation.

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2. (18) Consider again a straight wire as in the previous question, with J~ = σ E.
~

a) (6) Now suppose the current is driven through the wire at a high frequency ω, i.e, harmonic
fields varying as exp(−iωt). Apply Maxwell’s equations to such a situation to get the differ-
ential equation that Ez should solve inside the wire.

b) (6) Consider the case of a very good conductor. Explain physically why you should expect
~ to be nonuniform now within the wire. How should |E|
the electric field (and J) ~ vary with ρ
inside the wire? I am looking for a very approximate answer; it may not require a calculation.

c) (6) For a copper wire of radius a = 1.00 mm,  = 0 and µ = µ0 and σ = 5.95 × 107 (Ω · m)−1 ,
estimate the angular frequency ω above which one needs to account for this spatial variation
of the fields inside the wire.

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3. (16) Hydrogen gas (density = 1012 H2 molecules per cm3 ) is heated to a very high temperature
(kB T  13.6 eV) so that all molecules are broken apart and the atoms are ionized.

a) (8) Estimate the range of angular frequencies ω of propagating EM waves in the plasma. Give
a numerical result.

b) (8) Suppose a plane EM wave of amplitude E0 originally traveling in vacuum

√ is incident on
this plasma at normal incidence. The wave has a frequency ω = ωp / 2, where ωp is the
plasma frequency. Use the Fresnel formulas to find the amplitude of the electric field after it
travels a distance of 5λ into the plasma, where λ = 2πc/ω is the wavelength in vacuum.