課程名稱︰古典電力學一
課程性質︰物理系系定選修
課程教師︰朱國瑞
開課學院：理學院
開課系所︰物理學系
考試日期（年月日）︰2020年01月06日
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放
試題 :
Electrodynamics (I), Final Exam (6 problems) 1/6/2020
Note:(1) You may use Jackson. Any other materials (such as lecture notes, dic-
tionary, calculator, cell phone, and insertions of any kind) are NOT
allowed.
(2) You may use any eq. in Jackson without derivation unless you are
asked to derive it.
(3) Notationsm equation no, and page no follow Jackson.
1. Answer the following questions about Faraday's law:
∮\vec{E}‧d\vec{l}=-∫_s(∂\vec{B}/∂t)‧\vec{n}da
(a) If loop C is a closed wire with σ=∞ (hence \vec{E}=0 in the wire).
Then, ∮\vec{E}‧d\vec{l}=0 at all times even if a time-varying magnetic
flux is applied through the loop. Explain how this is possible. (4%)
(b) Consider a σ=∞ wire with a gap with end points a and b (see figure).
Let \vec{B}(\vec{x},t) be a known external field. What is the value of
∫^b_a\vec{E}‧d\vec{l}? (4%) ∫^b_a\vec{E}‧d\vec{l} depeneds on the
path from a to b. Why? (4%_
(c) Consider Case (b) again. If a point charge q (with negligible self fiels
) moves from a to b along the same path as in Case (b), the work done by
E on q is not equal to q times the ∫^b_a\vec{E}‧d\vec{l} in case (b).
Why? (4%)
╭──────────╮b
│\vec{B}(\vec{x},t)
│
╰──────────╯a
2. Inside a linear and uniform dielectric medium of permiitivity ε, there are
static free charge density [ρ_free(\vec{x})] and polarization charge dens-
ity [ρ_pol(\vec{x})].
(a) Show ρ_net/ε_0 = ρ_free/ε, where ρ_net = ρ_free + ρ_pol and ε_0
is the permittivity of free space. (4%)
(b) If ρ_free is a point charge given by ρ_free = qδ(\vec{x}-\vec{x_0}),
find ρ_pol. (4%)
(c) Discuss the physical implication of the result in (b) if ε>ε_0. (4%)
3. (a) Assume \vec{J}(t) = Re[\vec{J}_0e^{-iωt}] and \vec{E}(t) = Re[\vec{E}_0
e^{-iωt} ], where \vec{J}_0 and \vec{E}_0 are complex constants and ω
is a real constant. Show that the time-averaged value of \vec{J}(t)‧
\vec{E}(t) over one period is given by 1/2Re[\vec{J}_0\vec{E}^*_0]. (6%)
(b) Assume ε=ε'+iε'' is a complex number. Show Re√ε‧Im√ε = 1/2ε''.
(6%)
[Hint: Let ε(a+ib)^2]
4. A conducting surface has a static surface charge density σ. For simplicity,
assume that tje surface is flat. Use the Maxwell stress tensor [Jackson Eqs
(6.120) and (6.122)] to show that the force per unit area on the conducting
surface is σ^2/(2ε_0). (20%)
5. It's algebraically clear from observing (7.51) that we may neglect collisi-
ons and binding forces under the conditions ω>>γ_j and ω>>ω_j. As in the
lecture notes, do some simple calculations starting from (7.49) to find the
physical interpretations of these algebraic observations. (20%)
6. A plane wave of frequency ω and propagation constant k is propagating in
the +z direction in a uniform conducting medium characterized by a real
permeability μ and a complex permittivity: ε = ε_b + iσ/ω. Assume ε_b,
σ, and ω are all real.
(a) Write down the dispersion relation of the plane wave in terms of μ, ε,
k, and ω. (4%)
(b) Under what criterion is the medium a good conductor? Write it in terms
of ε_b, σ, and ω? (4%)
(c) Derive the skin depth (δ) of a good conductor in terms of σ, μ,and
ω. (4%)
(d) If E(z=0)=E_0, what is E(z) in terms of E_0, z, and δ? (4%)
(e) Calculate the Ohmic power/unit volume deposited into the medium by the
plane wave in terms of |E_0|, z, σ, and δ. (4%)