課程名稱︰電磁學一
課程性質︰電機系必修
課程教師︰江衍偉
開課學院：電機資訊學院
開課系所︰電機工程學系
考試日期（年月日）︰2018/06/29
考試時限（分鐘）：110
試題 :
1. In Fig. 1, medium 3 extends to infinity so that no reflected wave exists in
that medium. For a uniform plane wave having the electric field
Ei = (E0‧cos(2π휱0^8t-2/3πz), 0, 0) V/m
incident from medium 1 onto the interface z = 0, obtain the expressions for
the phasor electric- and magnetic-field componenets in all three media. (20%)
Medium 1 │ Medium 2 │ Medium 3
(μ0, ε0)│(μ0, 9ε0) │(μ0, 81ε0)
│ │
→→→ │ →→→ │ →→→ x
│ │ ↑
←←← │ ←←← │ │
z = 0 z = 1/4 m y⊙─→z
Fig. 1 for question 1.
2. In Fig. 2, there is a current sheet Js(t) = (-cos(2π휱0^8t), 0, 0) A/m in
the z = 0 plane which is the boundary media 1 and 2. (Hint: Make use of the
phasor electric and magnetic fields to satisfy the boundary conditions at z =
0
)
Medium 1 │ Medium 2 x
(σ = 0, μ0, 9ε0) │ (σ = 10 S/m, μ0, ε0) ↑
←←←←← │ →→→→→ │
↓ y⊙─→z
z = 0
Fig. 2 for question 2.
(a) (10%) From the loss tangent, please show that medium 2 is a good calculato
r
with α = β~√(πfμσ), η~√(2πfμ/σ)‧e^(jπ/4) and calculate their
values.
(b) (10%) Find E and H for both sides of the current sheet.
3. In Fig. 3, there is a current sheet Js(t) = (-cos(2π휱0^8t), 0, 0) A/m in
the z = 0 plane which is the boundary media 1 and 3. (Hint: Make use of the
phasor electric and magnetic fields to satisfy the boundary conditions at z =
0
)
Medium 1 │ Medium 3 x
(σ = 0, μ0, ε0) │ (σ = 10^(-3) S/m, μ0, ε0) ↑
←←←←← │ →→→→→ │
↓ y⊙─→z
z = 0
Fig. 3 for question 3.
(a) (10%) From the loss tangent, please show that medium 3 is and imperfect
dielectric with α ~ σ/2√(μ/ε), β ~ ω√(με), η ~ √(μ/ε) and
calculate their values.
(b) (10%) If the current sheet in the z = 0 plane is modulated with the
narrow-frequency-band signal envelope S(t) shown in Fig. 4(i.e., the modulated
J(t) = S(t)脟s(t)), plot the signal envelope of E and H at z = 100m and -100m,
respectively.
https://i.imgur.com/V2wxiMS.jpg
Fig. 4 for question 3(b)
4. A point charge Q is situated at the origin surrounded by a spherical
dielectric shell of uniform permittivity ε and having inner and outer radii a
and b, respectively. (a) Find the D and E fields in the three regions 0 < r <
a
, a < r < b, and r > b. (12%) (b) Find the polarization vector inside the
dielectric shell. (4%) (c) If the dielectric shell becomes lossy with finite
conductivity σ but still with the original permittivity ε, what is the
polarization vector inside the dielectric now? (4%)
5. Consider two large, plane, parallel, perfectly conducting plates (with area
A) occupying the planes x = 0 and x = d and kept at potentials V = 0 and V =
V0, respectively. Assume also that d << √A. The region between the two plates
is filled with two perfect dielectric media having permittivity ε1 for 0 < x
< t (region 1) and ε2 for t < x < d (region 2). (a) Find the potential
distributions in the two regions 0 < x < t and t < x < d. (8%) (b) Find the
capacitance of this system. (7%) (c) If the dielectric in region 2 becomes
lossy with a finite conductivity σ2 but still with the orginal permittivity
ε2, find the potential at the interface x = t. (5%)
6. A toroidal magnetic core has a cross-sectional area A and mean circumferenc
e
lc. (a) If a current I is passed through a filamentary wire of N turns wound
around the toroid by connecting an appropriate current source, a magnetic flux
Ψ is established in the core. Find the permeability μ of the core material.
(7%) (b) Now an air gap of width lg (lg << lc) is introduced and the wire
winding number N is unchanged. To maintain the same magnetic flux Ψ, find the
required new current I'. Here the fringing of flux in the air gap can be
neglected. (8%) (c) Does the accuracy of your answer for (b) increase or
decrease when the magnetic flux Ψ increases (with all other parameters being
fixed)? Explain briefly. (5%)