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）︰2018/04/27
考試時限（分鐘）：110
試題 :
1. (20%) In a region of uniform electric and magnetic fields E = (0, 20, 0) an
d
B = (0, 0, 2), respectively, a test charge q of mass m moves in the manner:
┌
│x = 10/ω(ωt - sinωt)
│y = 10/ω(1 - cosωt)
│z = 0
└
where ω = 2q/m. Find the forces acting on the test charge for the
following values of t:
(a) t = 0;
(b) t = π/(2ω);
(c) t = π/ω;
(d) t = 3π/(2ω); and
(e) t = 2π/ω;
2. (20%) Consider current distribution with density
┌
│(0, J2, J1); -a≦x≦0
J(A/m^2) = │
│-(0, J2, J1); 0≦x≦a
└
where J1 and J2 are constant values. Please find the magnetic flux density,
B, due to the current distribution. (hint: treat the two current components
separately to find magnetic flux density.)
https://i.imgur.com/OtafAz1.jpg
3. Evaluation of a closed surface integral in cylindrical coordinates.
Given A = (r*cosψ, -r*sinψ, 0) in cylindrical coordinates. One considers a
volume bounded by the plane surfaces ψ = 0, ψ = π/2, z = 0, z = 1 and the
cylindrical surface r = 2, 0 < ψ < π/2.
(a) (10%) Please find the unit vectors normal to these five surfaces. You need
to plot a figure to illustrate which surface you refer to by your solutions.
(b) (10%) Evaluate F = ∮A‧ds.
4. Current I flows along a straight line from a point charge Q1(t) located at
the origin to a point charge Q2(t) located at (0, 0, 1). We are to find the
line integral (mmf) of the magnetic field intensity H along a closed square
path from (1, 1, 0) to (-1, 1, 0), (1, -1, 0), and back to (1, 1, 0).
(a) (8%) Find the mmf if we apply the general Ampere's Law (including the
displacement current) using a planar square surface formed by above 4 vertices
but slightly above the xy-plane.
(b) (8%) Repeat (a) but using the surface slightly below the xy-plane.
(c) (4%) Do you get the same results for (a) and (b)? Explain the meaning of
your results.
5. (20%) For the electric field intensity E = (E0‧cos(ωt - αy - βz), 0, 0)
in a free space (J = 0), find the necessary condition (in terms of α, β, ω,
μ0, and ε0) for the field to satisfy both Maxwell's curl equations (Faraday'
s
Law and Ampere's Law).
6. A rigid conducting bar of length L, mass M, and electric resistance R rolls
down along two parallel conducting rails that are inclined at an angle α, as
shown below. The rails are of negligible resistance and friction, and are
joined at the bottom so that the total resistance of the loop is R. The entire
arrangement us situated in a region of uniform static magnetic field density B
= (0, 0, B0) directed downward. Assume the bar is observed rolling down along
the rail with a constant velocity v due to the influence of gravity g. Answer
the following questions:
(a) (7%) Find the induced emf
(b) (8%) Show that the v = MgR‧tanα/(B0^2‧L^2‧cosα)
(c) (5%) If the magnetic field is directed upward, is it possible to get the
constant v? Explain your answer.
https://i.imgur.com/j4lTvPr.jpg