課程名稱︰電磁學(一)
課程性質︰電機工程學系大二上必修
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）︰2019/11/4
考試時限（分鐘）：10:20-12:10
是否需發放獎勵金 : 是
試題 :
P1. (a) Show that A X (B X C) = (A‧C) B - (A‧B) C. (A,B,C are vectors) (10%)
(b) Verify Stokes' theorem for the vector field A = (zx,xy,yz) and the
close path comprising the straight lines from (0,0,0,) to (0,1,0),
from (0,1,0) to (0,1,1), and from (0,1,1) to (0,0,0). (10%)
P2. (a) Consider an infinite plane sheet of charge in the xy-plane with uniform
surface charge density ρ and apply Coulomb's law to find the electric
field everywhere. (10%)
(b) Consider two parallel infinite plane sheet held a distance d apart. The
surface charge density is ρ at z = 0 and is -ρ at z = d. Find the
electric field everywhere. (5%) Calculate the voltage between the two
plane sheets. (5%)
P3. (a) Consider a constant current I flowing straight from (0,0,0) to (0,0,∞)
. Apply the Biot-Savart law to find the magnetic flux density on the xy
-plane. (10%)
(b) Consider the current described in (a) and another constant current flow
I flowing straight from (d,0,∞) to (d,0,0). Find the magnetic flux
density at (x,0,0) where 0 < x < d. (5%)
(c) Assume that the two currents (the problem did not specify that the two
currents imply the two in (a) and (b). However, the problem can't be
solved if the above condition were not given) flow uniformly in solid
cylindrical wires of radius a. Add a movable wire of length d - 2a so
that a constant current I flows straight from (d-a,0,0) to (a,0,0).
Find the force on the movable wire. This is a simple model of a railgun
(5%)
P4. Consider a rectangular loop (width and length: a and b) in a time-varying
magnetic field environment, where
B = (0,Bo cos wt,0).
The induced emf can be found by Vemf = -d/dt(∫S B．dS), where S is the
area formed by the rectangular loop.
(a) If one would like to maximize Vemf, how will you set up the orientation
of the rectangular loop？ (5%)
What is the maximum peak value of Vemf？ (3%)
When does this happen (t=？)？ (2%)
(b) Under the condition of (a), which component of E will not be affected
by this setup along the rectangular loop？ Please provide detailed expl
-nation for justification. (5%)
(c) Under the condition of (a) and (b), please find ∫L E．dl, where L
represents the rectangular loop in the counter-clockwise. You should
also define L in your solution. (5%)
P5. Consider an electric field
E = (0,Eo cos(wt-αy-βz),0),
in free space (J = 0).
(a) Find the magnetic field, H = (Hx,Hy,Hz) from Maxwell's curl equation
▽ X E = -∂/∂t(μH. (6%)
(b) If the E and H also satisfy the other Maxwell's curl equation (▽ X H =
∂/∂t(ε0 E), what is the relationship between α, β,
ε0 and μ0？ (6%)
(c) One now re-writes the magnetic field in (a) as
H(vector) = H(scalar) h(hat), please find H and the unit vector h(hat).
Please show that E．H = 0. (4%)
(d) Find the ratio of η= E / H？ Is it independent of α and β？ (4%)
P6. A magnetic field , B = Bo (1,2,-2) exists at a point. For each of the
following velocities of a test charge q, find the electric field E at that
point for the following conditions.
(a) v = vo (1,-1,1) to make the acceleration experienced by the test charge
being zero. (5%)
(b) v = vo along the line y = -z = 2x to make the acceleration experienced
by the test charge being zero. (8%)
(c) v = vo (2,1,2) to make a net force -2qE. (7%)