課程名稱︰電磁學一
課程性質︰物理系必修
課程教師︰賀培銘
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰98/01/13
考試時限（分鐘）：180分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(15%) Two layers of materials of constant polarization and magnetization Pi
and Mi(i=1,2)
^ ^ ^ ^ ^ ^
P1 = x*Px + y*Py + z*Pz , M1 = x*Mx + y*My + z*Mz (0<z<a)
^ ^ ^ ^ ^ ^
P2 = x*P'x + y*P'y + z*P'z , M2 = x*M'x + y*M'y + z*M'z (a<z<b)
extends along the x-y plane. Find E,D,B,H for (i)0<z<a ,(ii)a<z<b , (iii)b<z
2.(15%) A spherical shell of inner radius a and outer radius b (b>a) is made
of linear dielectric with permittivity ε = 2ε0. It is placed in an
^
otherwise uniform electric field E = E0*z. Find the electric field in
the cavity (0<r<a).
3.(15%) A long cylinder of radius R with charge density ρ(s) = a/s rotates
around its axis, the z-axis, with angular velocity ω.(a is a constant)
The permeability of the cylinder is μ.
(a) Find the vector potential A inside and outside the cylinder, assume
▽．A = 0.
(b) A circular wire of radius L on the x-y plane, surrounding the cylinder
(L>R), has a constant line charge density λ. The circle is initially
at rest, and then the angular velocity of the cylinder is dropped from
ω to 0. Find the angular momentum of the circle after the cylinder
stops rotating, assuming that the circle rotates without friction.
^
4.(15%) A magnetized long cylinder of radius R with M = M0*x for constant M0
lies along the z-axis. Let H = -▽W for some function W. Find the
equation satisfied by W inside and outside the cylinder, and the
boundary condition W needs to satisfy on the surface. Then find the
magnetic field B inside the magnetized cylinder(s<R).
5.(a)(10%) Write down all Maxwell's equations.
(b)(5%) For the electric field given by the following standing wave
^
E = x*E0*cos(ωt)*cos(kz), find (i) the relation between ω and k,
(ii) the magnetic field B.(Assume that there is no charge nor
current in space.)
6.(20%) Which of the following statement(s) is(are) wrong?
(a) Lorentz force law can be derived from Maxwell's equations.
(b) If the magnetic monopole exists, it will be possible to violate the
energy conservation law.
(c) The electric field E exerts a force on a magnetic charge in the
direction of its motion.
(d) The fact that generators generate electric currents is an evidence of
Faraday's law.
(e) The flux rule ε = -dΦ/dt is invalid when the magnetic field is
changing and the loop is moving at the same time.
The general solution of Laplace equation with rotation symmetry in spherical
coordinates is
∞
V(r,θ) = Σ [Al*r^l + Bl/r^(l+1)]Pl(cosθ),
l=0
P0(x) = 1 , P1(x) = x , P2(x) = [(3*x^2)-1]/2,
1
∫dx Pm(x)Pn(x) = [2/(2*m+1)]δmn,
-1
and the general solution with translation symmetry in the z-direction in
cylindrical coordinates is
∞
V(s,ψ) = a + b*logs + Σ {s^k[ak*cos(kψ) + bk*sin(kψ)] + s^(-k)*[ck*cos(kψ)
k=1
+dk*sin(kψ)]}.
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