[試題] 99下 賀培銘 電磁學 期末考
課程名稱︰電磁學二
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）︰2011-06-23
考試時限（分鐘）：180min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(15%)An infinite straight wire carries the current
0 , for t≦t0
I(t)= I0, for t0≦t≦t1
0 , for t1≦t
This is, a constant current I0 is turned on abruptly at t = t0 and then
turned off abruptly at t =t1
(a)find the resulting vector potential A using the formula of retard potential
(b)check that A derives in the limit t0 → -∞. Does it mean that this result
is simply wrong, or how should we interpret the result?
2.(10%)Determine the Lienard-Wiechert potential V(x,t) for a point charge
in hyperbolic motion
^
w(t)=[b^2+(ct)^2]^1/2 x
Assume that the point of observation r is on the axis and to the right of the
charge.
3.(10%)An electric dipole is made of point charges q and -q with position
vectors
^ ^
w±q(t) = ±R(xcos(wt)+ysin(wt))
Find the time-averaged power of radiation.
4.(20%)Including radiation reaction, Newton's 2nd law for a charged particle
becomes
.
a =τa + F/m
where F is the external force acting on the particle. Let F be a constant F0
between t = 0 and t = T, and it vanishes for t<0 and t>T. Assume that you
exclude the runaway solution, find
(a) a(t) for all t∈(-∞,∞)
(b) v(t) for all t, assuming that v(-∞)=0
(c) Wext, the work done by the external force
(d) Using the Larmar formula, we can calculate the total energy radiated Wrad
(you do not have to calculate it) for t∈(-∞,T) and the kinetic energy
Wkin = (1/2)mv^2(T).
Do you expect the conservation of energy as Wert = Wrad + Wkin?
5.(10%)An electromagnetic plane wave of angular frequency w is traveling in
the x direction with speed v relative to the origin system S. Find the
_ _
electric and magnetic field in S and express them in terms of the S coordinate
_ _ _ _ _ _ _ _ _ _
E(x,y,z,t) and B(x,y,z,t).
^
6.(20%)A point charge q moves with the velocity v = vx in a constant electric
^ ^
field E = Ey and magnetic field background B = Bz
(a)Find the condition on E and B such that the magnetic field is absent in an
_
inertial frame S moving in the x direction with respect to the origin frame S
(b)Assuming that the condition in the precious question is satisfied, find the
_
relative velocity of S with respect to S
_
(c)Find the Lorentz force on the charge in S
(d)Show that the force in S is related to that in S through suitable Lorentz
transformations.
7.(15%)Which of the following(s) is(are) correct?(multiple choice)
(a)Az = Ax can be used as a gauge fixing condition.
(b)Whenever a point charge radiates (with radiation field taking energy to
the infinity), ut must also feel a nonzero radiation reaction.
(c)Energy conservation is broken by the radiation reaction Frad.
(d)The radiation reaction Frad leads to either runaway solution or acausal
preacceleration. At the scales at which classical electrodynamics, we should
avoid acausal preacceleration and accept runaway solution.
(e)
μν
F F is invariant under Lorentz transformations.
μν
=======================================================================
Equations you may or may not need:
Lienard-Wiechart potentials
1 qc
V(r,t) = ─── ────
4πε0 (rc-r‧v)
v
A(r,t) = ──V(r,t)
c^2
^
Electric dipole radiation for p = p0cos(wt)z
μ0 p0 w^2 sinθ ^
E = - ─────(──)cos[w(t-r/c)]θ
4πc r
1 ^
B = ─ E θ
c θ
↑E的下標
^
Magnetic dipole radiation for m = m0cos(wt)z
μ0 m0 w^2 sinθ ^
E = ─────(──)cos[w(t-r/c)]φ
4πc r
1 ^
B = - ─ E θ
c φ
↑E的下標
Time-averaged power of radiation for a linear dipole moment p0 with angular
frequenct w is
μ0 p0^2 w^4
P = ──────
12πc
Larmor formula
μ0 q^2 a^2
P = ──────
6πc
Radiation reaction
μ0 q^2 .
Frad = ──── a
6πc
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