[試題] 105-1 朱國瑞 古典電力學(一) 期中考
課程名稱︰古典電力學（一） Classical Electrodynamics (I)
課程性質︰物理學系選修、物理研究所碩士班必修、應用物理研究所碩士班選修
課程教師︰朱國瑞
開課學院：理學院
開課系所︰物理學研究所
考試日期（年月日）︰2016/12/09
考試時限（分鐘）：120 min.
試題 :
Electrodynamics (I), Mid-term Exam (6 problems) 12/9/2016
┌─Note──────────────────────────────────┐
│(1) You may use Jackson, a dictionary, and a calculator. Any other │
│ materials (including the lecture notes or insertions of any kind) are │
│ NOT allowed. │
│(2) You may use any equations in Jackson without derivation unless you are│
│ asked to derive it. │
│(3) Notations, equation no, and page no follow Jackson. │
└─────────────────────────────────────┘
1 ρ(x')(x-x')
1. Coulomd's law states E(x)＝──── ∫─────── dV'. Assume ρ(x')
4πε_0 │x-x'│^3
is a finite function of x'. Use Gauss' law and order-of-magnitude
considerations to explain why E(x)≠∞ as x→x' [x' is any point in ρ(x')].
(15%)
2. A hollow cube has six square sides. There are no charges inside. Five sides
are grounded. The sixth side, insulated from the others, is held at a
constant potential φ_0. Shoe that φ at the center of the cube is φ_0/6 by
using the physical argument in lecture notes, Sec.2.9[Do not use Eq.(1.44)].
(15%)
3. A spherical surface of radius a has a (static) surface charge density
σ(x)＝3 ε_0 E_0 cosθ (in spherical coordinates),
which corresponds to a volume charge density:
ρ(x)＝3 ε_0 cosθδ(r－a).
(a) Calculate the dipole moment p [＝∫xρ(x)dV].(10%)
Hint: Express x in spherical coordinates; dV＝r^2 dr dcosθ dφ.
(b) Calculate the potential φ (due to σ) at r＜a and r＞a (ε＝ε_0 in both
regions).(20%)
Hint: Use (1.17), (3.70), (3.53), (3.21), and P_1(x)＝x.
4. A point chsrge q is located in an infinite, uniform medium of permittivity
ε. Put the charge at the origin of coordinates. Calculate
(a) The potentialφ(x) everywhere.(5%)
(b) The polarization charge density ρ_pol(x)[＝－▽‧P(x)].(10%)
5. Explain why the derivation of the magnetic field energy in Jackson Sec.5.16
is valid only if the current J is established at an infinitesimally small
rate.(10%)
6. The two rails of a railroad track, insulated from each other and from the
ground, are connected by a voltmeter. What is the reading on the voltmeter
when a train is approaching at the speed of 180 km/hr, assuming that the
vertical component of the earth's magnetic field is 0.2 gauss and that tha
tracks are separated by one meter?(15%)
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※ 編輯: ycldingo (61.224.170.38), 01/20/2017 10:30:25