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推 amnotu:XD" 06/24 01:00
課程名稱︰普通物理學甲
課程性質︰
課程教師︰梁啟德
開課系所︰材料系
考試時間︰2 HR
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
Mid-term examination(17/4/2006)
1. A particle with eletric charge q moves along a straight line in a
uniform electric field E with a speed of u. the electric force exerted
on the charge if given by qE. The motion and the electric field ard
both in the x direction. Please show that the acceleration of the particle
in the x direction is given by (10%)
a= du/dt =qE*(1-u^2/c^2)^3/2/m
2. A thin nonconducting rod of length L has a positive charge of uniform
linear density λ as shown if Fig.1. Please determine the electric
potential V due to the rod at point P, a perpendicular distance d from
the ledt end of the rod. (15%)
Fig.1 一棒長L，一點P距其左端點d。
3. A sphere of radius 2a is made of a nonconducting material that has
a uniform volume charge density ρ. (Assume that the material does not
affect the eletric field.) A spherical cavity of radius a is now removed
from the sphere, as shown if Fig.2. Show that the electric field within
the cavity is uniform and is given by Ex=0 (10%) and Ey ρa/3ε0. (10%)
(Suggestion: The field within the cavity is the superposition of the
field due to the original uncut sphere, plus the field due to a sphere
the size of the size of the cavity with a uniform negative charge density
-ρ.)
Fig.2 一圓半徑2a，x軸正向上挖去一圓半徑a和大圓、x軸相切。
4. Twho spheres have radii a and b and their centers are a distance d
apardt. Show tha the capacitance of this system is given by
C = 4πε0/(1/a+1/b-2/d)
provided that d is large compared with a and b. (Suggestion: Because
the spheres are far apar, assume that the potential of each equals the
sum of the potemtials due to each sphere, and when calculating those
potentials assume that V = Q/4πε0r applies.) (10%) (b) Show that d
approaches infinity the above result reduces to that of two spherival
capacitors in series. (10%)
5. A material of resistivity ρ is formed into the shape of a truncated
cone of altutyde h as shown in Fig.3. The bottom end has radius b, and
the top end has radius a. Assume that the current is distributed uniformly
over any circular cross section of the cone, so that the current density
does not depend on radial position. (The current density does vary with
position along the axis of the cone.) Show that the resistance between
the two ends is described by the expression (10%)
R=ρh/πab
Fig.3 一上半被截去的圓錐，高h，上半部圓半徑a，底圓半徑b
6. The circuit in Fig.4 has been connected for a long time. (a) What
is the voltage across the capacitor? (5%) (b) If the battery is
disconnected, how long does it take the capacitor to discharge to one
tenth of its initial voltage? (5%)
Fig.4
┌─────┐
∣ ┌─┴──┐
∣ 1.00 8.00
10.0V Ω Ω
∣ ├1.00μF ┤
∣ 4.00 2.00
∣ Ω Ω
∣ └─┬──┘
└─────┘
7. A proton moves at 0.950c. Calculate its (a) rest energy, (5%) (b)
total energy, (5%) and (c) kinetic energy. (5%) Hint: the mass of a
proton is given by m = 1.67*10^-27 kg
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