課程名稱︰普通物理學甲下
課程性質︰必修
課程教師︰蔡爾成
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試日期（年月日）︰2008/4/15
考試時限（分鐘）：120mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (a) [5%] Prove that the static electric field is perpendicular to the
equipotential surface and (b)[5%] points to the direction of
decreasing electric potential.
2. [10%] A nonconducting solid sphere of radius R is centered at the origin.
The charge density ρ, which vanishes outside the sphere, is equal to
→ → →→
ρ(x)=ρ |x|/R inside the sphere. Find (a) [5%] the electric field E(x)
0
→
(b)[5%] the electric potential ψ(x) inside and outside the sphere.
3. [10%] A noncoducting solid sphere of radius a has a uniform charge density
ρ. A spherical cavity of radius b is hollowes out of the sphere,
as shown in the figure below. The center of the solid sphere and
the spherical cavity are distance d apart. Show the the electric
field in the hole is uniform.
圖同Halliday 第八版 chapter 29 788頁 65題 FIG. 29-78
4.[10%] A single isolated spherical conductor of radius a may be considered
as a capacitor with the other conductor being a sphere at infinity.
(a) [5%]What is the capacitance? (b)[5%]Suppose charge Q is uniformly
distributed on the sphere. What is the electrostatic energy?
5.[10%] An initially uncharged capacitor C is fully charged by a device of
constant emf ξ, in series with a resistor R. Show that final energy
stored in the capacitor is half the energy supplied by emf device.
6.[10%] Show that, according to the free-electron model of electrical
conduction in metals and classical physics, the resistivity of
metals should be proportional to √T, where T is temperature in
kelvins.
7.[10%] In a certain cyclotron , a particle of charge q and mass m moves in
a circle of radius R. The magnitude of the magnetic field is B. What
is the kinetic energy of the particle.
8.(a) [5%] Prove that the torque acting on a rectangular current loop by a
→ → →
constant magnetic field τ = μ × B where μ is the magnetic dipole
→ → →
moment. (b) [5%] Show the the τ = μ × B also holds for a closed loop
of any shape.
→
9.[10%] Let B be the magnetic field produced by a current loop of finite size
carrying a steady current i, show that according the law of Biot and
→
Savart, at a distance r far away from the current loop, r^2*B→0 as r→0.
10. (a) [5%] Show that the electromagnetic energy of a inductor L carrying
a current i is (L*i^2)/2. (b) [5%] Show that the energy density for
the magnetic field is (B^2)/2*μ by using solenoid as an example.
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