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. 0 --

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