※ [本文轉錄自 NTU-Exam 看板] 作者: ultimatePTT (終極俊) 看板: NTU-Exam 標題: [試題] 94下 蔡爾成 普通物理學甲 期末考 時間: Mon Jun 26 17:19:53 2006 課程名稱︰普通物理學甲下 課程性質︰系定必修 課程教師︰蔡爾成 開課系所︰大氣系 考試時間︰95/06/26 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. [10%] Show that the magnitude of the static magnetic field produced at the center of a rectangular loop of wire of length L and width W, carrying a current i, is 1/2 2μoi (L^2 + W^2) B = ─── ─────── π LW 2. (a) [5%] Two inductors L and L are connected in parallel, show that the 1 2 equivalent inductance is given by 1 1 1 ─── = ── + ── L L L eq 1 2 (b) [5%] Why must the separation of these two inductors be large for the above relationship to hold? 3. [10%] In an oscillating RLC circuit, show that the fraction of the energy lost per cycle of oscillation, ΔU/U, is given to a close approximation by 2πR/ωL. 4. [10%] Prove that the displacement current in a parallel-plate capacitor of capacitance C can be written as i = C(dV/dt), where V is the potential d difference between the plates. 5. [10%] Slits of unequal widths are used in a double-slit arrangement to produce an interference pattern on a distant screen. If only the narrower slit 1 is illuminated (the wilder slit 2 is covered), the light reaching the center of the pattern has amplitude Eο and intensity Iο. If only slit 2 is illuminated, the amplitude of light reaching the pattern center is 2Eο. When both slits are illuminated and a two slits interference pattern is on the screen, what is the intensity I(θ) in the pattern as a function of angle θ? 6. [10%] By macking use of the example of a solenoid, show that the energy density of the static magnetic field is 1 2 U = ── B B 2μo 7. [10%] The complex impedance is defined as the ratio of complex potencial over complex current for a device in an alternating circuit. Prove the complex impedance for capacitor C is 1/jωC and the complex impedance for inductor L is jωL, where j = (-1)^1/2. 8. [10%] Prove that the speed of the Electromagnetic wave c = (μoεo)^(-1/2) by giving an example of wave solution to the Maxwell equations in free space 9. (a) [5%] Write down the space-time transformation (Lorentz transformation) in special relativity for inertial frames with relative velocity v along the x-axis. (b) [5%] If two events (t ,x ), (t ,x ) occur at the same space point 1 1 2 2 x = x but at different times t ≠ t , prove that it is impossible to 1 2 1 2 find another inertial frame in which these two events occur simultaneously under special relativity. 10. For a point particle with mass m and speed v, (a) [5%] What are the relativistic expressions for its energy and momentum? (b) [5%] If the speed v is much less than the speed of light c, prove that the relativistic expressions for energy and momentum approach the corresponding classical expressions in the Newtonian mechanics. --

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