[試題] 100下 林敏聰 普通物理學甲下 期中考

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課程名稱︰普通物理學甲下
課程性質︰必帶
課程教師︰林敏聰
開課學院：工學院
開課系所︰材料系
考試日期（年月日）︰101.05.09
考試時限（分鐘）：10:20 ~ 13:00
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

1.Please describe in details how to synchronize the time clock in an inertial
 reference frame in special relativity.(5%) What is its difference from that
 in Newtonian Mechanics?(5%) Why?(5%)

2.
 (a) Please derive the kinetic energy K in special relativity.(10%)
 (b) Please derive the energy-momentum relation in relativity:
     E^2 = c^2p^2 + (m_0 c^2)^2. (5%)
 (c) Please show that a particle whose mass at rest is zero must travel with
     velocity equal to c. (5%)

3.
 (a) Show that for an extremely relativistic particle, the particle speed, u
     differs from the speed of light, c, Δu = c - u = (c/2)(m_0 c^2/E)^2,
     where m_0 is the rest mass and E is the energy. (10%)
 (b) Find Δu for electrons produced by
     1) NSRRC in Hsinchu Science Park, Taiwan, where E = 1.5 GeV.(5%)
     2) LHC in CERN, Switzerland, where E = 7000 GeV.(5%)

4.In an RC circuit as shown (a) Calculate the time constant. (b) Find the
 maximum charge on the capacitor during charging.(15%)

                      R_1
                 ┌─/\/\/\─┬───┐
                 │          │      │
                 │          │      │
                 ●          /       ┴ C
                 ε          \ R_2   ┬
                 ●          /       │
                 │          \       │
                 │          │      │
                 └─────┴───┘

5.Four particles form a square. The charges are q_1 = q_4 = Q and q_2 = q_3 = q
 (a) What is Q/q, if the net electrostatic force on particles 1 and 3 is zero?
 (b) Find the electric field at particle 2 by using Coulumb's law.
 (c) Find the electric field at particle 2 by using Gauss' law. (15%)

                     ↑
                     │
               q_1   │    q_2
               ●    │    ●
                     │
                     │
                     │
         ──────┼─────→x
                     │
                     │
               q_3   │    q_4
               ●----┼----●
                     │ a
                     │

6.Consider a cylindrical capacitor. The figure shows the cross section view. A
 cylindrical capacitor of length L formed by two coaxial cylinders of radii a
 and b, filled by two different dielectric materials with the dielectric
 constant κ_1 and κ_2 (see the figure). Assume L >> b, please calculate the
 capacitance. (15%)


                       ┌───┬───┐
                       │      │      │
                       │  ┌─┴─┐  │  (請把它想像成圓形)
                       │  │ a↑  │b │
                       │  │  └─┼→│
                       │  │      │  │
                       │  └─┬─┘  │
                       │ κ_1 │ κ_2 │
                       └───┴───┘

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