課程名稱︰普通物理學甲下 課程性質︰物理系必修 課程教師︰張寶棣 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰2012/05/15 考試時限（分鐘）：200 min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Please answer the following questions. a. Describe the Gauss' Law. If our space is four dimentional, what will you expect the r dependence of the Coulomb's law? (5 %) b. What is the origin of electric resistance in material? Please give an argument to exlain why capacitance depends on the geometry of the capacitor. (5 %) c. Why was the Maxwell displacement current introduced? Suppose that the space between a capacitor is filled with a dielectric, what is the change of the Maxwell displacement current inside the capacitor? (5 %) 2. The charge density in a region of space of spherically symmetric is given by ρ(r) = C e^(-r/a) when r < R and ρ= 0 when r > R. Find the electric field as a function of x. (10 %) 3. Three concentric conducting spherical shells have radii a, b, and c such that a < b < c. Initially the inner shell is uncharged, the middle shell has a positive charge Q, and the outer shell has a negative -Q. (a) Find the electric potential of the three shells. (b) If the inner and outer shells are now connected by a wire that is insulated as it passes through the middle shell, what is the electric potential of each of the three shells, and what is the final charge on each shell? (15 %) 4. A parallel plate capacitor of area A and separetion d is charged to a potential difference V and is then removed from the charging source. A dielectric slab of constant κ= 2, thickness d, and area 1/2 A is inserted as shown in Fig. 1. Let σ_1 be the free charge density at the conductor - dielectric surface and σ_2 be the free charge density at the condutor-air surface. (a) Why must the electric field have the same value inside the dielectric as in the free space between the plates? (b) Please find relation beween σ_1 and σ_2. (c) Find the new capacitance and new potential difference. (15 %) --------------------- / / ----------------------- | | | κ | ----------- | | / ---------------------- Fig. 1 5. A digital voltmetre of internal resistance r is used to measure the voltage across a capacitor in an RC circuit after the switch in Fig. 2 is closed. Because the metre has finite resistance, part if the current supplied by the battery passes through the metre. (a) Apply Kirchhoff's rules to this circuit, and use the fact that i_C = dq / dt to show that this leads to a differential equation dq q r R_eq ---- + - = ----- ε, dt c r+R where R_eq = rR/(r+R). (b) Solve the differential equation using the method described in the class to obtain q and V_C as a function of t. (10 %) Voltmetre ----V---- | | | | ------R-------------C------- | | | | | ╱ | ---- -------------│｜------- S ε Fig. 2 6. A Hall-Effect probe operates with a 120-mA current. Whem the probe is placed in an uniform magnetic field of magnitude 0.080 T, it produces a Hall voltage of 0.700 μV. (a) When it is measuring an unknown magnetic field, the Hall voltage is 0.330 μV. What is the magnitude of the unknown field? (b) The thickness of the probe in the direction of B is 2.00 mm. Find the density of charge carriers, each of which has charge of magnitude e. (10 %) 7. The rectangular frame in Fig. 3 is free to rotate about the axis A-A on the horizontal shaft. The frame is 10 cm long and 6 cm wide and the rods that make up the frame have a mass per unit length of 20 g/cm. An uniform magnetic field B = 0.2 T is directed as shown. A current may be sent around the frame by means of the wires attached at the top. (a) If no current passes through the frame, what is the period of this physical pendulum for small oscillation? (b) If a current of 8.0 A passes through the frame in the direction indicated by the arrow, what is then the period of this physical pendulum? (c) Suppose the direction of the current is opposite to that shown. The frame is displaced from the vertical by some angle θ. What must be the magnitude of the current so that this frame will be in equilibrium? (15 %) |← wire → | | | ＼＼ | ＼＼ ↙ ↙ | ↙ / ＼＼ | 6 / ↙＼＼ ↙ ↙| / ＼＼ | cm / ↙ ＼＼↙ | / ↙ ＼＼ | Fig. 3 ↗ / ↙ ＼＼| / ↙ ＼＼ ╲ ↙ ↙ / ＼＼ ↙ ╲ / ＼＼ ╲ ↙ ↙ / A ↙ ╲ ↙ / B↙ ↙ ╲ / ↙ ↙ ╲ ↙ / ↙ ╲ / 8. A long cylindrical conductor of radius a has two cylindrical cavities of diametre a through its entire length, as shown in Fig. 4. A current I is directed out of the paper and is uniform through a cross section of the conductor. Find the magnitude and direction of the magnetic field in terms of μ_0, I, r, and a at (a) point P_1 and (b) point P_2. (10 %) --------------P_1 ↑ | r ↑ | a ↓ ↓ -------------x-------------P_2 ↑ | a | ↓ | |←----r----→| Fig. 4 --

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