課程名稱︰普通物理學甲上 課程性質︰必修 課程教師︰梁啟德 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰04/22 考試時限（分鐘）：120 是否需發放獎勵金：是,謝謝 （如未明確表示，則不予發放） 試題 : Mid-term examination (22/4/2010) 1. Two spheres have radii a and b and their centers are a distance d apart. Show that the capacitance of this system is given by 4πε0 C = ———————— provided that d is large compared with a and 1 1 2 — + — + — a b d b (Suggestion: Because the spheres are far apart, assume that the potential of each equals the sum of the potentials due to each sphere, and when calculating those potentials assume that Q V = ————— applies.) (10%). (b) Show that as d approaches 4πε0 infinity the above result reduces to that of two spherical capacitors in series. (10%) 2. A sphere of radius R surrounds a particle with charge Q, located at its center. (a) Show that the electric flux (電通量) through a circular cap of half-angle θ (Fig. 1) is given by Q ΦE = ————(1-cosθ) (10%). What is the flux for (b) θ= 90°? 2ε0 (5%) and (c) θ= 180°? (5%) ▄▄▄▄▄▄▄▄▄ ▄▄. ▄ ▄ .▄▄ ▄ ＼'--┬θ'／ ▄ ▄ R＼ ︱ ／ ▄▄ ▄▄▄ ＼／ ▄▄▄ ▄▄▄▄Q ▄▄▄▄ ▄▄▄▄▄▄▄▄▄ ▄▄▄▄▄▄▄▄▄ ▄▄▄▄▄▄▄▄ Figure 1 7me 3. Please calculate the electric field (電場) E(z) (10%) and the electric potential (電位) V(z) (10%) at points at the center axix of a plastic disk of radius R that has a uniform charge density of one surface. Here z corresponds to the distance between any point at the cnter axis of the disk and the center of the disk. The surface charge density of the disk is given by σ (in units of C/㎡). 4. A certain X-ray tube operates a current of 7.0×10^(-3) Ampere and a potential difference of 8.0×10^3 Volt. What is its power in Watt? (10%) 5. RC circuit. Figure 2 shows that when switch S is closed on a, the capacitor is charged through the resistor. Please show that the charge on the capacitor q(t) is given by q = Cξ(1-e^(-t/RC)) where t is time (10%) [Hint: q(t = 0) = 0] ┌───a S │ ───^^^^^^─┐ │ b R ┴ C ─┴─ │ ┬ ┬ │ │ │ │ │ └───┴──────┘ Figure 2 6. A rectangular coil consists of N = 100 closely wrapped turns and has dimensions a = 0.400m and b = 0.300m. The coil is hinged along the y axis, and its plane makes an angle θ= 30° with the x axis (Fig. 3). What is the magnitude of the torque (力矩大小) exerted on the coil by a uniform magnetic field B = 0.800T directed along the x axis when the current is I = 1.20A in the direction shown? (10%) y │ ＿│↖ ↑ ║ 0.300m ｜ ║＼ ↘ 0.400m║ ↘ ︳ I = 1.20A ｜ ║ ＼︳ ↓＿╙──-║-x ／ ＼30°║ ／ ＼ ║ z ＼║ Figure 3 7. A charged cork ball of mass 1.0×10^(-3)kg is suspended on a light string in the presence of a uniform electric field as shown in Fig. 4. When E = (3.00i + 5.00j)×10^(-5)N/C, the ball is in equilibrium at θ= 37.0°. Find (a) the charge on the ball and (5%) (b) the tension in the string (5%). i and j are unit vectors for the x- and y-axis (right-hand rule). ████████ ▕＼ ▕θ＼ ↗ E y ▕ ╳ ↗ │ ▕ ／ ╳ ┼─x ▕ ／ ○ q Figure 4 e = 1.6×10^(-19) Coulomb ε0 = 8.85×10^(-12) C^2 / N‧㎡ , g = 9.8 m/s^2 --

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