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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