課程名稱︰普通物理學甲下
課程性質︰理組必修
課程教師︰趙元
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2018/06/28
考試時限（分鐘）：1020-1230
試題 :
Mid-term Exam for General Physics 1007
1. Terminologies: definitions and comparisons.
a. The four principle forces (interactions) of the physics.
b. What is a conductor? A semiconductor? An insulator? A superconductor?
Compare the mobility and density of the charged carrier; draw the
tempaerature dependence of resistance.
2. Four charged particles of charge q is put on tje corners of a tetrahedron
with each side of size a. The Coulomb's law F = (1/4πε)(q_1q_2/r^2),
F = qE.
a. What is the net force, field, potential on one of the particle?
b. If one of the four is missing, the net force on the rest?
3. Derive the Coulomb's Law about the electric field due to a charges
point-like particle using the Gauss' Law:
a. Write down the defination of eletric field flux from the surface integral
of electric filed E and differential area dA.
b. Find a surface that the electric field is identical on it as the Gauss
surface based on symmetry.
c. From Gauss' Law εΦ = q_enc, drive the electric field E(r) caused by a point
charge q from a distance of r.
4. Derive the electric field due to a charged disk with total charge q using
Coulomb's Law:
a. Consider a charged ring of radius R with charged density λ, write down
the differential field due to a segmet ds from a distance Z.
b. Integrate the differntial field to get the total field as
E = (1/4πε)(qZ/[R^2+Z^2]^(3/2)).
c. Consider that a disk is an integration of rings with thickness dr, show
that the electric field E = (σ/2ε)(1 - [Z/sqrt(Z^2+R^2)].
d. Taking an approximation that Z≧0,R→∞, showing that from (c) the
electric field due to an infinite sheet is independenct of Z.
5. A spherical conducting shell has a inner radius a = 10cm and an outer radius
b = 20cm with charge Q = +5e. At the center of the shell, a charged particles
with charge q = -3e is placed. The quantiity of the electron charge
e = 1.6*10^-19C and ε = 8.85*10^-12C^2/N-m^2.
a. Derive the electric field at r=5cm, r=15cm and r=30cm.
b. How much is the charge induced on the inner and outer surface of the shell
using the Gauss' Law?
c. What are the charges on both surfaces of the shell if grounded on the outer
one?
6. Two conducting plates arre placed in parallel at a distance d and each plate
has an area of A. Please derive the following:
a. The amount of charge accumulated on the plate is q for each. What is the
electric field between the two plates?
b. What is the new capacitance and the work needed if two di-electric material
κ_1 and κ_2 each with thickness b = 0.5d is inserted.
7. Rutherfold's scattering experiment uses an alpha-particles source shooting
at a gold foil target. A few of them are found to have very large scattering
angle. It is later identified as the nuclei of helium (He2+). If the initial
kinetic energy of the alpha-particle is 50MeV shooting on lead, what is the
closest distance to the lead nuclei could be reached? The atomic number(Z)
of lead is 82.(1 Joule = 6.242*10^18eV)
8. Microscopic view of Ohm's Law:
a. From Newton's law, find the accerlation a of the charged carrier of charge
e and mass m in an elactric field E. Derive the drift speed v = aτ, where
τ is the mean free time, in terms of e, m, E, τ.
b. As current density J = nev and E = ρJ, show the resistivity ρ = m/nτe^2.
9. The equivalent resistance/capacitance of resistors/capacitors connected.
a. Write down the equivalent resistance of three resistors R_1, R_2, R_3
connected in series and in parallel.
b. Write down the equivalent capacitance of three capacitors C_1, C_2, C_3
connected in series and in parallel.
10. For an RC circuit, find the current and charge on the capacitors as a
function of time t. A general from the solution is
q(t) = q_p + Ke^(-t/τ).
a. Draw the circuit and write down the equation of the potential charge:
an EMF device ε, a resistor R, and a capacitor C.
b, For t=∞, the capacitor is fully charged. One can obtian q_p?
c. At time t=0, assuming that on the capacitor q(t=0) = 0, find K?
d.Put the solution of K and q_p back to obtain time constant τ and i(t).
11. The electric dipole moment p = qd and the magnetic dipole moment μ = NiA,
while the torque on a dipole is τ = p×E or τ = μ×B.
a. From dU = -dW = -τdθ, derive the potential energy U(θ) = -p‧E.
b. What's the work to flip a circular coil of 1500 turns with an area of
5.04*10^-4m^2 and a current of 100μA by degree(90)?
c. What's the work to flip a water molecular of dipole moment 3.1*10^-30 C-m
under an eletric field of 1.5*10^4 N/C by degree(180)?