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 :
Final Exam for General Physics 1007
1. Biot-Savart's Law dB = (μ/4π)(ids×r/r^2) and Ampere's Law:
a. Derive the magnetic field due to a long straight wire with current i as
B(R) = μi/2πR.
b. From Ampere's law, derive the same magnetic field based on symmetry.
c. Derive the force between two parallel current: F_ab = [μL(i_a)(i_b)]/2πd.
2. An RL circuit with current i(t):
a. What are potential changes across a resistor R and an inductor L.
b. Write down the equation of a battery ε, L, R following the loop rule.
c. Consider the initial voltage of the V_L(t=0) = ε, write down the loop
equation, show that a solution i(t) = Ie^(-t/τ), derive and solve the
current amplitude I and time constant τ with ε, R, L.
3. Am RCL circuit has electrical and magnetic energies:
U_E = (q^2)/2C amd U_B = (Li^2)/2
a. From the energy conservation, explain and write down the time differential
equation for an RLC circuit as a function of q, t, R, L, C.
b. A general solution to this equation is q = Qe^(-Rt/2L)cos(ω't+φ).
Derive that ω'= sqrt(ω^2-(R/2L)^2) where ω = 1/sqrt(LC).
4. An RLC circuit with external driven potential ε = Εsin(wt), the impedance
of the circuit is Z = Ε/sqrt(R^2+(X_L-X_C)^2) where X_L = wL, X_C = 1/wC
and tanφ = (X_L-X_C)/R. Explain the loading condition fot (1) X_L > X_C
(2) X_L = X_C (3)X_L < X_C. Showing the phase differnce φ by drawing
qualitative curves of I(t) and ε(t).
5. Write down the four Maxwell's equation about electric and magnetic fields.
a. The corresponding equation of Guass' Law for electricity and magnetism,
Faraday's Law, Ampere-Maxwell's Law. (explain symbols, no derivation
needed)
b. Explain why Guass' Law for magnetism is zero? Explain what is the displaced
current, i = ε(dΦ_E/dt).
6. A comet goes around the sun. At point A, the neutral dust particles of its
tail goes in the tangent direction of its orbit.
a. Assuming dust is in spherical shape, what would be the size if the density
is 2.5*10^2 kg/m^3 and the sun light is totally absorbed?
b. If the particles are fiber, what would be the path curve?
(P_s = 3.9*10^26W, M_s = 1.99*10^30kg, G = 6.67*10^-11N-m^2/kg^2
7. The refraction of an incident light on the refraction between two medias.
a. Write down the Snell's law of refraction with refraction index n_1, n_2 and
incident angle θ_1 and θ_2.
b. A total reflection happens when the refraction angel θ_r>π/2. if the
incident angle θ_i > θ_c. Derive the critical angle θ_c with n_1 and
n_2.
c. Sir D. Brewster found that when θ_i/B + θ_r = π/2, the reflected light
will be fully polarized, Derive θ_B with n_1 and n_2.
8. Particle Physics:
a. Write down the three generations of quarks, leptons and mediators.
b. From the charge and strangeness, derive the qrark compounds of p+, n0,
K±/0 and π±/0.
9. The interference of two coherent lights goes to maximimum if their light
path is integer times of their wavelength. To measure the refraction index
of their air, we use Michelson's interferometer with two identical
transparent boxes of internal width 10cm. We gradually make one into vacuum
and sees 120 bright spot changes. Draw the light path of a Michelson's
interferometer. Assuming the wavelength of the light source is λ = 590nm,
what's the refraction index of the air?
10. To make a portable light spectrometer, one can make a Blueray disc.
a. The track pitch of BD is 0.23μm, refracting index of the coating is 1.55,
calculate the angle of the first maximum for light λ = 400nm.
b. Would it be possible to distinguish between two characteristic wavelengths
λ = 589.59nm and 589nm of the sodium light if the light beam size is 0.5mm?
11. Einstein's special relativity:
a. Write down the two assumption of Einstein's S.R..
b. Write down the total energy E^2 with p, m and light speed c.
c. Calculate the total energy in Joule of a proton with 8 GeV kinetic energy.
d. The matter wavelength of this proton?
12. Secondary particles from cosmic ray:
a. Explain/derive the time dilation with light clocks.
b. The energetic cosmic ray hits earth atmosphere could generate muons, with
lifetime 2.2 μs. If a muon is produced at 25 km above the surface with a
apeed of 0.995c, could it reach the earth surface before dacaying?
Calculate the time needed in muon rest frame.