課程名稱︰普物甲下
課程性質︰系定必修
課程教師︰梁啟德
開課系所︰生機系
考試時間︰94/6/22 10:20-12:10
試題 :
Final examination (22/6/2005)
1. Heisenberg's uncertainty principle tells us that (delta)px(delta)x>=h/(2pi)
where (delta)px and (delta)x represent the intrinsic uncertainties in the
measurements of the x component of r and p and h is the Planck constant.
Please prove that in a one-dimensional system, Eq. 1 is equivalent to
(delta)E(delta)t>=h/(2pi) where (delta)E and (delta)t represent the
instrinsic uncertainties in the measurements of the energy and time,
respectively. (10%) [Hint: Consider the kinetic energy of the particle
(delta)E.]
2. Please write down Maxwell's equations. [Hint: There are four of them in
total.] (20%)
3. We wish to use a glass plate with index of refraction n=1.57 to polarize
light in air. (a) At what angle of incidence is the light reflected by the
glass fully polarized? (10%) (b) What is the corresponding angle of
refraction? (10%) [Hint: Brewster's angle and polarization by reflection]
4. A capacitor in an LC oscillator has a maximum potential difference of 17V
and a maximum energy of 160uJ. When the capacitor has a potential
difference of 5V and an energy of 10 uJ, what are (a) the emf across the
inductor(10%) and (b)the energy stored in the magnetic field(10%)?
5. A charge q is distributed uniformly around a thin ring of radius r. The
ring is rotating about an axis through its center and perpendicular to its
plane, at an angular speed w. Please whow that the magnetic moment due to
the rotating charge is u = qwr^2/2 (10%)
6. Michael Jordan is 198cm tall. How tall must a vertical mirror be if he is
to be able to see his entire length in it? (10%)
7. Quantum Hall effect. Let us start with the classical Hall effect. (a)
Please prove that in a two-dimensional(2D) electron system, the Hall
resistance is given by R =V / I = B/(ne), where V is the Hall potential
H H H
difference, I is the current, B is the applied perpendicular magnetic
field, n is the carrier concentration (in cm^-2) and e is the electron
charge, respectively. (5%) (b) It has been shown that at extremely low
temperatures and high magnetic fields, the Hall resistance R shows
H
quantized values. From simple calculations we know veB/h = n where v is the
Landau level filling factor, h is the Planck constant and n is the carrier
density. Please prove R =h/(e^2v). (5%). This is the Nobel-prize winning
H
integer quantum Hall effect. [Hint: In 2D, the current density is given by
J=I/w where w is the width of the 2D system. Qhm's law is given by E=p J
H
where p =R is the Hall resistivity in 2D.]
H H
The permittivity constant etta0 = 8.85 X 10^-12 C^2/N/m^2
The permeabillity constant u0 = 4pi X 10^-7 T m/A
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