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1 . (a Find the eigenvalues and eigenfunctions of problem: (b The differential equation ( ( ( =, - =, =. (8% - - = has one basis solution =. Show that the other basis solution on the interval - < < is = ln -. A forced oscillator is governed b (, (, ( r t A B = = =,.(% where r( t is the unspecified driving force. Show that the solution can be epressed as in B = *. (% convolution form, t ( Acos( t sin ( t sin ( t rt ( 3. Find a basis of eigenvectors and then diagonalie the following matri (% Write down the (a Divergence Theorem, (b Stokes s Theorem, and (c Green s Theorem in the Plane. (d Laplace s Equation, =, in Clindrical Coordinate. (% a 5. Find the Fourier transform of the function e -, where a >. (% w w 6. Consider semi-infinite string that satisfies the wave equation = c, c is a constant. t This elastic string also satisfies the following conditions: (i It is initiall at rest on the -ais from = to. (ii For time t > the left end of the string is moved in a given fashion, sint t namel, w(, t = f ( t =. (iii Furthermore lim wt (, = for t. otherwise Please use the Laplace transform to solve the wave equation. (% 7. Find the integral (a 4 4 d, (b - sin d. (% 8. (i Show that i = are the branch points of f ( branch cut(s of this comple function. (% =. (ii Plot and discuss the possible

2 . The Earth has moved around the Sun 5 ears with velocit v = 3 4 m/ s, what is the time difference between two clocks in the Earth and the Sun. (%. Two spin one half particles ( Sand S interact to each other, the Hamiltonian is H = JS S what is the ground state and ground state energ forj > and J <, respectivel. (% 3. H = H HI = and perturbation interaction H = with <<. Please, H 3 calculate the Eigen value to second order and Eigen function to first order. (5% I 4. A three dimensional simple harmonic oscillator has the p Hamiltonian H = k(, please write down the degenerac number of the m states from the first to the forth ecitations. (% 5. The Hamiltonian for an aiall smmetric rotator with angular momentum L= L L L ish =, what are the Eigen values of H. (% I I 3 6. An electron is confined to a one-dimensional infinite potential well of widtha. Calculate the epectation values of and for the ground state. (5% d 7. The one dimensional harmonic oscillator, H = - k, its ground state trial wave m d function is ( = Aep( -B. What is the value of B (% 8. The one dimensional quantum sstem has a delta potential well, V( = - av (, what is the bond state energ (% 9. Radiation of wavelength = 9 nm falls on a metal surface for which the work function isw= 4. 5eV. What potential is needed to stop the most energetic photoelectrons - 34 ( h = 6.63 J s (%

3 . (5% Find the interior and eterior magnetic fields of an ideal solenoid with radius R, length L and N turns while a current i is applied.. (3% Describe Hall effect. What kind of phsical quantities can be obtained from Hall effect Describe their significance. 3. (5% Describe the Young s interference eperiment and interpret its phsical meaning/significance. 4. (% Give phsical definitions to conductor, insulator, semiconductor, and superconductor.

4 . Briefl give the definition of following terms, please deliver relation formulas if need: (a. Channel length modulation effect (3% (b. Show four amplifier tpes, including circuit model, gain, ideal characteriation of input/output resistance. (6% (c. Transmission frequenc. (3% (d. Concentration profile of the minorit carriers of pnp transistor operating in active mode. (5%. Fig. shows a discrete-circuit CS amplifier. Answer following questions. (In figure, capacitors are ver large, shown as infinite (a. If the transistor has V t =V, and k n W/L= ma/v, verif that the bias circuit establishes V gs =V, I D =ma, and V D =7.5V. That is, assume these values, and verif that the are consistent with the values of the circuit components and the device parameters. (% (b. Find g m and r if V A =V. (6% (c. Draw a complete small-signal equivalent circuit for the amplifier assuming all capacitors behave as short circuit at signal frequencies. (6% (d. Find R in, v gs /v sig, v /v gs, and v /v sig. (6% 3. Fig. shows a transistor amplifier, please determine its voltage gain. Assume =. (% 4. Comparison of the MOSFET and BJT, briefl answer following questions. (a. Circuit Smbol. (% (b. DC bias for active mode. (% (c. current-voltage characteriation formula in the active region. (3% (d. Low-frequenc hbrid- model. (4% (e. High-frequenc Model. (4% Fig. Fig.

5 . (6 Points The electromagnetic wave travels from medium (permeabilit, refraction inde n to medium. (permeabilit, refraction inde n with oblique incidence. The plane forms the boundar between that two linear non-conducting media. The Fresnel s equations for the case of polariation in the plane of incidence are a - b a b = ( a b = ( R I T I Where R and T I are the comple amplitudes of reflected transmitted and incident waves, respectivel, and a cosq cosq T I mn mn b. (a. Calculate the Brewster s angle B if ~. (b. Calculate the reflection and transmission coefficients, and then prove the energ conservation.. (4 Points A uniforml magnetied sphere of radius R with magnetiationm= M is situated at the origin. Answer the following questions (a. Calculate the magnetic induction B inside the sphere. (Hint: The magnetic induction B inside a spherical shell, of radius R, carring a uniform surface charge velocit, is = ms Rw 3 B and spinning at angular (b. Calculate the magnetic induction B at point (,,R in term of M. (Hint: The magnetic m induction B caused b a magnetic dipole m is B = [3(m rr - m] 3 4 p r

6 (c. If now the measured magnetic induction and auiliar field inside the sphere are B and H, respectivel, and then a spherical cavit is hollowed out of that sphere. Find the auiliar field H at the center of the cavit in terms of H and M.. (4 Points If we consider the monochromatic electromagnetic wave confined to the interior of a rectangular wave guide (assumed to be a perfect conductor and a > b and propagated along the ais as shown in the Fig., the fields can be epressed as (,,,t = i(k- wt (, e B ~ (,,,t = B ~ i(k- wt (, e where = E E E and B ~ = B B B, and also E i E = (k w c - k B E i E = (k w c - k - B i B B = (k - c - k w c E i B B = (k c - k w c E The uncoupled equations for E and B are : [ c - k ]E = ( / c k ]B [ w - = (a. Calculate the E (, solution of TM mn mode. (b. Find the lowest TM cutoff frequenc for this wave guide. a b Fig.

7 . ( Points A sphere of homogeneous linear dielectric material with the electric susceptibilit c e is placed in an uniforml eternal electric field E. (a. Find the electric potential and electric field inside the sphere b solving Laplace s equation. (b. Find the electric field produced solel b the polariation P inside the sphere.. (6 Points The magnetic field at a distant s from a straight segment of wire carring a stead I current I is (sin - sin, where and are the initial and final angles, respectivel. 4 s (a. Find the magnetic field at the center of a square loop, which carries a stead current I. Let R be the distance from center to side, as shown in Fig. (a. (b. If an infinite straight wire carring the same stead current I is placed near that square loop, as shown in Fig. (b. Find the force on that square loop. Fig.