医学物理历年试题TestPaper05A - A

Score

D. Filling in the following blanks with your calculating results (6

problems, 4 marks for each problem)

Planck constant: h?6.626?10Light speed: c?3.0?10m,

8?34J?s ?4.136?10?15eV?s

Boltzmann constant k = 1.38 ? 105 J/K

1. There are two kinds of radioactive nuclides whose half-life are 6 days and 12 hours

respectively. Suppose that their radioactivities (also called radioactive intensity) are the same. At a particular moment, if the number of the latter (后者的) radioactive nuclide

is 3?10, Find the number of the former （前者）radioactive nuclide. Solution:

18I?N?

N1I1/?1?2T16?24?????12 N2I2/?2?1T212

N1?3?1018?12?36?1018

2. There are 1000 cars in a motor match. If each car produce 55 db noise level, what is

the sound intensity level produced by all the cars at the same time. Answer: ___85db or 8.5B___. Solution:

10lg1000III?10(lg1000?lg)?30?10lg?30?55?85 I0I0I03. A hyperopic patient has a near point of 50 cm. In order to let him read the paper 25 cm

in front of him, calculate the lens power required to remedy this defect. Solution: Using the thin lens equation, now u = 0.25m, v = -0.6m, so we have

11111?????2.0 fuv0.250.6 So the lens power is 2D

4. Silicon has an atomic number of 14. That is Z = 14 for this atom. Write the

ground-state configuration for this atom (ground-state configuration of atoms is also

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called electronic configuration, or the distributions of electrons in atoms). __1s2 2s2 2p6 3s2 3p2__.

5. A lens made from glass with a refractive index of 3/2 is put in air that has a refractive

index of 1. The focal length of the optical system is __0.5____ if the lens has two convex surfaces with curvature radii 2/5 cm and 2/3 cm respectively. Solution:

?n?n0f???n?0?11?????r?r???2???1?1?(0.5*(5/2?3/2))?1?1 26. The wavelength of light used in Compton Effect experiment is 0.51nm. The scattered wavelength at a scattered angle of 1800 is _________ . (Compton wavelength ?c = 0.00243nm) Solution: Score

E. Proving and Calculating problems (Total marks: 16)

????????2?csin2???0.51?2?0.00243?(1)?0.51486nm

?2? 1. (4 marks) Consider a single particle moves in a central force field. At a particular

moment, it is at the position r with momentump. Prove that in such a central force

????LLfield, the particle’s angular momentum is conserved. (Hint: in order to prove ??is conserved, you should prove that L is time-independent or L is a constant; you

should give brief explanations for crucial (关键的) steps.). Proof:

???L?r?p

??????????dLdr??dp???p?r??v?(mv)?r?F?m(v?v)?r?F dtdtdt6

?? As the force F in the central force field has the same direction with radius vector r,

????so r?F?0, Of course v?v?0. Therefore we have

?dL?0, dt So the angular momentum is a constant or conserved.

2. (6 marks) Eight identical raindrops (雨滴) of radius 1 cm coalesce (接合) to form a single raindrop. (Hint: Consider the rain drops are spherical in shape. Surface tension coefficient of water = 7.0 ? 10-2 Nm-1). (1) Find the energy released (释放出的)

(2) Find the difference between the additional pressures within the small and big

raindrops. (Hint: the additional pressure is caused by the surface tension of the raindrop)

Solution: (1) Suppose that the radius of the small raindrop is r, the one of the big raindrop is R, according to the liquid property that liquid is incompressible. So the volume of the eight small raindrops is equal to the volume of the combined one.

448??r3??R3?R?2r

33 The surfaces of the eight identical raindrops and the big raindrop are different, so they

have different surface energy. The energy released should be

?16?0.012??1.12??10?4J ?S???(8*4?r?4?R)??16?r????43.52?10J?222

(2) ?p?2?2?????7.0Nm?2?7.0Pa rRr

3. (6 marks) Suppose that a particle is limited in a one-dimensional infinity deep potential

well and its wave function and energy are given respectively by

?3?2?3?x?9?2?2sin?. ? (0 < x < L) andE3?2mL2L?L?Find (1) the positions corresponding to the highest probabilities of finding the particle;

(2) the De Broglie wavelength of the particle in such a state.

(3) the probability of finding the particle within the range of [0, L/6]

(Hint: please give your calculating procedures) Solution:

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d?33?x?1??3?x??0;?cos???n?????0,?dxL2??L??(1).

115?x?L,L,L626

P2h29h22(2) E3??????L De222m2m?De8mL3

2(3) ?|?3|2dx?L0

Set y?L/6L/6?013?x11(1?cos(2?))dx??2L6LL/6?0cos(2?3?x)dx L

6?xL,?dx?dy L6?so when x=0 to L/6, y = 0 to ?, therefore the second part in the above formula

gives

1LL/6?03?x1L1?cos(2?)dx??cos(y)dy?siny|0?0 ?LL6?06??Therefore, the probability of finding the particle in the region [0,L/6] is 1/6

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