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Appendix Brief Review to Medical Physics Chapter 1. Introduction to Physics In this chapter and for the first part, you should be able to answer some of simple questions about physics, fundamental quantities in SI unit, standard units used in physics, rounding off numbers and scientific notation. In the second part of this chapter, you should be able to do the dot product and cross product of two vectors and of course you should also be able to do some differentiation to some simple functions as you have learned advanced mathematics. Important concepts and formulae: Example 1.1 The fundamental quantities in physics are time, length, mass, Electric current, temperature, amount of a substance and luminous intensity, and their corresponding units in SI system are second, meter, kilogram, Ampere, Kelvin, Mole and candela. Example 1.2* Suppose we have two vectors: A 2i 4 j 2k , B 8i 2 j 4k , find: (1) the magnitude of the two vectors; (2) the dot product of the two vectors; (3) the angle between the two vectors (4) the sum of the two vectors; (5) magnitude of the cross product of the two vectors. If A a x i a y j a z k , B bx i b y j bz k , in order to finish the above problem, the formulas you need are: Magnitude of vector A : Dot product: A A a x2 a y2 a z2 (1.1) A B a x bx a y b y a z bz (1.2) A B Included angle between vectors: A B AB cos cos AB if F A B then Fi Ai Bi Sum of vectors: (1.3) (1.4) Cross product: A B (a y bz a z b y )i (a z bx a x bz ) j (a x b y a y bx )k (1.5) Amplitude of cross product: A B AB sin (1.6) where in (1.6) can be determined by (1.3) or calculated using (1.5). Example 1.3 About the differentiations, you should understand that the meaning of 153 the differentiation to a function can be the slope of a function at a particular point, the speed or acceleration of an object depending on the property of the function, and should remember the following formulae: d n x nx n 1 ; dx d sin( x 4) cos(x 4); dx d cos(x 4) sin( x 4) dx (1.7) Chapter 2. The fundamental laws of Mechanics In this chapter, more contents are included. You should know that (1) the description methods of motion, (2) the concepts of displacement, velocity, acceleration, angular velocity, angular acceleration, tangential velocity and acceleration and angular momentum, (3) Newton’s law of motion and rotational law*, (4)* conservational law of momentum, energy, and angular momentum. Important concepts and formulae: 1. Equation of motion: r (t ) x(t )i y(t ) j z (t )k (2.1) The instantaneous velocity and acceleration are given as dr (t ) dx dy dz v (t ) i j k, dt dt dt dt dv (t ) d 2 r (t ) d 2 x d 2 y d 2 z a (t ) 2 i 2 j 2 k dt dt 2 dt dt dt (2.2) (2.3) Here you should be clear that the displacement, the velocity and acceleration are vectors. They are determined by both of magnitudes and directions. 2. Newton’s laws of motion dp d dv (mv ) m ma a) Fnet 0. b) Fnet (2.4) dt dt dt c) FA due to B FB due to A (2.5) 3. Work and Energy a) Work: b Aab F dr a b) Kinetic energy and potential energy (see Chinese text book on page 9-10) c) Conservation of energy in a system E K U = Constant. 4. Rotational motion 154 a) Angular displacement , d b) angular velocity , dt c) tangential velocity = curvature radius angular velocity v r d) Angular acceleration: (2.6) (2.7) d d 2 2 dt dt (2.8) e) tangential acceleration = curvature radius angular acceleration: (2.9) a r f) Angular momentum: L r p , the amplitude of the momentum is L r p r p sin (2.10) Chapter 3. Fluid Dynamics In this chapter, there are several concepts in hydrostatics you should know. They are the pressure, the Pascal’s principle, Buoyancy and Archimedes’ principle. However, Hydrodynamics is much more important than the hydrostatics for you. The concepts and equations involved in hydrodynamics are steady flow, ideal fluid, continuity equation, Bernoulli’s equation, laminar flow, viscosity, Reynolds number, Poiseuille’s law and Stokes law: Important concepts and formulae: F A 1. Pressure: p 2. Pascal’s principle F1 F2 A1 A2 3. Archimedes’ principle: FB g V is density of fluid, g is the gravitational acceleration and V is the volume of the body in liquid. 4. Ideal fluid: The properties of ideal fluid are a) non-viscous, b) incompressible and c) moving in a streamline motion. 5. Continuity equation: Av = Constant. The velocity of the fluid is inversely proportional to the cross-sectional area. 6. Bernoulli’s equation: 155 1 v 2 gh = Constant. 2 For a horizontal flow, h can be chosen as zero, we have 1 p v 2 = Constant 2 When v is big, the pressure will be small, vice versa. 7. *Applications to Bernoulli’s equation: a) Kinemometer, b) The Pitot tube 8. *The flow of viscous fluid: a) Laminar flow, b) Reynolds number, c) Poiseuille’s Law and Stokes’ Law. p Chapter 4. Vibrations and Waves In this chapter, you need to master (1) the basic laws of simple harmonic motion, (2) the method of the superposition of two simple harmonic motions, (3) the propagation regularity of waves, (4) the physical meaning of wave equation, and (5) the phenomenon and laws of the interference of waves, and to understand (a) the formation of standing waves, (b) the basic concept of acoustics (声学), (c) the sound intensity level and loudness level, and (d) the characteristics of ultrasound and its applications to medicine. Important concepts and formulae: 1. The equation of SHM x A cos(t ) or x A sin( t ) 2. The velocity and acceleration of SHM (1) Velocity of SHM: dx v A sin( t ) dt (2) Acceleration of SHM a dv d 2 x A 2 cos(t ) dt dt 2 (4.1) (4.2) (4.3) 3. The characteristic quantities of SHM (1) The amplitude: A; the angular frequency: ; and the initial phase: . (2) The period and frequency of SHM: (a) The period is the time taken for completing one complete vibration, denoted by T. Obviously 2 (4.4) T 156 (b) The frequency, denoted by f or , is the number of complete vibrations in one second. Therefore, 1 f (4.5) T 2 (3) The amplitude and initial phase can be determined by initial conditions which are initial displacement of SHM x0 and initial speed v0: v02 A x 2 0 2 v x 0 arctan - , (4.6) 4. The energy of SHM (1) The kinetic energy of SHM 2 Ek 1 2 1 dx 1 mv m m 2 A 2 sin 2 t 2 2 dt 2 (4.7) (2) The potential energy of SHM 1 2 1 2 kx kA cos 2 t 2 2 1 m 2 A 2 cos 2 t 2 Ep (4.8) k m (3) The total energy of the SHM system 1 1 E E k E p m 2 A 2 kA2 2 2 The total energy does not change with time, so it is conservative. 5*. Damped vibration, forced vibration and resonance (1) Features of damped vibration: (a) The amplitude decreases where 2 (b) The period increases T 2 2 0 2 . (4.9) (4.10) (2) Forced vibration: (a) External force: F Fm cos t (b) Equation of damped vibration: x Acost (c) Amplitude of damped vibration: A Fm m 02 2 4 2 2 2 2 2 2 0 arctan (d) The initial phase: (3) Resonance: 157 (4.11) (4.12) (a) Angular frequency of resonance: esonance 02 2 2 , (b) Amplitude of resonance: Aresonance Fm (4.13) (0) (4.14) 2m 02 2 6. Composition of SHM (1) The addition of two vibrations with same vibrational directions and same frequencies (a) Two original vibrations: x1 A1 cost 1 , x2 A2 cost 2 (b) The resultant vibration: x A cos(t ) with A12 A22 2 A1 A2 cos(1 2 ) , A tan (4.15) A1 sin 1 A2 sin 2 . A1 cos 1 A2 cos 2 (4.16) (c) Two special cases: 1 2 2k (k =0, 1, 2, …), A A1 A2 1 2 (2k 1) , (k =0, 1, 2,…), (4.17) A A1 A2 (4.18) 7. The composition of two vibrations with the same frequency but orthogonal vibrational directions (1) Two original vibrations: x A1 cost 1 , y A2 cost 2 (2) The compositional orbital equation in x-y plane is given as x 2 y 2 2 xy 2 cos( 2 1 ) sin 2 2 1 2 A1 A2 A1 A2 (a) 2 1 2k y (k = 0, 1, …) A2 x A1 (b) 2 1 (2k 1) y (4.20) (k = 0, 1, …) A2 x A1 1 (c) 2 1 (2k ) 2 (4.19) (4.21) (k = 0, 1, …) 158 (4.22) 2 2 x y 1 A1 A2 (4.23) (d) Go back to the original equations to determine the rotational direction for the particle. 8. Wave motion and propagation (1) Wave front, wave line, amplitude A, wave period T, wave frequency f, the propagating speed of waves u, the angular frequency of wave, and wave number k. They have the following relations. 2 λ 2 2 T , u , f , k . (4.24) T (2) Wave equation has quite a few forms. These forms are obtained by the above relations between the characteristic quantities: x 2 x S A cos t A cos t u T u t t x x A cos 2 A cos 2 T Tu T 2 A cos t x A cos t kx 9. *Wave energy and wave intensity: 1 Ek E p VA2 2 sin 2 t kx 2 Density of energy: w Ek E p V 1 2 2 2 A sin t kx 2 (4.25) (4.26) (4.27) T 1 1 wdt A2 2 T 0 2 Average energy in a period: E Wave intensity: I wu 1 2 2 A u 2 (4.28) (4.29) 10. Huygens’ principle: Every point of a wave front may be considered the source of secondary wavelets, which spread out in all directions with a speed equal to the speed of propagation of the waves. 11. Superposition principle of waves: Several waves from different sources can propagate in the same medium independently but at the meeting point, the resultant vibration of the particle is the vectorial superposition of the vibrations of all waves. 12. Interference of waves: s1 A1 cost kx1 1 s2 A2 cost kx2 2 159 (1) Interference conditions: Two coherent waves should have the same frequency, the same vibrational direction and the same initial phase or constant phase change. (2) Resultant amplitude and phase changes: A12 A22 2 A1 A2 cos1 2 A (4.30) 1 2 1 2 k x1 x2 1 2 2 (a) 2m (m 0,1,2,...) (b) 2m 1 (c) 1 2 (4.31) x1 x2 (m 0,1,2,...) x1 x2 m x1 x2 2m 1 2 A A1 A2 (4.32) A A1 A2 (4.33) x1 x2 2 (4.34) A A1 A2 (4.35) (m 0,1,2,...) A A1 A2 (4.36) (m 0,1,2,...) 2 13* Standing waves: s s1 s2 Acost kx Acost kx 2 Acos kx cost (a) Nodes: cos kxm 0 , xm 2m 1 (b) Loops: cos kxm 1 , xm m 2 4 (4.37) (4.38) (4.39) 14. Sound waves: (1)* Sound pressure: (2)* Acoustic impedance: (3)* Sound intensity: (4). Intensity level: L 10 lg I (dB) I0 (4.40) where I0 = 10-12Wm-2 is the standard reference sound intensity. Please note that for the noises made by car on the street, the intensity made by several cars should be the intensity of one car multiplying the number of cars. Substituting the resultant intensity into the above formula, you will get the intensity level made by many cars. Look at the examples given in your English text book on page 89. 15. Doppler effect 160 f observer vsound vobserver f source vsound vsource (4.41) Besides remembering the above formula, there is still one thing you should learn by heart. Whenever the observer and wave source move closer, plus sign will be taken in numerator and minus sign will be taken in denominator in the above formula. Therefore, the frequency becomes higher to the observer. On the other hand, when they move apart, the minus and plus signs should be taken in numerator and denominator respectively in the above formula. So the receiver get a lower frequency. 16*. Ultrasound and its applications to medicine: (1) Properties of ultrasonic wave (a) Directionality, (b) High transmission, (c) Reflection. (2) Special actions of ultrasonic wave (a) Mechanical action, (b) Cavitation, (c) Heat effect (3) Application of ultrasound in medicine (a) A-ultrasound diagnosing instrument. (b) B- ultrasound diagnosing instrument. (c) M-ultrasound; (d) Ultrasonic Doppler diagnosing instrument; (e) Color Doppler ultrasonic blood-flow imaging instrument. Chapter 5. Special relativity and General relativity In this chapter, you should understand that the difference between Galileo transformation and Lorentz transformation, two basic postulates, time dilation, Lorentz contraction, the relativity of simultaneity (同时的相对性), the mass-speed relation, the mass-energy relation, the energy-momentum relation, the two hypotheses of general relativity, the characteristic of time-space in the gravitational force field (引力场的时空特性). Important concepts and formulae: 1. The difference between the Galileo and Lorentz transformation: (a) Galileo transformation describes the classical Newton’s mechanics. It represents the relative principles of the classical mechanics and the absolute outlook of time and space. Its obvious feature is that the space and time are completely independent. (b) Lorentz transformation agrees with the two postulates of special relativity. The time and space are not independent any more. This transformation is the mathematical expression of special relativity. All the phenomena, kinematical and mechanical, happened for objects with high speed can be explained by such a transformation. 161 x ( x ut ) y y z z u t t 2 x c With 1 1 u2 c2 (6.1) . Using the following relation: 1 u2 c2 ds ds ds dt dt v , v v , u dt dt dt dt dt 1 2 vx c we have the velocity relations: v u v x x u 1 2 vx c vy (6.2) 1 u2 c2 v y u 1 2 vx c vz 1 u2 c2 v z u 1 2 vx c 2. Postulates of special relativity: (a) The relativity principle: All the laws of physics have the same form in inertial reference frame. (b) Constancy of the speed of light: Light propagates through empty space with a definite speed c independent of the source or observer. 3. Time dilation, Lorentz contraction and the relativity of simultaneity: (a) Moving clock is measured to run slowly. 0 , (6.3) where 0 is proper time and is called the time expansion factor. (b) Moving ruler becomes shorter. L 1 L0 , (6.4) where L0 is proper length and 1/ is called length contraction factor. (c) The relativity of simultaneity: (t1 = t2) x x u u t 2 t1 t 2 2 x 2 t1 2 x1 u 1 2 2 c c c 4. Mass-speed, mass-energy and energy-momentum relations: 162 (6.5) (a) Mass-speed relation: m (b) Mass-energy relation: E (c) Energy-momentum relation: While m0 = 0, we have (d) kinetic energy m0 1 u2 c2 m0 c 2 1 u c 2 2 m0 (6.6) m0 c 2 E0 (6.7) E 2 p 2 c 2 m02 c 4 (6.8) E pc (6.9) E k mc 2 m0 c 2 (6.10) 5. The two hypotheses of general relativity: (a) Equivalent principle: For all physical process, the reference frame with uniform acceleration is equivalent to the local region of gravitation and the inertial force is equivalent to the local region of gravitation. (b) General relativity principle: Physics law has the same form in all reference frames, no matter inertial or non-inertial. 6. The characteristic of time-space in the gravitational force field: In the general theory of relativity of 1915, Einstein extended his earlier work to include accelerated system, which led to his analysis of gravitation. He interpreted the universe in terms of a four-dimensional space-time continuum in which the presence of mass curves space in such a way that the gravitational field is created. This explains that the mass curves space or the presence of gravitational field will also curve space. Chapter 6. The surface phenomenon of liquid In this chapter, you need to understand the concepts of surface tension and surface energy, and to know the concepts of the additional pressure of a curved surface, capillarity and air embolism. Important concepts and formulae: 1. Surface tension: Total forces pointing to the inner liquid act on molecules in the surface layer. So the surface is in tension. It is called surface tension. 2. Surface energy: The work done by increasing liquid surface per unit area is called surface energy. (Jm-2) Its value is equal to the surface tension coefficient F . (5.1) 2L Where F is the applied force and L is the width of the metal frame. 3. The additional pressure caused by a curved surface: 2T 2 p For a spherical elastic pellicle: (5.2) R R 163 4. Capillarity: The tube with very small diameter is called a capillary tube. When such a tube is put into liquid, the liquid surface in it will change. The phenomenon is called Capillarity. The level change of the liquid surface can be expressed by h 2 cos , rg (5.3) where h is the height of liquid surface in tube, r is the radius of the tube, is the contact angle between the liquid and tube wall, is the surface tension coefficient and is the density of liquid. 5. Air embolism: When fluid flows in a capillary tube, liquid will be blocked if there are air bubbles in the tube. This phenomenon is called air embolism. Because of this, when we transfuse patients, we have to avoid air embolism in the transfusing tube. Especially, we have to avoid an air embolism in the syringe (注射器) when injecting veins. Chapter 7. Ray (or geometric) optics In this chapter, you have to know the imaging principles of a single spherical surface, the calculation methods and sign rules; to understand the imaging principles of the coaxial spherical system and thin lens, and their formulas; to master the optical system of human eyes, know how to calculate the lens power for remedying sight problem. Important concepts and formulae: 1. The laws of reflection and refraction and total reflection: (a) The law of reflection: Incident angle = reflected angle (b) The law of refraction: (c) The total reflection angle: n1 sin i1 n2 sin i2 sin ic n2 n1 (7.1) (7.2) 2. Refraction equation of a single spherical surface: n1 n2 n2 n1 u v r (7.3) 3. Sign rules: (a) Object distance (物距): If the direction from object to spherical surface is same as the direction of light, the object distance u is positive. Otherwise it is negative. (实物物距为正,虚物物距为负). 164 (b) Image distance: If the direction from spherical surface to image is the same as the direction of light, the image distance is positive. Otherwise it is negative. (实像像距为正,虚像像距为负). (c) Curvature radius: If the direction from spherical surface to its center point is the same as the direction of light, the curvature radius is positive. Otherwise it is negative. 4. The thin lens equation and the lens power: (a) The thin lens equation: 1 1 1 u v f (b) The lens power: D (7.4) 1 (1/m) f (7.5) 1 AngularSiz e (7.6) 5. *The vision of the human eyes: vision Angular where ObjectSize ObjectDist ance (minute ) 6. Corrective eyeglasses for visual defects (a) Correction of myopia; (b) Correction of hypermetropia. (Read the examples in English textbook on page 152-153) 7. *Magnifying glasses, fibro-scope and microscope: Chapter 8. Wave optics In this chapter, the important concepts are the interference and diffraction of light, double-slit interference, thin film interference, single slit diffraction, diffractions grating and so on. Important concepts and formulae: 1. Interference conditions: of light: Interference conditions: they are the same with the coherent waves. Coherent light should also have the same frequency, the same vibrational direction and the constant phase change. 2. Double –slit interference (Yong’s experiments): (1) Constructive condition = m (8.1) m xm L or (m = 1, 2, …) (8.2) d 165 where is the light path difference; xm is the distance from the mth-order bright fringe to the zeroth-order bright fringe; d is the slits separation; L is the distance from the double-slit to the screen; is the wavelength of light used in the experiment. (2) Destructive conditions: or 1 2 m (8.3) 1 x m m L 2 d (8.4) (3) Spacing of two bright fringes (n 1) n L x x n 1 x n L (8.5) d d d 2. Lloyd mirror: (1) Lloyd mirror experiment proved the abrupt phase () change on the boundary of two mediums; (2) The abrupt phase change occurs when the light wave initially traveling in a less optically dense medium is reflected at an interface with a more optically dense medium. 3. *Interference in thin films, air wedges and Newton’s ring. 4. The Fraunhofer single slit diffraction: (1) Dark fringe positions: m sin m (m =1, 2, 3, …) (8.6) a m xm f (m =1, 2, 3, …) (8.7) a where a is the width of slit, xm is the height from mth order dark fringe to the center bright fringe, f is the focal length of the lens used in the experiment. (2) Bright fringe positions: 1 a sin m m 2 (m =1, 2, 3, …) (8.8) (m =1, 2, 3, …) (8.9) (3) The dark fringe height from 5. The diffraction grating: Bright fringe condition: d sin m m where d is the grating spacing or grating constant. Chapter 9. Introduction to Quantum Mechanics In this chapter, the important concepts are the blackbody radiation, Planck’s 166 hypothesis, Planck-Einstein quantization law, photoelectric effect, Compton effect, the wave-particle duality of light, energy quantization in atoms, De Broglie wave, the Heisenberg uncertainty principle Important concepts and formulae 1. Blackbody radiation: (1) Blackbody: it is defined that it can absorb all electromagnetic radiations fallen on it; (2) Planck’s hypotheses; (3) Planck-Einstein quantization law: hc E h (9.1) 2. Photoelectric effect equation. 1 2 h W0 mvmax (9.2) 2 3. Compton effect: 2 h sin 2 mc 2 (9.3) 4. De Broglie Wave p h 5. Line spectrum and energy quantization in atoms. (1) General Balmer series: 1 1 R 2 2 n k 1 (k = 1, 2, …; n = k +1, k+2, …) (9.4) (9.5) (2) Bohr’s atomic theory (1) Quantization condition of orbital angular momentum h 2 (2) Transition postulate: mvrn n (n = 1, 2, …) h En Ek (9.6) (9.7) (3) Important result: Using the quantization condition of orbital angular momentum and Newton’s law and Coulomb’s law; the energy of the atomic system could be obtained. En me 4 1 13.6 2 E1 2 eV 2 2 2 8 0 h n n n h2 rn 0 2 n 2 r1 n 2 5.3 10 11 n 2 (m) me 167 (9.8) 6. Heisenberg uncertainty principle h (9.9) p x x 2 7. Schrödinger equation: (1) The interpretation of wave functions: it is probability wave function. The function must have three properties: single value, continuous, and limited. In addition to these conditions, it could also be normalized. (2) Schrödinger equation: i Hˆ t Hˆ E h . 2 1. One dimensional infinity potential well: with 2 2n2 (1) Energies: En (2) Wave function in well: x 2mL2 2 n sin x using this function, the L L distribution of the particle in the well can be determined. For a given n, it is easy to find the maximum position of its distribution. 2. Hydrogen-like atom: (1) Energy quantization: the principal quantum number n: me 4 Z 2 Z2 13.6 Z 2 E n 2 2 2 2 E1 eV 8 0 h n n n2 (n = 1, 2, …) (2) Angular momentum quantization: angular quantum number l: h L l (l 1) (l = 0. 1, 2, …, (n – 1)) 2 (3) Space quantization: Magnetic quantum number m: Lz m ( m = 0, 1, 2, …, l ) (4) Spin quantization: LS s(s 1) , Lsz ms (ms = -s, -s +1, …, s) n 1 (5) Total number of states: Z n 2(2l 1) 2n 2 l 0 Chapter 10~12 Laser, X-rays, Nuclear Physics and radioactivity 1. Laser: (1) Absorption, spontaneous emission 168 (2) Population inversion (3) Stimulated emission. (4) The characteristic of laser: (a) Good directionality; (b) High brightness and high intensity; (c) Monochromatic; (d) Spatially coherent; (e) Good polarization; 2. X-rays: (1) The characteristic of x-rays: (a) The hardness of x-rays: The hardness of x-rays denotes the ability of penetration of x-rays. It depends on the energy of x-photon (the wavelength of x-rays) only and has nothing to do with the number of x-photons. (b) Ionizing function : x-ray enables the atoms and molecules to be ionized and this characteristic is quite useful in medical treatment. (c) Fluorescence function, photo-chemistry function, biological effect and penetration capability. 3. Nuclear physics and radioactivity: (1) The structures of nucleus: (a) The atomic mass number A; (b) Atomic number Z, which is the number of proton in an atom; (c) Number of neutrons N; A = Z + N; (d) Symbol: A Z X (2) Nuclear forces and nuclear energy levels (3) Nuclear binding energy and mass defect (4) The decay type of the atomic nucleus: (a) decay: A Z X ZA22Y 24 Q A Z X Z A1Y Q - decay A Z X Z A1Y Q + decay A Z X Z A1Y Q electron capture (b) decay: (5) decay and inner transmission: In this process, a photon is emitted. 169