PreHealth Physics Review Johnny B Holmes Ph D

Pre-Health Physics Review Johnny B. Holmes, Ph. D.

Fundamentals MKS system of units: M for meters (distance) K for kilograms (mass) S for seconds (time) Also add Q in Coulombs (electric charge) Other units are combinations of these fundamental units: Speed in m/s; Acceleration in m/s 2; Force in Nt=kg*m/s 2; Energy in Joules = Nt*m (convert to calories, BTUs); Power in Watts = Joules/sec Pressure in Pascals = Nt/m 2 (convert to psi, atm, mm of Hg, bars); Current in amps = Coul/sec; Voltage in volts = Joule/Coul.

Common Prefixes milli = 10 -3 (m) micro = 10 -6 ( ) nano = 10 -9 (n) pico = 10 -12 (p) femto = 10 -15 (f) Kilo = 103 (k) Mega = 106 (M) Giga = 109 (G) Tera = 1012 (T)

Vectors Space is three-dimensional, so many quantities in physics have directions as well as magnitudes. For 3 -D, need three numbers to specify the quantity (called a vector). Common forms for vectors are rectangular (x, y, z) in 2 -D (x, y) spherical (r, f) in 2 -D (r, ) cylindrical (r, , z) To add vectors, must add in rectangular!

Motion Velocity: v = dx/dt; for constant velocity, have x = xo + vo*t (straight line) v is the slope of the x vs t curve. Acceleration: a = dv/dt; for constant acc, have y = yo + vo*t + ½*a*t 2 (parabola) a is the slope of the v vs t curve. For 2 & 3 dimensions, work in rectangular components. For trajectories, ax = 0 and ay = -g = -9. 8 m/s 2

Circular Motion Work and think in rectangular (x, y) Convert to polar (r, ) for easier equations: r = constant changes in time vr = 0 w d /dt = constant for uniform CM v = w*r = constant for uniform CM ar = -w 2*r directed towards the center due to direction of velocity changing a = a*r where a = dw/dt is angular acc = 0 for uniform CM Also, w 2 f and 1/f = T (f is frequency, T is period)

Newton’s Laws of Motion SF = m*a (vector equation so work in rectangular components) The equation above is Newton’s 2 nd law. Newton’s 1 st law is a special case of the 2 nd : if F=0, then a=0. Newton’s 3 rd law: for every action there is an equal and oppositely directed action (or, you can’t push yourself). The forces in Newton’s 2 nd law are the ones ON the object, not the ones BY the object.

Forces • Contact: Fc balances, perpendicular to surface • Friction: Ff <= m*Fc parallel to surface • Tension: T is same along the string; directed parallel to the string • Weight: near earth’s surface, W = Fgravity = m*g directed down. • Gravity far from earth’s surface: Fgravity = G*M*m/r 2 with G = 6. 67 x 10 -11 Nt*m 2/kg 2

Example: Circular Orbits For a satellite to orbit the moon at a height of 100 km, how fast should it go and how long will it take to make one orbit? Radius of Moon is 1, 700 km; mass of moon is 7. 2 x 1022 kg.

Satellite Problem – cont. Recognize: 1. Circular orbit, so ar = w 2*r, v = w*r, w=2 /T 2. Gravity is the only force: F = G*M*m/r 2 3. Use SF = m*a 4. Recognize that r = Rmoon + h = 1, 700 km + 200 km = 1, 900 km. 5. Therefore: G*M*m/r 2 = m*w 2*r 6. Recognize that m (mass of satellite) doesn’t matter, solve for w then use v = w*r, w=2 /T to get both v and T.

Energy and Conservation of Energy Work = Force through a distance (a scalar) W = F*s*cos( Fs) Energy is the capacity to do work (in ideal situations) Kinetic energy = KE = ½*m*v 2 Gravitational Potential energy: near the earth, PEgravity = m*g*h in general, PEgravity = -G*M*m/r Spring Potential energy: PEspring = ½k*(x-xo)2

Example: Escape Speed S Energiesinitially = S Energiesfinally. KEi + PEi = KEf + PEf (1/2)mvi 2 - Gmearthm/ri = (1/2)mvf 2 - Gmearthm/rf (note that this is equivalent to saying KE = PE, or ½mvf 2 – ½mvi 2 = -Gmem/rf - -Gmem/ri) We see that m is in each term, so we cancel it. (1/2)*(vi)2 - (6. 67 x 10 -11 Nt*m 2/kg 2) *(6. 0 x 1024 kg)/(6. 4 x 106 m) = (1/2)*(0 m/s)2 - (6. 67 x 10 -11 Nt*m 2/kg 2) *(6. 0 x 1024 kg)/(infinity) We again have one equation in one unknown (vi).

Power We now know what Force and Energy are, but what is Power? The definition of Power is that it is the rate of change of Energy from one form into another: Power = Energy t. The units of power are: Joule/sec = Watt. Another common unit is the horsepower, hp. The conversion factor is: 1 hp = 746 Watts.

Review of Rotational Equations Basically we replace F with t, m with I, v with , a with and p with L (where L is the angular momentum): S F = ma S t = I Work = = F ds Work = t d Power = F v Power = t KE = (1/2)mv 2 KErotation = (1/2)I 2 p = mv L = I F = p/ t S t = L/ t.

Example: Your elbow Let’s consider as an example of torque how your muscles, bones and joints work. Consider holding up a ball of weight 5 lb. How does this work? First we draw a diagram: biceps triceps weight = elbow rw r b

Your elbow From F = 0 we have: -Fc + Fb - W = 0 And from t 0 and measuring from the elbow gives: Fc*rc + Fb*rb - W*rw = 0. We have two equations and we have two unknowns (Fc and Fb).

Your elbow • We can use the torque equation first, since rc=0 eliminates one of the unknowns, Fc. Fc*rc + Fb*rb - W*rw = 0 or Fb = W*rw/rb. • Then we can use the force equation to find Fc : -Fc + Fb - W = 0, or Fc = Fb - W.

Example: Object rolling down an incline Conservation of Energy (KEregular + KErotational + PEgravity)initial = (KEregular + KErotational + PEgravity)initial + Elost 0 + mgh = (1/2)mv 2 + (1/2)I 2 + 0. Substituting I=(2/5)mr 2 and w=v/r gives: mgh = (1/2)mv 2 + (1/2)[(2/5)mr 2][v 2/r 2] , or mgh = (1/2)mv 2 + (1/5)mv 2 = (7/10)mv 2.

Momentum and Conservation of Momentum p = m*v (p and v are vectors!) Conservation of Momentum Fxext on 1 + Fxext on 2 = (px 1 + px 2) / t. If the external forces are small, or if the time of the collision, t, is small, then we have: (px 1 + px 2) = 0. This can be re-written as: (px 1 + px 2)i = (px 1 + px 2)f. This is called Conservation of Momentum. This is a vector law, so a similar equation holds for each component of momentum.

1 -D Collisions In one dimensional collision cases, we can apply two laws: Conservation of Momentum and Conservation of Energy (here we assume there are no PE’s that change): (1/2)m 1 v 1 i 2 + (1/2)m 2 v 2 i 2 = (1/2)m 1 v 1 f 2 + (1/2)m 2 v 2 f 2 + Elost m 1 v 1 i + m 2 v 2 i = m 1 v 1 f + m 2 v 2 f These are two equations with 7 quantities: m 1, m 2, v 1 i, v 2 i, v 1 f, v 2 f, Elost. Hence if we know five, we can solve for the other two.

Explosions Normally, in an explosion the initial object is in one piece and at rest. After the explosion, one piece goes forward. Conservation of Momentum says the other piece must then go backwards. (If we brace ourselves, we can compensate for that backwards push and not fall over. ) 0 + 0 = m 1 v 1 f + m 2 v 2 f or v 2 f = - m 1 v 1 f / m 2.

Pressure P (pressure, not power or momentum) P = Force/Area (definition) (force is perpendicular to area, not parallel to it) units of pressure: – Nt/m 2 – 1 atmosphere = 1. 01 x 105 Nt/m 2 = 14. 7 lb/in 2 – 1 bar = 1. 00 x 105 Nt/m 2 – 1 Torr = 1 mm of Hg, 760 Torr = 1 atmosphere

Ptop Pressure Pside effects of gravity: h W=mg – consider little cube of fluid – consider forces on the fluid in y direction Pbottom 1. weight acts down 2. pressure underneath pushes up 3. pressure on top pushes down – Fy = -m*g + Pbottom*Abottom - Ptop*Atop = 0 , – where m = V = *A*h, and A = Abottom = Atop so: Pbottom*Abottom - Ptop*Atop = m*g = *A*h*g , or: Pbottom - Ptop = *g*h. Buoyant Force = P*A = *g*V

Fluid Flow Conservation of Energy: (1/2)mvi 2 + mghi + Won = (1/2)mvf 2 + mghf + Wby divide each term by Volume, and note m/V= , also note W = F*s, F=P*A, A*s=V, so Work = P*V: (1/2) vi 2 + ghi + Pi = (1/2) vf 2 + ghf + Plost examples: • lift on wing of airplane • coffee pot • siphon • oil well

Fluid Flow Viscous Force: F dv/ds ; for tubes (cylindrical hoses) with constant velocity (Fapplied = Fresisted, F = P*A ) P r 2 r. L) dv/dr Q = P)R 4 / (8 L) wwhere P = Plost and Q is the volume per time flowing. Power = Work/t = F*s/t = Pressure*A*s/t = Pressure*V/t = Pressure*Q

Reynolds Number Have laminar flow (previously assumed layer over layer flow) as long as flow is slow enough; otherwise have turbulent flow Reynolds number: R = 2 vavgr = 2 Q /( r ) (dimensionless!) If R < 2, 000, then laminar; If R > 2, 000, then turbulent.

Ideal Gas Law P*V = N*k*T. We further define R = Na*k, where Na = 6. 02 x 1023 = 1 mole. Thus we have: P*V = n*R*T , where n = N/Na = number of moles in volume, V; T must be in Kelvin, Not o. F or o. C ! k = experimental constant = 1. 38 x 10 -23 J/K ; R = Na*k = 8. 3 Joules/mole*Kelvin.

HEAT CAPACITY The amount of energy necessary to heat a material per temperature change is what we call the heat capacity: C (heat capacity) = Q/ T where Q is the energy to raise temperature of an amount of material by T. Usually we specify the heat capacity in one of three ways: per object, per mole (usually for gases), and per mass (usually for liquids and solids).

Heat Capacity of Air Cmolar-constant P = Cmolar-constant V + R Air is made up mostly of N 2 and O 2. These gases act approximately as diatomic ideal gases. Usually, when we heat air it is NOT in a contained volume but expands to keep its pressure constant. This means that most of the time, the heat capacity of air is: Cmolar - air - constant P = (5/2)R + R = (7/2)R.

Heat Capacity of Materials By definition, a calorie is the energy necessary to raise the temperature of 1 gram of water up 1 o. C. Cwater = 1 cal/gm-o. C = 4. 186 J/gm-o. C Cethyl alcohol = 2. 400 J/gm-o. C Cwood = 1. 700 J/gm-o. C Cglass = 0. 837 J/gm-o. C Ccopper = 0. 387 J/gm-o. C Since liquids and solids don’t expand to fill the space like gases do, we don’t usually distinguish between heat capacities at constant pressure versus constant volume.

Latent Heat For water, the latent heat of fusion (heat needed to melt ice to water) is 79. 7 cal/gm. For water, the latent heat of vaporization (heat needed to boil water) is 540 cal/gm. For alcohol, the latent heat of vaporization is less at 204 cal/gm.

Heat Transfer There are four ways of moving heat: • Evaporation (using latent heat) • Convection (moving heat with a material) • Conduction (moving heat through a material) • Radiation (light, usually mainly in the infrared, both emitted and absorbed)

Heat Transfer: Conduction Power = Q/t = k*A* T/L where k is a constant that depends on the material, called thermal conductivity; where A is the cross sectional area; LL where L is the distance from the hot end to the cold end; A hot and T is the temperature difference k Thi between the hot and cold ends. cold Tlow

Blackbody Radiation: Experimental Results Ptotal = AT 4 where = 5. 67 x 10 -8 W/m 2 *K 4 peak = b/T where b = 2. 9 x 10 -3 m*K Intensity (log scale) UV blue yellow wavelength red IR

Thermodynamics The First Law of Thermodynamics is a fancy name for the Law of Conservation of Energy applied to thermal systems. It says: U = Q - W where U indicates the change in the internal energy of the system. This internal energy is related to the temperature and heat capacity of the system; Q is the heat energy added to the system; and W is the work done by the system.

Second Law of Thermodynamics Entropy is a measure of the probability of being in a state. Since things tend to go to their most probable state, we can write the 2 nd Law of Thermodynamics as: systems tend to have their entropy increase.

Efficiency is a measure of how much you get out versus how much you put in. For heat engines: Efficiency = = Work done / Heat Added By the first law, the work done is simply the difference in the heat going into the engine minus the heat coming out of the engine. The total heat added is the heat going into the engine. = (Qhot - Qcold) /Qhot. For the most efficient engine possible: Carnot = (Thot - Tcold) / Thot

Oscillations with a mass on a spring? y = A sin( t + o) The oscillation speed, , describes how fast the mass oscillates. But what does this oscillation speed (ω=d phase/dt ) depend on? By putting in our solution for y into Newton’s Second Law (the differential equation), we can get a prediction: = (k/m). For stiffer springs and lighter masses, the frequency of the oscillation increases. Note: the Amplitude does NOT affect the frequency!

Energy: Amplitude and frequency Since Energy = (1/2) m 2 A 2 , as the frequency goes up (ω), to keep the same energy the amplitude (A) needs to go down. Can you make sense of that relationship? Since kinetic energy depends on velocity (squared), and since v = dx/dt , a higher frequency means that for the same distance (amplitude) we have a smaller dt. To keep the same v, we need a smaller distance (amplitude) to go with the smaller dt (higher frequency).

Waves (in general) y = A sin( ) where is a phase angle in a moving wave, changes with both – time (goes 2 radians in time T) and – distance (goes 2 radians in distance ) so = (2 / )*x +/- (2 /T)*t – where 2 /T = and – where 2 / = k and so phase speed: v = distance/time = /T = f = /k

Standing Waves To create what are called standing waves (we will play with these in the last lab), we need to create constructive interference from both ends. This leads to the following condition: #( /2) = L , which says: we need an integer number of half wavelengths to “fit” on the Length of the string for standing waves. We can vary the wavelength by either varying the frequency or the speed of the wave: recall that phase speed: v = distance/time = /Tperiod = f. For a wave on a string, recall that v = f = (Ttension)/m) where m m/L.

Standing Waves For stringed instruments (piano, guitar, etc. ), the string vibrates with both ends fixed. However, with wind instruments (trumpet, trombone, etc. ), we can have the situation where both ends are free and a different situation where one end is free and one end is fixed. 1. If both ends are free, we get the same resonance condition as for both ends fixed: #( /2) = L. 2. If one end is free and the other end is fixed, we get a different condition: #odd( /4) = L, where #odd is an odd number (1, 3, 5, etc. ).

Sound Intensity I(d. B) = 10*log 10(I/Io) where Io = 10 -12 W/m 2 The weakest sound intensity we can hear is what we define as Io. In decibels this becomes: I(d. B) = 10*log 10(10 -12 W/m 2 / 10 -12 W/m 2) = 0 d. B. The loudest sound without damaging the ear is 1 W/m 2, so in decibels this becomes: I(d. B) = 10*log 10(1 W/m 2 / 10 -12 W/m 2) = 120 d. B.

Electric Force To account for repulsive and attractive forces, we find that like charges repel, and unlike charges attract. We also find that the force decreases with distance between the charges just like gravity, so we have Coulomb’s Law: Felectricity = k q 1 q 2 / r 122 where k, like G in gravity, describes the strength of the force in terms of the units used.

Electric Force Charge is a fundamental quantity, like length, mass and time. The unit of charge in the MKS system is called the Coulomb. When charges are in Coulombs, forces in Newtons, and distances in meters, the Coulomb constant, k, has the value: k = 9. 0 x 109 Nt*m 2 / Coul 2. (Compare this to G which is 6. 67 x 10 -11 Nt*m 2 / kg 2 !)

Fundamental Charges When we break matter up, we find there are just a few fundamental particles: electron, proton and neutron. (The proton and neutron are now thought to be made up of more elementary particles called quarks, while the electron remains elementary. ) electron: qe = -1. 6 x 10 -19 Coul; me = 9. 1 x 10 -31 kg proton: qp = +1. 6 x 10 -19 Coul; mp = 1. 67 x 10 -27 kg neutron: qn = 0; mn = 1. 67 x 10 -27 kg (note: despite what appears above, the mass of neutron and proton are NOT exactly the same; the neutron is slightly heavier; however, the charge of the proton and electron ARE exactly the same - except for sign)

Electric Field for a point charge If I have just one point charge setting up the field, and a second point charge comes into the field, I know (from Coulomb’s Law) that Fon 2 = k q 1 q 2 / r 122 and Fon 2 = q 2 * Eat 2 which gives: E at 2 due to 1 = k q 1 / r 122 for a point charge.

Electric Potential Energy Since Coulomb’s Law has the same form as Newton’s Law of Gravity, we will get a very similar formula for electric potential energy: PEel = k q 1 q 2 / r 12 Recall for gravity, PEgr = - G m 1 m 2 / r 12. Note that the PEelectric does NOT have a minus sign. This is because two like charges repel instead of attract as in gravity.

Voltage Just like we did with forces on particles to get fields in space, (Eat 2 due to 1 = Fon 2/ q 2) we can define an electric voltage in space (a scalar): Vat 2 due to 1 = PEof 2 / q 2. We often use this definition this way: PEof 2 = q 2 * Vat 2.

Voltage and Field V = - E s , or Ex = - V / x. Note also the minus sign means that electric field goes from high voltage towards low voltage. Note also that this means that positive charges will tend to “fall” from high voltage to low voltage (like masses tend to fall from high places to low places) , but that negative charges will tend to “rise” from low voltage to high voltage (like bubbles tend to rise) !

Review F 1 on 2 = k q 1 q 2 / r 122 PE 12 = k q 1 q 2 / r 12 Fon 2 = q 2 Eat 2 PEof 2 = q 2 Vat 2 Eat 2 = k q 1 / r 122 Vat 2 = k q 1 / r 12 use in S F = ma KEi + PEi = KEf +PEf +Elost VECTOR scalar Ex = - V / x

Electric Circuits In electricity, the concept of voltage will be like pressure. Water flows from high pressure to low pressure (this is consistent with our previous analogy that Voltage is like height since P = gh for fluids) ; electricity flows from high voltage to low voltage. But what flows in electricity? Charges! How do we measure this flow? By Current: current = I = q / t UNITS: Amp(ere) = Coulomb / second

Resistance By experiment we find that if we increase the voltage, we increase the current: V is proportional to I. The constant of proportionality we call the resistance, R: V = I*R Ohm’s Law UNITS: R = V/I so Ohm = Volt / Amp. The symbol for resistance is W (capital omega).

Resistance The resistance depends on material and geometry (shape). For a wire, we have: R= L/A where is called the resistivity (in Ohmm) and measures how hard it is for current to flow through the material, L is the length of the wire, and A is the cross -sectional area of the wire. The second lab experiment deals with Ohm’s Law and the above equation.

Electrical Power The electrical potential energy of a charge is: PE = q*V. Power is the change in energy with respect to time: Power = PE / t. Putting these two concepts together we have: Power = (q. V) / t = V( q) / t = I*V.

Capacitance We define capacitance as the amount of charge stored per volt: C = Qstored / V. UNITS: Farad = Coulomb / Volt Just as the capacity of a water tower depends on the size and shape, so the capacitance of a capacitor depends on its size and shape. Just as a big water tower can contain more water per foot (or per unit pressure), so a big capacitor can store more charge per volt.

Capacitance While we normally define the capacity of a water tank by the TOTAL AMOUNT of water it can hold, we define the capacitance of an electric capacitor as the AMOUNT OF CHARGE PER VOLT instead. There is a TOTAL AMOUNT of charge a capacitor can hold, and this corresponds to a MAXIMUM VOLTAGE that can be placed across the capacitor. Each capacitor DOES HAVE A MAXIMUM VOLTAGE.

Review: Capacitors: C = Q/V PE = ½CV 2; C// = KA/[4 kd] Series: 1/Ceff = 1/C 1 + 1/C 2 Parallel: Ceff = C 1 + C 2 series gives smallest Ceff , parallel gives largest Ceff. Resistors: V = IR Power = IV; R = L/A Series: Reff = R 1 + R 2 Parallel: 1/Reff = 1/R 1 + 1/R 2 series gives largest Reff , parallel gives smallest Reff.

Magnetic Force Law magnitude: Fmagnetic = q v B sin( v. B) direction: right hand rule: thumb = hand fingers Point your right hand in the direction of v, curl you fingers in the direction of B, and the force will be in the direction of your thumb; if the charge is negative, the force direction is opposite that of your thumb (or use you left hand).

Magnetic Force and Motion Since the magnetic field is perpendicular to the velocity, and if the magnetic force is the only force acting on a moving charge, the force will cause the charge to go in a circle: SF = ma, Fmag = q v B, and a = 2 r = v 2/r gives: q v B = mv 2/r, or r = mv/q. B. This is the basis of the mass spectrometer and the cyclotron.

Torque on rectangular current loop Recall that torque is: t = r F sin( r. F). For magnetic force, F = qv. B becomes F = ILB. In the figure below we can see that r = w/2. Thus the Fleft gives a torque of (w/2)ILB, and the Fright also gives a torque of (w/2)ILB. r N F I w B F I S L

Lenz’s Law V = [ (N B A cos( BA) ] / t The above formula is for determining the amount of voltage generated. But what is the direction of that voltage (what direction will it try to drive a current)? The answer is Lenz’s Law: the direction of the induced voltage will tend to induce a current to oppose the change in magnetic field through the area.

RMS Voltage and Current In order to work with AC circuits just as we did with DC circuits, we create a voltage and current called rms (root mean square). Vrms = Vo (1/2)1/2 and Irms = Io (1/2)1/2 so that we have Pavg = Irms Vrms and Vrms = Irms R. Note that the power formula and Ohm’s Law are the same for DC and for AC-rms, but NOT for instantaneous AC.

Review of Circuit Elements Resistor: VR = R I where I = q/ t Capacitor: VC = (1/C)q (from C = q/V) Inductor: VL = -L I/ t We can make an analogy with mechanics: q is like x; V is like F; t is like t; L is like m; I = q/ t is like v = x/ t; C is like 1/k (spring); I/ t is like a = v/ t; R is like air resistance.

AC Circuits A resistor obviously limits the current in a circuit. But, as we just saw, a capacitor and an inductor also limit the current in an AC circuit. However, the reactances do not just add together. Using the fundamental relations and the calculus, we come up with the concept of impedance, Z: V = IZ where Z takes into account all three reactances: XR=R, XL= L and XC= 1/ C: Z = [R 2 + ( L - 1/ C)2]1/2. Power, however, is still: Pavg = I 2 R (not P=I 2 Z).

Property 1: Speed of Light particle (photon): wave (E&M): no prediction Maxwell’s Eqs. in vacuum: v = [1 / { o o}]1/2 where o = 1/{4 k} = 1 / {4 * 9 x 109 Nt-m 2/Coul 2} o = 4 * 1 x 10 -7 T-s /Coul v = [4 *9 x 109 / 4 *1 x 10 -7 ]1/2 = 3 x 108 m/s = c

Property 2: Color Experiment: – invisible as well as visible – total spectrum order: • • • radio microwave IR visible UV x-ray and gamma ray

Property 2: Color Experiment: – visible order: • • • red orange yellow (yellow) green blue violet

Property 3: Reflection A white paper is rough on a microscopic level, and so a beam of light is reflected in all directions: Blue is incoming, red is outgoing rough paper smooth mirror A mirror is smooth on a microscopic level, and so a beam of light is all reflected in just one direction.

Property 4: Refraction Snell’s Law: n 1 sin( 1) = n 2 sin( 2) • NOTE: If n 1 > n 2 (v 1 < v 2), THEN 1 < 2. • NOTE: All 2 values (in the faster medium) between 0 & 90 degrees work fine. • NOTE: Not all values of 1 (in the slower medium) work! Example: If n 1 = 1. 33, n 2 = 1, and 1 = 75 o, then 2 = inv sin [n 1 sin( 1) / n 2] = inv sin [1. 28] = ERROR - this is called total internal reflection

Refraction and Thin Lenses We break the THIN LENS equation: (nglass – nair) 1 1 *{ + } = + nair R 1 R 2 s s’ Into the LENS MAKERS equation and the LENS USERS equation: (nglass – nair) * { 1 + 1 } = 1 & 1 = 1 1 + nair R 1 R 2 f f s s’ where f is a distance called the focal length.

Refraction and the Lens-users Eq. For s>f (lens used with camera or projector) – Note that a real image is formed. – Note that the image is up-side-down. object image f f ray 1 ray 3 ray 2

Refraction and the Lens-users Eq. For s<f (lens used as a magnifying glass) Notice that: s’ is on the “wrong” side, which means that s’ < 0 , and that |s’| > |s| so f > 0. ray 1 h’ f s s’ f ray 2 ray 3 Example: 1 / 5 cm = 1 / 4 cm + 1 / -20 cm

Refraction and the Lens-users Eq. Notes on using a lens as a magnifying glass: • hold lens very near your eye • want IMAGE at best viewing distance which has the nominal value of 25 cm so that s’ = -25 cm.

Microscope M 1 = -s 1’/s 1 M 2 = -s 2’/s 2 Mtotal = M 1 * M 2 = (s 2’*s 1’) / (s 2*s 1) s 1’ objective lens s 2 L = s 1’ + s 2 eyepiece s 2’ 1/s 1 + 1/s 1’ = 1/f 1 1/s 2 + 1/s 2’ = 1/f 2 Object 1 Image 2 Image 1 Object 2 NOTE: s 2’ = -25 cm so Mtotal < 0 !

Property 5: Shadows Double Slit Experiment n = d sin( n) d tan( n) = d (xn / L) bright x dim d bright L SCREEN dim bright

Diffraction: single slit REVIEW: -2 -1 0 1 2 • For double (and multiple) slits: n = d sin( n) for MAXIMUM (for ALL n) -2 -1 0 1 2 • For single slit: n = w sin( n) for MINIMUM (for all n EXCEPT 0)

Diffraction: circular opening If instead of a single SLIT, we have a CIRCULAR opening, the change in geometry makes: the single slit pattern into a series of rings; and the formula to be: 1. 22 n = D sin( n).

Rayleigh Criterion: a picture In this case: limit = sin-1(1. 22 /D) = tan-1(x’/s’) = tan-1(x/s). D lens x x’ s s’

Limits on Resolution: • Imperfections in the eye (correctable with glasses) • Rayleigh Criterion due to wavelength of visible light • Graininess of retinal cells (Note that in low light where only the rods are activated, we cannot resolve very well because the rod cells are not packed as closely as the cone cells are. Also in low light we only see in black and white – not in color. )

Cone cells & Color Recognition Cone cell sensitivity to different wavelengths 400 450 500 550 600 650 700 (in nm) If only the “blue” cone is activated, the color is violet. If both the “blue” and “green” cones are activated, and the “blue” gives a stronger signal, the color is blue. If both the “blue” and “green” cones are activated, and the “green” gives a stronger signal, the color is green.

Polarization: Wave Theory Three polarizers in series: Sailboat analogy: North wind sail force on sail boat goes along direction of keel

Polarization: Wave Theory #2 Polarization by reflection – Brewster Angle: when refracted + reflected = 90 o – Sunglasses and reflected glare incident ray vertical horizontal reflected ray no problem with horizontal almost no vertical since vertical is essentially longitudinal now surface refracted ray vertical can be transmitted

Interference: Thin Films • Recall that the light is in the FILM, so the wavelength is not that in AIR: f = a/nf. air film water reflected red interferes with refracted/reflected/refracted blue. t

Interference: Thin Films • reflection: no difference if 180 degree difference if • distance: no difference if 180 degree difference if n f < n w; n f > n w. t = a/2 nf t = a/4 nf • Total phase difference is sum of the above two effects.

Interference: Thin Films • Total phase difference is sum of the two effects of distance and reflection • For minimum reflection, need total to be 180 degrees. – anti-reflective coating on lens • For maximum reflection, need total to be 0 degrees. – colors on oil slick

Photons and Colors • Electron volts are useful size units of energy 1 e. V = 1. 6 x 10 -19 Coul * 1 V = 1. 6 x 10 -19 J. • radio photon: hf = 6. 63 x 10 -34 J*s * 1 x 106 /s = 6. 63 x 10 -28 J = 4 x 10 -9 e. V = 4 ne. V • red photon: f = c/ 3 x 108 m/s / 7 x 10 -7 m = 4. 3 x 1014 Hz, red photon energy = 1. 78 e. V • blue: = 400 nm; photon energy = 3. 11 e. V. • X-ray photon with = 1 nm; photon energy = 1, 243 e. V = 1. 24 ke. V

Photoelectric Effect Light hits a metal plate, and electrons are ejected. These electrons are collected in the circuit andlight form a current. ejected electron A - + V

Photoelectric Effect Put into a nice equation: • hf = W + e*Vstop – where f is the frequency of the light – W is the “WORK FUNCTION”, or the amount of energy needed to get the electron out of the metal – Vstop is the stopping potential • When Vstop = 0, f = fcutoff , and hfcutoff = W.

Rutherford Scattering The results of the scattering were consistent with the alphas scattering off a tiny positive massive nucleus rather than the diffuse positive pudding. The results indicated that the positive charge and heavy mass were located in a nucleus on the order of 10 -14 m (Recall the atom size is on the order of 10 -10 m).

The Bohr Theory: angular momentum, radius and Energy are all quantized (with quantum number, n) r = n 2 2/(meke 2) = (5. 3 x 10 -11 m) * n 2 (for n=1, this is just the right size radius for the atom) and E = [-mek 2 e 4/2 2]*(1/n 2) = -13. 6 e. V / n 2 (where 1 e. V = 1. 6 x 10 -19 Joules). This says the electron energy is QUANTIZED.

The Bohr Theory - an example E = hf = [-13. 6 e. V]*[(1/nf 2) - (1/ni 2)] Example: In the case of ni = 3, and nf = 2, E = (-13. 6 e. V)*(1/4 - 1/9) = 1. 89 e. V E = hf = hc/ , so in this case, emitted = hc/ E = (6. 63 x 10 -34 J-sec)*(3 x 108 m/s)/(1. 89 x 1. 6 x 10 -19 J) = 658 nm (red light).

The Bohr Theory • Note that we have quantized energy states for the orbiting electron. • Note that for all nfinal = 1, we only get UV photons. • Note that for all nfinal > 2, we only get IR photons.

De. Broglie Hypothesis Problem with Bohr Theory: WHY L = n ? • have integers with standing waves: n( /2) = Length • consider circular path for standing wave: n = 2 r , and so from Bohr theory: L = mvr = nh/2 get 2 r = nh/mv = n which means = h/mv = h/p.

Quantum Theory What we are now dealing with is the Quantum Theory: • atoms are quantized (you can have 2 or 3, but not 2. 5 atoms) • light is quantized (you can have 2 or 3 photons, but not 2. 5) • in addition, we have quantum numbers (L = n , where n is an integer)

Heisenberg Uncertainty Principle A formal statement of this (from Fourier analysis) is: x * k (where k = 2 / , and indicates the uncertainty in the value) But from the De. Broglie Hypothesis, = h/p, this uncertainty relation becomes: x * (2 / ) = x * (2 p/h) = 1/2 , or x * p = /2.

Heisenberg Uncertainty Principle A similar relation from Fourier analysis for time and frequency: t * = 1/2 leads to another part of the Uncertainty Principle (using E = hf = ): t * E > /2. There is a third part: * L > /2 (where L is the angular momentum value). All of this is a direct result of the wave/particle duality of light and matter.

Nuclear Physics Stability: see sheet detailing stable isotopes Radiations: 1) a, - + g are all emitted; 2) protons and neutrons are NOT emitted, except in the case of mass numbers 5 and 9; 3) alphas are emitted only for mass numbers greater than 209, except in the case of mass number 8.

Alpha ( ) decay 234 + g example: 92 U 238 Th 90 2 (it is not obvious whethere is a gamma emitted; this must be looked up in each case) Mass is reduced! NOTE: 1. subscripts must be conserved (conservation of charge) 92 = 90 + 2 2. superscripts must be conserved (conservation of mass) 238 = 234 + 4

Beta minus b-) decay 14 + b 0 + u 0 example: 6 C 14 N 7 -1 0 (a neutron turned into a proton by emitting an electron; however, one particle [the neutron] turned into two [the proton and the electron]. Charge and mass numbers are conserved, but since all three (neutron, proton, and electron) are fermions [spin 1/2 particles], angular momentum, particle number, and energy are not! Need the anti-neutrino [0 u 0] to balance everything!

Positron (b+) decay 11 + 0 + u 0 example: 6 C 11 B b 5 +1 0 a proton turns into a neutron by emitting a positron; however, one particle [the proton] turned into two [the neutron and the positron]. Charge and mass numbers are conserved, but since all three are fermions [spin 1/2 particles], angular momentum, particle number, and energy are not! Need the neutrino [0 u 0] to balance everything!

Nuclear Physics General Rules: 1) a emitted to reduce mass, only emitted if mass number is above 209 2) - emitted to change neutron into proton, happens when there are too many neutrons 3) + emitted (or electron captured) to change proton into neutron, happens when there are too few neutrons 4) g emitted to conserve energy in reaction, may accompany a or .

Mass Defect & Binding Energy Similarly, the missing mass was converted into energy (E=mc 2) and emitted when the carbon-12 atom was made from the six protons and six neutrons: m = 6*mproton + 6*mneutron - m. C-12 = 6(1. 00782 amu) + 6(1. 008665 amu) - 12. 00000 amu =. 099 amu; BE = m*c 2 = (0. 099 amu)*(1. 66 x 10 -27 kg/amu)*(3 x 108 m/s)2 = 1. 478 x 10 -11 J*(1 e. V/1. 6 x 10 -19 J) = 92. 37 Me. V

Activity Review: N(t) = No e- t A = N = Aoe- t T(half life) = ln(2) / . If the half life is large, is small. This means that if the radioactive isotope will last a long time, its activity will be small; if the half life is small, the activity will be large but only for a short time!

Radioactivity around us an example For 1 gram of carbon in a living plant, Ao = 15. 0/min. Also, carbon-14 has a half life of 5, 730 years. If a 1 gram carbon sample from a dead plant has an activity of 9. 0/min, then using: A = Aoe- t , we have 9. 0/min = 15. 0/min * e-(ln 2/5730 yrs)t , or -(ln 2/5730 yrs)*t = ln(9/15) , or t = 5730 years * ln(15/9) / ln(2) = 4, 200 years.

Radioactivity around us Another radioactive isotope found in the earth is 238. Since it is well above the 209 mass limit, it U 92 gives rise to a whole series of radioactive isotopes with mass numbers 238, 234, 230, 226, 222, 218, 214, 210. The 226 isotope is 88 Ra 226, which is the isotope that Marie Curie isolated from uranium ore. The 222 isotope is 86 Rn 222 which is a noble gas. Other radioactive isotopes found in nature are 232 , 235, and 40. Both 232 and 235 Th U K Th U 90 92 19 90 92 have decay chains that lead down to 82 Pb (lead).

X-rays example Eionization = 13. 6 e. V * (Z-1)2 where the -1 comes from the other inner shell electron. If the electrons have this ionization energy, then they can knock out this inner electron, and we can see the characteristic spectrum for this target material. For iron with Z=26, the ionization energy is: 13. 6 e. V * (26 -1)2 = 1 e * 8, 500 volts.

X and g ray penetration I = Io e-mx where depends on the material the x-ray is going through and the energy of the x-ray. In a similar way to half lives, we can define a half-value-layer, hvl, where hvl = ln(2)/m.

Measuring Radioactivity • How do we measure radioactivity? The Bq (dis/sec) and Curie (1 Ci = 3. 7 x 1010 Bq) measure how many decays happen per time. However, different radioactive materials emit different particles with different energies. • What is the source of the health effects of radiation? Radiation ( , b, g) ionizes atoms. Ionized atoms are important to biological function, and so radiation may interfere with biological functions. • Can we devise a way to measure the health effects of radiation?

Measuring Health Effects Can we devise a way to measure the health effects of radiation? A unit that directly measures ionization is the Roentgen (R) = (1/3) x 10 -9 Coul created per cc of air at STP. This uses air, since it is relatively easy to collect the charges due to ionization. It is harder to do in biological material, so this method is best used as a measure of EXPOSURE dose.

Measuring Health Effects Can we devise a way to measure the health effects of radiation? In addition to measuring ionization ability in air, we can also measure the energy that is absorbed by a biological material: Rad =. 01 J/kg MKS: Gray (Gy) = 1 J/kg = 100 rads. This is called an ABSORBED dose. Generally, one Roentgen of exposure will give one rad of absorption.

Measuring Health Effects This difference in penetrating ability (and localization of ionization) leads us to create an RBE (radiation biological equivalent) factor and a new unit: the rem. The more localized the ionization, the higher the RBE. # of rems = RBE * # of rads. This is called an EFFECTIVE dose. RBE for gammas = 1; RBE for betas = 1 to 2; RBE for alphas = 10 to 20.

Levels of Radiation and Measurable Health Effects 200 millirems/year: background Here are some more benchmarks based on our experience with acute (short time) doses: 20, 000 millirems: measurable transient blood changes; 150, 000 millirems: acute radiation sickness; 200, 000 millirems: death in some people; 350, 000 millirems: death in 50% of people.

Chain Reactions In some cases, a very heavy nucleus, instead of undergoing alpha decay, will spontaneously split in two. Example: 238 U 92 129 + 106 + 3 n 1 + energy Sn Mo 50 42 0 The amount of energy coming from this reaction is on the order of 200 Me. V, which is about 200 million times more than a chemical reaction. This fissioning of uranium does not always result in these two resultant atoms - there is a whole range of resulting atoms. But it always gives a few neutrons. In some cases a neutron can stimulate a heavy nucleus to split in two. This, if properly set up, can cause a chain reaction. This chain reaction is the basis for both the nuclear bomb and the nuclear power station.

Fusion 1 + H 1 1 1 + D 2 H 1 1 1 + T 3 H 1 1 2 + 0 + + energy D 1 +1 3 + 0 + + energy T 1 +1 4 + energy He 2 so we have four hydrogens becoming one helium, with about 24 Me. V of energy and two neutrino’s produced plus two positrons which will combine with the extra two electrons from the 4 H’s to give another 2 Me. V’s of energy.