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7Creview - SHM Wave basics Interference Interference...

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Unformatted text preview: SHM Wave basics Interference Interference summary Standing waves Ray optics Ray optics summary Thin lenses Thin lens summary Gravity Gravitational field Gravitational force Acceleration due to gravity Gravitational potential energy Electrostatics Fields and forces Potential and potential energy Field lines Equipotentials Dipoles Magnetostatics Fields and forces Magnetic field line Right Hand Rule #1 Right Hand Rule #2 Dipoles Induction Electromagnetism Polarization Quantum mechanics De Broglie wavelength Particle in a box Harmonic oscillator Photons Hydrogen atom Line spectra Pauli exclusion principle Saturday, 13 March 2010 1 TOC Simple Harmonic Motion (SHM) Conditions for SHM * Must have an equilibrium * Must have a restoring force when displaced from equilibrium * Force must take the form where Equilibrium y-y0 m Equilibrium • y0 is equilibrium ￿ F = ma = −k (y − y0 ) • y is the current position • k is a constant characteristic of the system. For a mass spring system k is the spring constant, for a pendulum k = mg/L Motion described by ￿ y (t) = A sin 2π t + φ0 T • A “amplitude”: max. displacement • t “time” • φ0 “fixed phase”: tells us about initial conditions • T “period”: time to repeat ￿ ￿ ￿ + y0 y-y0 y-y0 Saturday, 13 March 2010 1 f= 2π k , m 1 T = = 2π f m k indep of A, phi0 2 TOC Wave basics y (x, t = 3s) λ Amplitude x (m) To describe a wave we need • An amplitude A • A period T • A fixed phase constant φ0 • A direction (±) • A wavelength λ • A wave speed v = λf = λ/T y (x, t) = A sin(Φ(x, t)) + y0 y (x = 1.5m, t) Amplitude period WRONG!!! period Saturday, 13 March 2010 where 2π 2π Φ(x, t) = t± x + φ0 T λ t (s) Note: pick x constant – then you just have π SHM with “constant phase” φ0 ± 2λ x This case: y (x, t) = (3m) sin 6 s t + + See if you can show it from the graphs! 2π 4 mx ￿ 2π π 4 ￿ 3 TOC Source 1 d Waves-Interference Source 2 General goal: Figure out if the interference will be constructive, destructive or partial at the “point of interest”. Strategy: Look at the difference in total phase ∆Φ = Φ2 − Φ1 . We have three cases: • ∆Φ = (even)π – constructive • ∆Φ = (odd)π – destructive • ∆Φ ￿= (integer)π – partial Tool: Phase chart, which lists all sources of total phase. 2π t xi Source - 2π + φi = Φi Ti λi Source 2 + = Source 1 + = Source (2-1) + = x1 x2 x point of interest Special case #1: ∆Φ (determines type of interference) is in yellow bubble. If T1 = T2 then time term cancels. ∆Φ independent of time! Special case #2: Type of interference independent of time, but If T1 ￿= T2 then time term changes with t. depends on place! ∆Φ dependents of time! Some places have constructive, some places At a location, interference “beats” between have destructive interference. constructive and destructive. Saturday, 13 March 2010 4 TOC Source 1 d 2 d 2 Waves-Interference Source 2 If we are a long way from the sources (technically, x1 ￿ d,x2 ￿ d) then IMPORTANT: The interference type repeats once we increase ∆x by λ (if the waves have the same frequency). This, and ∆x, is typically MUCH smaller than either x1 or x2 . In lab we were using xi ∼ O(cm) to measure wavelengths λ ∼ O(100nm). If we have beats then once 1 1 − T2 T1 ⇒ ∆Φ = 2π (f2 − f1 ) t + ∆Φ(t = 0) ∆Φ = 2π ￿ ￿ t + ∆Φ(t = 0) contains path length, phi0 ∆x = x2 − x1 ≈ d sin θ x1 x2 θ x point of interest For ∆Φ to increase by 2π at the same location we have to wait for an amount of time 1 Tbeat = ⇒ fbeat = |f2 − f1 | |f2 − f1 | f1 + f2 fcarrier = If sound, the frequency we hear is the average frequency of the two waves. 2 time Tbeat Saturday, 13 March 2010 Tcarrier 5 TOC Waves-Interference (Summary) Source 1 d 2 d 2 Strategy: Look at the difference in total phase ∆Φ = Φ2 − Φ1 . We have three cases: • ∆Φ = (even)π – constructive • ∆Φ = (odd)π – destructive Tool: Phase chart, which lists all sources of total phase. 2π t xi Source - 2π + φi = Φi Ti λi Source 2 + = Source 1 + = Source (2-1) + = Approximation (useful in “far field”) ∆x = x2 − x1 ≈ d sin θ x1 Source 2 x2 • ∆Φ ￿= (integer)π – partial θ x point of interest ∆Φ lives here If T1 = T2 interference only a function of position. If T1 ￿= T2 we get beats T1 ￿= T2 : Carrier frequency is fcarrier = T1 ￿= T2 : Beat frequency is fbeat = |f1 − f2 | f1 +f2 2 Saturday, 13 March 2010 6 TOC Optics (Reflection & Refraction) Law of reflection: θi = θreflect (as measure from the normal) Law of refraction: n1 sin θi = n2 sin θrefract normal θi n1 n2 θreflect Rays going from a fast medium (n1 ) to a slow medium (n2 ) n2 > n1 θrefract normal θrefract ⇒ “bend closer to the normal” (Technically θi > θrefract ) Same materials, but incident ray from medium 2: n2 > n1 ⇒ “bend away from the normal” (Technically θi < θrefract ) n1 n2 θi Saturday, 13 March 2010 θreflect 7 TOC Optics (Reflection & Refraction) Law of reflection: θi = θreflect (as measure from the normal) Law of refraction: n1 sin θi = n2 sin θrefract normal normal normal n1 n2 n1 n2 n1 n2 incoming ray outgoing rays outgoing rays incoming ray Both the refracted and the reflected rays are on the opposite side of the normal to the incident ray (i.e. “ingoing” and “outgoing” sides) WRONG!!!! Saturday, 13 March 2010 8 TOC Optics (Reflection & Refraction) Law of reflection: θi = θreflect (as measure from the normal) Law of refraction: n1 sin θi = n2 sin θrefract normal n1 n2 θc incoming ray outgoing rays The critical (incident) angle is when the refracted angle θrefract = 90◦ . n2 sin θc = n1 sin 90◦ = n1 There is only reflection if θinc > θc n1 n2 θinc > θc normal θi = θB θreflect This light is completely polarized Occurs when angle between the reflected and refracted rays is 90◦ tan θBrewster nrefracting medium = nincident medium n1 n2 (Condition for completely polarized reflecting ray) θrefract Saturday, 13 March 2010 9 TOC Optics (Reflection & Refraction) - summary normal θi n1 n2 Law of reflection: θi = θreflect (as measure from the normal) Law of refraction: n1 sin θi = n2 sin θrefract speed of light in vacuum n= ≥1 speed of light in medium As n gets higher, light travels slower Total Internal reflection can only occur going from a slow medium to a fast one (we “run out of room” to bend away from the normal) TIR occurs for θinc > θc , where sin θinc nrefract = nincident θreflect θrefract normal θrefract n1 n2 θi θreflect Make sure the refracted ray and reflected ray are on the opposite sides of the normal to the incident ray! Make sure you know how to locate the normal! The Brewster’s angle θB is the angle from which reflected light is completely polarized. Note that we get a Brewster’s angle going from slow → fast or fast→ slow. The Brewster’s angle is found via tan θBrewster Saturday, 13 March 2010 n1 n2 θc n1 n2 nrefracting medium = nincident medium 10 TOC o Thin lenses Solid – real light rays Dashed – virtual rays |i| Virtual rays crossing – virtual image (i < 0) o −f i f Magnification m: m= −i o Real rays crossing – real image (i > 0) |m| > 1 : image larger m > 0 : image upright |m| = 1 : image same size m < 1 : image inverted |m| < 1 : image smaller Saturday, 13 March 2010 11 TOC Single lens “cheat sheet” Lens Diverging Converging Object distance All o > 2f f < o < 2f o<f Thin lenses (summary) Image size Smaller Smaller Larger Larger Image type Virtual Real Real Virtual Image Orientation Upright Inverted Inverted Upright For a single lens, o > 0 and so the magnification has the opposite sign to i. i.e. can only have upright virtual images or inverted real images. NOT true for multiple lenses. Sign conventions: i>0 i<0 f >0 f <0 m>0 m<0 Real image Virtual image Converging lens Diverging lens Upright image Inverted image Locating images: • Ray tracing 11 1 • Thin lens equation + = o i f Magnification m: −i m= o |m| > 1 : image larger |m| = 1 : image same size |m| < 1 : image smaller Optometry: Goal: to place a (virtual) image between near and far points To make images closer (|i| < o) we need to make them smaller. Use cheat sheet -> Diverging lens! To make images further away (o > |i|) we need to make them larger. Use cheat sheet --> Converging lens (object must be within focal distance of lens) Saturday, 13 March 2010 12 TOC Electric and Gravitational potential Saturday, 13 March 2010 13 TOC Magnetic field and force µ0 I B (r) = 2π r Currents create magnetic fields Field from a straight wire: Field lines are circles around the wire. To find direction of field, use RHR #1 force I The magnetic field lines curve, but the magnetic field is a vector (i.e. straight) at every point. To find the direction of the B vector, it is “tangent to” the field line. |FB on charge | = qvB sin θ Saturday, 13 March 2010 14 TOC Induction Effective Area A￿ = A cos θ (θ angle between loop normal and B field) Intuitively the flux Φ is the number of field lines through the loop. Quantitatively Φ = BA cos θ θ To create a voltage ε we need a change in flux through the loop. Quantitatively dΦ ∆Φ ε=− ≈− dt ∆t Using Ohm’s Law: “∆V ” = ε = IR i.e. an induced current I flows. To find the direction of the induced current we use Lenz’s law: The direction of the induced current set up a flux that will oppose the change in flux that created it Saturday, 13 March 2010 15 TOC Electromagnetic waves Electric field (direction of polarization) Direction wave is moving Magnetic field E B • E and B fields are both waves 2π = E0 sin t± T ￿ 2π = B0 sin t± T ￿ 2π x + φ0 λ ￿ 2π x + φ0 λ ￿ • The E and B fields at a given point in space at a particular time are perpendicular. • We only look at the E field to discuss polarization. The field keeps oscillating so we talk about a vertically polarized ray, not an “up” polarized ray (E field will point up and down!) λ c = λf = T • The amplitudes of the E and B fields are related. E0 = cB0 • The E and B fields are in phase. Saturday, 13 March 2010 16 TOC Quantum mechanics Saturday, 13 March 2010 17 ...
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This note was uploaded on 11/14/2010 for the course PHY 7C taught by Professor Mahmud during the Fall '08 term at UC Davis.

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