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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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