Spring 2016 PHYS 306: Final Exam Review Exam date – May 2nd 9:45-11:45 Two final practical exams – one due 26th in class and the other for yourself Last lecture – April 26th on chapter 8: nonlinear harmonic oscillator 4/21/2016 Phasor Presentation Rotating Arrow + Phase Angle t t 4/21/2016 Representation of a complex number in terms of real and imaginary components Additions of Two Parallel SHOs in 1D 1=5, 2 =6 1=11, 2 =12 4/21/2016 Damped SHOs b2 0 4 2 0 4/21/2016 x(t ) bx (t ) x(t ) 0 2 0 0 A2 F0 (m 02 ) 0 2 1 ( ) 2 Q 0 4/21/2016 20 Q tan 1/ Q 0 0 Variation of Pav with 4/21/2016 0 Q Coupled oscillators and normal modes x t 4/21/2016 Coupled oscillators and normal modes 4/21/2016 Wave Motion as Coupled Oscillations Oscillation of infinite no. of coupled particles (lattice/medium): 2 y 2 y 2 c 2 t x 2 y ( x, t ) f ( x ct ) yn An cos(nt kx) 4/21/2016 Coupled Oscillations of a Loaded String y T a x 4/21/2016 Reflection Wave eqations and atConditions an interface Wave Equations & Boundary y A yB y A y B x x y1 t x v A y3 t x vB y5 t x vC y2 t x v A y4 t x vB y A B x combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components 4/21/2016 hence derive fractional11 C transmission and reflection 2E 2E 2 0 2 x t y 2 M2 y x, t M1 1 T δx x 4/21/2016 T x+δx 2 2 y T 2 y 2 y c 2 2 t x x 2 x Light is an Electromagnetic Wave E 2 E 2 0 t 2 B 2 B 2 0 dt 2 ~ i ( kx t ) E y (r , t ) E y e ~ i ( kx t ) Bz ( r , t ) Bz e 1. The electric field, the magnetic field, and the k-vector are all perpendicular: EB k 2. The electric and magnetic fields are in phase. 4/21/2016 c 1 0 0 3 *108 m / s The Fresnel’s problem ki Er Ei Bi kr ni y Br i r Interface z t Et Bt kt I r Ar R Reflected Power / Incident Power I i Ai I t At T Transmitted Power / Incident Power 4/21/2016 I i Ai nt x Note that R+T =1 R Reflected Power / Incident Power wi i r ni nt I r Ar I i Ai wi R r2 I t At T Transmitted Power / Incident Power I i Ai wi ni nt i t 4/21/2016 wt nt cos t 2 T t ni cos i r E0 r / E0i ni cos( i ) nt cos( t ) / ni cos( i ) nt cos( t ) t E0t / E0i 2ni cos( i ) / ni cos( i ) nt cos( t ) ki Er Ei Bi Interface Br i r t Et Bt 4/21/2016 kr kt ni nt r|| E0 r / E0i ni cos(t ) nt cos(i ) / ni cos(t ) nt cos(i ) t|| E0t / E0i 2ni cos(i ) / ni cos(t ) nt cos(i ) ki Bi Ei Br i r kr ni × Er Interface Beam geometry for light with its electric field parallel to the plane of incidence (i.e., in the page) 4/21/2016 t Et Bt kt nt The Tangential Field Components are Continuous Parallel polarization Perpendicular polarization 1.0 1.0 T T .5 .5 R R 0 0 0° 30° 60° Incidence angle, i 4/21/2016 90° 0° 30° 60° Incidence angle, i 90° Calculate R and T for normal incidence from an air-glass interface. nt ni R nt ni 2 T 4 nt ni nt ni For an air-glass interface (ni = 1 and nt = 1.5), R = 4% and T = 96% 4/21/2016 2 Energy and energy transport in waves U UE UB E2 2E 2E 2 0 2 x t E(x,t) = A cos[( t- k x + ] y 2 M2 y x, t M1 1 T δx x 4/21/2016 T x+δx 2 2 y T 2 y 2 y c 2 2 t x x 2 x Light is an Electromagnetic Wave 1. The electric field, the magnetic field, and the k-vector are all perpendicular: EB k 2. The electric and magnetic fields are in phase. 3. The light density an EM field: 1 Bz ( x, t ) E y ( x, t ) c So the electrical and magnetic energy densities in light are equal. 4/21/2016 The Energy Transport in Waves EM waves: U B 11 2 1 E E 2 U E 2 2 U UE UB E2 4/21/2016 S c 2 E B k Group Velocity Light-wave beats Etot(x,t) = 2E0 cos(kavex–avet) cos(kx–t) This is a rapidly oscillating wave: [cos(kavex–avet)] with a slowly varying amplitude [2E0 cos(kx–t)] The phase velocity comes from the rapidly varying part: v = ave / kave What about the other velocity—the velocity of the amplitude? Define the "group velocity:" vg /k In general, we define the group velocity as: vg d /dk 4/21/2016 vg dE /dp Group vs. Phase Velocity V Vg c n c dn n d So the group velocity equals the phase velocity when dn/d = 0, such as in vacuum. Otherwise, since n increases with , dn/d > 0, and vg < vphase. 4/21/2016 Group and phase velocity Quiz: vg and vph directions? 4/21/2016 Fourier transform We desire a measure of the frequencies present in a wave. This will lead to a definition of the term, the "spectrum.“ Plane waves have only one frequency, . F ( ) f (t ) exp(it ) dt FourierTransform f (t ) 4/21/2016 1 2 F ( ) exp(i t )d Inverse Fourier Transform Long vs. Short Pulses Long pulse Short pulse Time-bandwidth product (The uncertainty principle) 4/21/2016 CB ~ 0.5 in most cases t 2cB Example: the Fourier Transform of a rectangle function: rect(t) 1/ 2 1 [exp( i t )]1/1/2 2 F ( ) exp( i t ) dt i 1/ 2 1 [exp( i / 2) exp(i i exp(i / 2) exp( i sin( 2i F ( sinc( Imaginary Component = 0 4/21/2016 The Fourier Transform of the Triangle function, (t), is sinc2(/2) Convolution f (t ) g (t ) f ( x) g (t x) dx g ( x= ) dxF ( w {f (t) f (gt –(xt))} ) G ( ) 4/21/2016 rect(x) * rect(x) = (x) We can perform convolution visually. 4/21/2016 Interference and diffraction Division of amplitude 4/21/2016 Division of wavefront Matter of scale Superposition of Waves The irradiance is given by: I I1 c Re E1 E2 I 2 * I 2 I 0 2 I 0 cos(2kx) Fringes (in delay) I I 2 I 0 2 I 0 cos[ ] ‐ 4/21/2016 Spatial Fringes The fringe spacing, : Large angle: 2 /(2k sin ) /(2sin ) sin /(2) 0.5 m / 200 m Small angle: 1/ 400 rad 0.15 0.1 mm is about the minimum fringe spacing seen by eye 4/21/2016 Diffraction and Fourier Transform Fraunhofer Diffraction from a slit is simply the Fourier Transform of a rect function, which is a sinc function. The irradiance is then sinc2 . 4/21/2016