* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Physics 261 - Purdue Physics
Survey
Document related concepts
Time in physics wikipedia , lookup
Introduction to gauge theory wikipedia , lookup
Circular dichroism wikipedia , lookup
Anti-gravity wikipedia , lookup
Potential energy wikipedia , lookup
Magnetic field wikipedia , lookup
Work (physics) wikipedia , lookup
Field (physics) wikipedia , lookup
History of electromagnetic theory wikipedia , lookup
Magnetic monopole wikipedia , lookup
Superconductivity wikipedia , lookup
Maxwell's equations wikipedia , lookup
Electromagnet wikipedia , lookup
Electromagnetism wikipedia , lookup
Electric charge wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Transcript
Lecture 1-1 Coulomb’s Law • Charges with the same sign repel each other, and charges with opposite signs attract each other. • The electrostatic force between two particles is proportional to the amount of electric charge that each possesses and is inversely proportional to the distance between the two squared. q1q2 F1,2 k 2 rˆ1,2 r1,2 r1,2 by 1 on 2 q1 q2 r • Coulomb constant: k 1 4 0 8.99 109 N m 2 / C 2 where 0 is called the permittivity constant. Lecture 1-2 Electric Field Define electric field, which is independent of the test charge, q2 , and depends only on position in space: F E q Electric Field due to a Point Charge Q F 1 Q E rˆ 2 q 4 0 r Lecture 1-3 Dynamics of a Charged Mass in Electric Field For -Q<0 in uniform E downward: F ma ( Q ) E QE a ay j j (E E j) m 1 2 y (t ) a y t , x (t ) v x t 2 -Q 2 1 x QEx 2 y ay 2 vy2 = at = qE/m t vx >>0 v 2 mv x x QEt v (t ) v v y (t ) v m tan y 2 xv 2 2 x 2 • Oscilloscope • Ink-Jet Printing • Oil drop experiment vx http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/electric_single_particle/applet.html Lecture 1-4 Electric Field from Coulomb’s Law Bunch of Charges r + + + - qi+ - - 1 i P + - Continuous Charge Distribution P r dq + k qi E rˆ 2 i 4 0 i ri Summation over discrete charges http://www.falstad.com/vector3de/ 1 dq E rˆ d E 2 4 0 r dV dq dA dL (volume charge) (surface charge) (line charge) Integral over continuous charge distribution Lecture 1-5 Gauss’s Law: Quantitative Statement The net electric flux through any closed surface equals the net charge enclosed by that surface divided by ε0. E ndA E Qenclosed 0 How do we use this equation?? The above equation is TRUE always but it doesn’t look easy to use. BUT - It is very useful in finding E when the physical situation exhibits a lot of SYMMETRY. Lecture 1-6 Charges and fields of a conductor • In electrostatic equilibrium, free charges inside a conductor do not move. Thus, E = 0 everywhere in the interior of a conductor. • Since E = 0 inside, there are no net charges anywhere in the interior. Net charges can only be on the surface(s). 0 The electric field must be perpendicular to the surface just outside a conductor, since, otherwise, there would be currents flowing along the surface. Lecture 1-7 Electric Potential Energy of a Charge in Electric Field • Coulomb force is conservative => Work done by the Coulomb force is path independent. • Can associate potential energy to charge q0 at any point r in space. dl U (r ) It’s energy! A scalar measured in J (Joules) dW q0 E d l dU dW q0 E d l Lecture 1-8 Electric Potential • U(r) of a test charge q0 in electric field generated by other source charges is proportional to q0 . • So U(r)/q0 is independent of q0, allowing us to introduce electric potential V independent of q0. U ( r ) V ( r ) q0 • [Electric potential] = [energy]/[charge] SI units: J/C = V (volts) 1J U (r) V (r) q0 taking the same reference point Scalar! Potential energy difference when 1 C of charge is moved between points of potential difference 1 V E from V We can obtain the electric field computes V from E from the potential V by inverting the integral that E: r r V (r ) E d l ( Ex dx E y dy Ez dz ) V Ex x Expressed as a vector, V Ey y E is the negative gradient of V E V V Ez z Electric Potential Energy and Electric Potential positive Lecture 1-10 charge High U (potential energy) High V High V (potential) Low U negative charge Low U Low V Low V High U Electric field direction Electric field direction Lecture 1-11 Two Ways to Calculate Potential - • Integrate E from the reference point at (∞) to the point (P) of observation: V r E dl Q r P P A line integral (which could be tricky to do) If E is known and simple and a simple path can be used, it may be reduced to a simple, ordinary 1D integral. • Integrate dV (contribution to V(r) from each infinitesimal source charge dq) over all source charges: q1 P q2 Q P q3 q4 Lecture 1-12 Capacitance The two conductors hold charge +Q and –Q, respectively. • Capacitor plates hold charge Q • The capacitance C of a capacitor is a measure of how much charge Q it can store for a given potential difference ΔV between the plates. Expect Q V Q Let C V Capacitance is an intrinsic property of the capacitor. C Q Coulomb F V Volt (Often we use V to mean ΔV.) farad Lecture 1-13 Steps to calculate capacitance C 1. 2. 3. 4. Put charges Q and -Q on the two plates, respectively. Calculate the electric field E between the plates due to the charges Q and -Q, e.g., by using Gauss’s law. Calculate the potential difference V between the plates b due to the electric field E by Vba E dl a Calculate the capacitance of the capacitor by dividing the charge by the potential difference, i.e., C = Q/V. Lecture 1-14 Energy of a charged capacitor How much energy is stored in a charged capacitor? Calculate the work required (usually provided by a battery) to charge a capacitor to Q Calculate incremental work dW needed to move charge dq from negative plate to the positive plate at voltage V. dW V (q) dq q / C dq Total work is then Q 1 1 Q2 U dW qdq C0 2 C 2 1 QV Q U CV 2 2 2 2C Lecture 1-15 Dielectrics between Capacitor Plates free charges +Q • Electric field E between plates can be calculated -Q from Q – q. E (Q q) / A neutral -q +q Polarization Charges ± q 0 , V Ed Q Q C V (Q q)d / 0 A 0 A d 1 q 1 Q Lecture 1-16 Capacitors in Parallel V is common q1 q2 q3 V C1 C2 C3 Equivalent Capacitor: C q V where q q1 q2 q3 q1 q2 q3 Ceq C1 C2 C3 V Lecture 1-17 Capacitors in Series q is common q C1V1 C2V2 C3V3 Equivalent Capacitor: C q V where V V1 V2 V3 1 V1 V2 V3 1 1 1 Ceq q C1 C2 C3 Lecture 1-18 Electric Current Current = charges in motion Magnitude q dq I lim x 0 t dt rate at which net positive charges move across a cross sectional surface Units: [I] = C/s = A (ampere) Current is a scalar, signed quantity, whose sign corresponds to the direction of motion of net positive charges by convention I J dA A J = current density (vector) in A/m² Lecture 1-19 Ohm’s Law Current-Potential (I-V) characteristic of a device may or may not obey Ohm’s Law: I V (J E) or V = IR with R constant Resistance tungsten wire V V R I A gas in fluorescent tube diode (ohms) Lecture 1-20 Energy in Electric Circuits • Steady current means a constant amount of charge ΔQ flows past any given cross section during time Δt, where I= ΔQ / Δt. Energy lost by ΔQ is V U Q (Va Vb ) I t V => heat So, Power dissipation = rate of decrease of U = dU P IV I 2 R V 2 / R dt Lecture 1-21 Resistors in Parallel i i iR i R i R i R 1 12 23 3 12 3e q Devices in parallel has the 1 1 1 1 same potential drop o r R R R 1 2R 3R e q R e qR 1 2R 3 Generally, 1 1 Req i Ri ••• R L A Lecture 1-22 Kirchhoff’s Rules Kirchhoff’s Rule 1: Loop Rule When any closed loop is traversed completely in a circuit, the algebraic sum of the changes in potential is equal to zero. V0 Coulomb force is conservative i loop Kirchhoff’s Rule 2: Junction Rule The sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction. I I i in j o u t Conservation of charge In and Out branches Assign Ii to each branch Lecture 1-23 Galvanometer Inside Ammeter and Voltmeter Galvanometer: a device that detects small currents and indicates its magnitude. Its own resistance Rg is small for not disturbing what is being measured. galvanometer Ammeter: an instrument used to measure currents shunt resistor Voltmeter: an instrument used to measure potential differences galvanometer Lecture 1-24 Galvanometer Inside Ammeter and Voltmeter Galvanometer: a device that detects small currents and indicates its magnitude. Its own resistance Rg is small for not disturbing what is being measured. galvanometer Ammeter: an instrument used to measure currents shunt resistor Voltmeter: an instrument used to measure potential differences galvanometer Lecture 1-25 Discharging a Capacitor in RC Circuits 1. Switch closed at t=0. Initially C is fully charged with Q0 2. Loop Rule: 3. Convert to a differential equation dQ I dt 4. Q IR 0 C Q dQ R 0 C dt Solve it! Q Q0 e t / RC dQ Q0 t / RC I e dt RC I Lecture 1-26 Charging a Capacitor in RC Circuits 1. Switch closed at t=0 C initially uncharged, thus zero voltage across C. I0 / R 2. Loop Rule: IR Q 0 C 3. Convert to a differential equation dQ I dt 4. dQ Q R 0 dt C Solve it! Q C 1 e t / RC dQ t / RC , I e dt R (τ=RC is the time constant again) Lecture 1-27 Magnetic Field B • Magnetic force acting on a moving charge q depends on q, v.Vary q and v in the presence of a given magnetic field and measure magnetic force F on the charge. Find: F varies sinusoidally as F v direction of v is changed F qv Fq vB (q>0) direction by Right Hand Rule. B is a vector field This defines B. F v , B F q v B s i n F NN B T ( t e s l a ) q v m / sA m C vB 1 T = 104 gauss (earth magnetic field at surface is about 0.5 gauss) If q<0 Lecture 1-28 Magnetic Force on a Current A • Consider a current-carrying wire in the presence of a magnetic field B. • There will be a force on each of the charges moving in the wire. What will be the total force dF on a length dl of the wire? • Suppose current is made up of n charges/volume each carrying charge q < 0 and moving with velocity v through a wire of cross-section A. • Force on each charge = • Total force = • Current = qv B dF n A(dl ) qv B I n Av q For a straight length of wire L carrying a current I, the force on it is: dF Idl B F IL B Lecture 1-29 Both B and E present F q v B u p m F q E d o w n e E v B when balanced velocity selector No deflection when E=3 kV/m, B=1.4 G 4 v 3 0 0 0 / 1 . 4 1 0 0 7 2 .1 4 3 1 0 ( m /) s http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/exb_thomson/applet.html Lecture 1-30 Magnetic Torque on a Current Loop Definition of torque: rF abut a chosen point • If B field is parallel to plane of loop, the net torque on loop is 0. B • If B is not zero, there is net torque. n magnetic moment direction n so that n is twisted to align with B Lecture 1-31 Potential Energy of Dipole • Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field. • Define a potential energy U (with zero at position of max torque) corresponding to this work. B x F F . θ θ U τdθ U μB sin θdθ 90 90 Therefore, U μB cos θ 90 θ U μB cos θ U B Lecture 1-32 Sources of Magnetic Fields • Moving point charge: μ0 q v r dB 4π r 2 also N 0 4 107 2 T m / A A Permeability constant • Bits of current: I The magnetic field “circulates” around the wire. μ0 I d l r dB 4π r 2 Biot-Savart Law http://falstad.com/vector3dm/ Lecture 1-33 Gauss’s Law for Magnetism sources Gauss’s Law E S E da qS 0 Gauss’s Law for Magnetism No sources B S B da 0 Lecture 1-34 Ampere’s Law in Magnetostatics Biot-Savart’s Law can be used to derive another relation: Ampere’s Law The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop, C Bt dl 0 (i1 i2 ) C B d l 0 I C • Choosing a direction of integration. • A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration. Lecture 1-35 Potential Energy of Dipole • Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field. • Define a potential energy U (with zero at position of max torque) corresponding to this work. B x F F θ U τdθ U = +μ B cos θ 90 Therefore, U μB cos θ U μB cos θ 90 θ . Lecture 1-36 Faraday’s Law of Induction The magnitude of the induced EMF in conducting loop is equal to the rate at which the magnetic flux through the surface spanned by the loop changes with time. dΦB ε dt wher B e S B ndA N Minus sign indicates the sense of EMF: Lenz’s Law • Decide on which way n goes Fixes sign of ΔϕB • RHR determines the positive direction for EMF N Lecture 1-37 Ways to Change Magnetic Flux B BA cos • Changing the magnitude of the field within a conducting loop (or coil). • Changing the area of the loop (or coil) that lies within the magnetic field. • Changing the relative orientation of the field and the loop. motor generator Lecture 1-38 Motional EMF of Sliding Conductor Induced EMF: Lenz’s Law gives direction counter-clockwise Faraday’s Law FM decelerates the bar dv B 2l 2 v m dt R dv B 2l 2 dt v mR v(t ) v 0 e B 2l 2 t mR dB dx Bl Blv dt dt This EMF induces current I Blv I R R Magnetic force FM acts on this I B 2l 2 v FM I lB R Lecture 1-39 Self-Inductance • As current i through coil increases, magnetic flux through itself increases. This in turn induces counter EMF in the coil itself • When current i is decreasing, EMF is induced again in the coil itself in such a way as to slow the decrease. Self-induction B L i NB L (if flux linked) i L T m2 / A Wb / A H (henry) Faraday’s Law: dB dI L dt dt Lecture 1-40 1. 2. Energy Stored By Inductor Switch on at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + Loop Rule: dI IR L 0 dt - 3. Multiply through by I dI I I R LI dt 2 Rate at which battery is supplying energy Rate at which energy is stored in inductor L dU m dI LI dt dt Rate at which energy is dissipated by the resistor Um 1 2 LI 2 Lecture 1-41 1. RL Circuits – Starting Current Switch to e at t=0 As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. + I0 0 2. Loop Rule: - dI IR L 0 dt 3. Solve this differential equation I R 1 et /( L / R ) , VL L τ=L/R is the inductive time constant dI e t /( L / R ) dt T m2 / A T m2 / A s L / R V/A Lecture 1-42 Alternating Current (AC) = Electric current that changes direction periodically ac generator is a device which creates an ac emf/current. A sinusoidally oscillating EMF is induced in a loop of wire that rotates in a uniform magnetic field. B NBA cos NBA cos t dB NBA sin t dt where 2 2 f T ac motor = ac generator run in reverse Lecture 1-43 Resistive Load Start by considering simple circuits with one element (R, C, or L) in addition to the driving emf. Pick a resistor R first. + I(t) Kirchhoff’s Loop Rule: t I t R 0 , t peak sin t vR t I t R peak sin t V peak peak I t Ipeak Vpeak R sin t vR(t) and I(t) in phase -- Lecture 1-44 Capacitive Load Loop Rule: + -- q(t ) (t ) 0 C peak sin t q(t ) peak sin t C dq (t ) d (t ) I (t ) C dt dt C peak cos t v(t ) I(t) leads v(t) by 90o (1/4 cycle) Power: p(t ) I (t )v(t ) I peak cos t VC , peak sin t VC , peak I peak 2 sin 2t Pav 0 Lecture 1-45 Inductive Load Kirchhoff’s Loop Rule: + -- dI (t ) (t ) L 0 dt (t ) peak sin t dI (t ) vL (t ) L peak sin t dt peak I (t ) cos t L vL(t) leads I(t) by 90o (1/4 cycle) Power: p(t ) I (t )vL (t ) I peak cos t VL , peak sin t VL, peak I peak 2 sin 2t Pav 0 Lecture 1-46 (Ideal) LC Circuit • From Kirchhoff’s Loop Rule Q dI L 0 C dt • From Energy Conservation 2 Q 2 1 2 Q peak E LI 2C 2 2C dQ I dt dE 0 dt same const. Q dQ dI LI 0 C dt dt d 2Q 1 Q 0 2 dt LC Q Q peak cos(0t ) Q dI L 0 C dt harmonic oscillator with angular 1 frequency 0 LC Natural Frequency Lecture 1-47 Impedance in Driven Series RLC Circuit peak I peak R X L X C 2 2 Z 1 Z R L C 2 2 impedance, Z 1 L VL VC R C tan , cos VR R Z ϕ Lecture 1-48 Resonance For given ε peak , R, L, and C, the current amplitude Ipeak will be at the maximum when the impedance Z is at the minimum. peak I peak R X L X C 2 2 Z X L XC res L 1 resC ε and I in phase i.e., load purely resistive This is called resonance. Resonance angular frequency: , Z R, and I peak peak 1 res LC R Lecture 1-49 Transformer • AC voltage can be stepped up or down by using a transformer. • AC current in the primary coil creates a timevarying magnetic flux through the secondary coil via the iron core. This induces EMF in the secondary circuit. Ideal transformer (no losses and magnetic flux per turn is the same on primary and secondary). (With no load) d B V1 V2 turn dt N1 N 2 N1 N 2 V1 V2 N1 N 2 V1 V2 step-up step-down With resistive load R in secondary, current I2 flows in secondary by the induced EMF. This then induces opposing EMF back in the primary. The latter EMF must somehow be exactly cancelled because V1 is a defined voltage source. This occurs by another current I1 which is induced on the primary side due to I2. Lecture 1-50 Maxwell’s Equations S E dA Qinside 0 dB C E d l dt S B dA0 dE C B d l 0 I 0 0 dt Basis for electromagnetic waves! The equations are often written in slightly different (and more convenient) forms when dielectric and/or magnetic materials are present. Lecture 1-51 Electromagnetic Wave Propagation in Free Space So, again we have a traveling electromagnetic wave Em c Bm k c Em 1 Bm 0 0c 0 4 107 (T m / A) 1 0 0 speed of light in vacuum 0 8.85 1012 (C 2 / N m2 ) B 1 E 2 x c t E B x t Ampere’s Law Faraday’s Law 2 B 1 2 B Wave Equation 2 2 2 x c t c 3.00 108 (m / s) Speed of light in vacuum is currently defined rather than measured (thus defining meter and also the vacuum permittivity). Lecture 1-52 Maxwell’s Rainbow Light is an Electromagnetic Wave f c Lecture 1-53 ©2008 by W.H. Freeman and Company Lecture 1-54 Snell’s Law of Refraction n1 sin θ1 = n 2 sin θ2 Lecture 1-55 Summary: Laws of Reflection and Refraction Law of Reflection • A reflected ray lies in the plane of incidence • The angle of reflection is equal to the angle of incidence Medium 1 1 1 Law of refraction • A refracted ray lies in the plane of incidence • The angle of refraction is related to the angle of incidence by n2 sin 2 n1 sin 1 Snell’s Law c / ni c 1 i f f ni ni Where λ is the wavelength in vacuum Medium 2 n2 n1 Lecture 1-56 Total Internal Reflection All light can be reflected, none refracting, when light travels from a medium of higher to lower indices of refraction. medium 2 e.g., glass (n=1.5) to air (n=1.0) sin 2 n1 1 sin 1 n2 2 1 But θ2 cannot be greater than 90O ! medium 1 In general, if sin θ1 > (n2 / n1), we have NO refracted ray; we have TOTAL INTERNAL REFLECTION. c sin 1 n2 / n1 Critical angle above which this occurs. Lecture 1-57 Polarization of Electromagnetic Waves Polarization is a measure of the degree to which the electric field (or the magnetic field) of an electromagnetic wave oscillates preferentially along a particular direction. Linear combination of many linearly partially polarized rays of polarized random orientations unpolarized linearly polarized Looking at E head-on components equal y- and zamplitudes unequal y- and zamplitudes Lecture 1-58 ©2008 by W.H. Freeman and Company Lecture 1-59 Focal Point of a Spherical Mirror When parallel rays incident upon a spherical mirror, the reflected rays or the extensions of the reflected rays all converge toward a common point, the focal point of the mirror. Distance f is the focal length. concave mirror: Real focal point: the point to which the reflected rays themselves pass through. This is relevant for concave mirrors. f convex mirror Virtual focal point: the point to which the extensions of the reflected rays pass through. This is relevant for convex mirrors. Rays can be traversed in reverse. Thus, rays which (would) pass through F and strike the mirror will emerge parallel to the central axis. f Lecture 1-60 Mirror Equation and Magnification (f = r/2) 1 1 1 s s' f y' s' m y s m s' s • s is positive if the object is in front of the mirror (real object) • s is negative if it is in back of the mirror (virtual object) • s’ is positive if the image is in front of the mirror (real image) • s’ is negative if it is in back of the mirror (virtual image) • m is positive if image and object have the same orientation (upright) • m is negative if they have opposite orientation (inverted) • f and r are positive if center of curvature in front of mirror (concave) • f and r are negative if it is in back of the mirror (convex) Lecture 1-61 Locating Images only using the parallel, focal, and/or radial rays. Real images form on the side of a mirror where the objects are, and virtual images form on the opposite side. Lecture 1-62 Thin Lenses nomenclature • A lens is a piece of transparent material with two refracting surfaces whose central axes coincide. A lens is thin if its thickness is small compared to all other lengths (s, s’, radii of curvature). 1 1 1 n 1 f r1 r2 f>0 Convergent lens • Net convex – thicker in the middle • Parallel rays converge to real focus. r1>0 r2<0 •f>0 f<0 Divergent lens • Net concave – thinner in the middle • Parallel rays diverge from virtual focus. •f<0 r1<0 r2>0 Lecture 1-63 Signs in the Lens Equation for Thin Lenses 1 1 1 p q f q m p • p is positive for real object • p is negative for virtual object • q is positive for real image yi f q q yo f p • q is negative for virtual image • m is positive if image is upright • m is negative if image is inverted • f is positive if converging lens • f is negative if diverging lens Lecture 1-64 Geometric Optics vs Wave Optics • Geometric optics is a limit of the general optics where wave effects such as interference and diffraction are negligible. Geometric optics applies when objects and apertures involved are much larger than the wavelength of light. In geometric optics, the propagation of light can be analyzed using rays alone. • Wave optics (sometimes also called physical optics) - wave effects play important roles. Wave optics applies when objects and apertures are comparable to or smaller than the wavelength of light. In wave optics, we must use the concepts relevant to waves such as phases, coherence, and interference. Lecture 1-65 Thin-Film Interference-Cont’d (Assume near-normal incidence.) Path length difference: l 2t destructive m (m 1/ 2) constructive where 0 n • ray-one got a phase change of 180o due to reflection from air to glass. • the phase difference due to path length is: •then total phase difference: = ’+180. ' l 2 l ' n 2 Lecture 1-66 Interference Fringes For D >> d, the difference in path lengths between the two waves is L d sin • A bright fringe is produced if the path lengths differ by an integer number of wavelengths, d sin m , m 0, 1, • A dark fringe is produced if the path lengths differ by an odd multiple of half a wavelength, d sin (m 1/ 2) , m 0, 1, Lecture 1-67 Intensity of Interference Fringes-Cont’d For Young’s double-slit experiment, the phase difference is 2 d sin 2 m, for maxima 2 ( m1/2), for mimina 2 4 I 0 cos 2 Lecture 1-68 Dark and Bright Fringes of Single-Slit Diffraction Lecture 1-69 Intensity Distribution 1 2 sin 2 2 a sin I ( ) I max where 2 maxima: 0 central maximum because or sin sin x 1 as x 0 x 1 1 or a sin ( m ) 2 2 minima: sin 2 0 or a sin m m 1, 2, 3,... m 0 Lecture 1-70 Intensity Distribution from Realistic DoubleSlit Diffraction I 4 I 0 cos2 double-slit intensity sin replace by I m 2 sin 2 I ( ) I m (cos ) d sin single-slit intensity envelope 2 a sin d sin m2 How many maxima will fit d m between central max and 2 a first envelope min?: Lecture 1-71 Diffraction Gratings • Devices that have a great number of slits or rulings to produce an interference pattern with narrow fringes. • One of the most useful optical tools. Used to analyze wavelengths. D Types of gratings: • transmission gratings • reflection gratings up to thousands per mm of rulings Maxima are produced when every pair of adjacent wavelets interfere constructively, i.e., D d d sin m , m 0, 1, mth order maximum