PHYSICS II PHYS 1200
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This 12 page Class Notes was uploaded by Justine Nitzsche on Monday October 19, 2015. The Class Notes belongs to PHYS 1200 at Rensselaer Polytechnic Institute taught by Scott Dwyer in Fall. Since its upload, it has received 17 views. For similar materials see /class/224883/phys-1200-rensselaer-polytechnic-institute in Physics 2 at Rensselaer Polytechnic Institute.
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Date Created: 10/19/15
Oscillations and Waves Period The period T isthe time required for one complete oscillation or cycle It is related to the frequency by T fi Simple Harmonic Motion In simple harmonic motion SHM the displacementxt of a particle from its equilibrium position is described by the equation x t xm coswt 43 In which xm is the amplitude ofthe displacement wt is the phase of the motion and f is the phase constant he angular frequency is related to the period and frequency ofthe motion a 211 2nf Sinusoidal Waves A sinusoidal wave moving in the positive direction of an x axis has the mathematic form yxt xm sinkx 7 wt re ym isthe amplitude of the wave k is the angular wave number a isthe angular frequency and kx 7 wt is the phase The wavelength l is related to k by k Lquot 1 Finally the wave speedvis related to these other parameters by k 739 39 Sound Waves are longitudinal mechanical waves that can travel through solids liquids or gases The speed 17 ofa sound wave is a medium having bulk modulusBand densityp is 17 5 speed of sound p In air at 20 C the speed of sound is 343 ms Sound Intensity the intensi of a sound wave at a surface isthe average rate per unit area at which energy is transferred by the wave through or ontothe surf e 1 5 Where P is the time rate of energy transferpower ofthe sound wave and A is the area of the surface intercepting the sound Electromagnetic Waves Electromagnetic Waves An electromagnetic wave consists of oscillating electric and magnetic fields 1h 39 39 reque 39 Ul 39 spectrum a small part ofwhich is visible light An electromagnetic waves travelling along an x axis has an electric field and a magnetic field with magnitudesthat depend on x and E 5l sinkx 7 wt and B 8l sinkx 7 mt Where 5l and 3l are the amplitudes of EandB The electric field inducesthe magnetic field and vice versa The speed ofelectromagnetic wave in vacuum isc which can be written as E C 7 B lumen Where Eand B are the simultaneous magnitudes ofthe fields Ener FlowThe rare per unit area atyv ich energy istransported via an electromagnetic wave is given by the Poynting vectorS a 1 a a s M DE X B The direction of and thus oLthe wave stravel and the energytransport is perpendicular to the directions of oth E X T e 39meraveraged rate per unit area at which energy is transported is a which is called the intensityI ofthe wave I WhereEmu Em 7 A nnint nurrP oI 39 isotropicallyrthat is with equal intensity in all directions The intensity ofthe waves at istance rfrom a point source 0 is p 44 Polarization 39 39 iftheir eleulll liclu quot39 single plane called the plane of oscillation Light waves from common sources are not polarized that is they are unpolarize o arized random y Polarizing Sheets W en a polarizing sheet Is place 39 the path oflight only electric field m onents ofthe light parallel to the sheet s polarl ng irection are transmitted by the 39 39 39 39 L L J The light that s ee m e emerges from a polarizing sheet is polarized parallelto the polarizingdirection ofthe h s e lfthe original light is initially unpolarized the transmitted intensityI is halfthe original intensity ID I z lfthe original light is initially polarized the transmitted intensity depends onthe angle 19 between the polarization direction ofthe original light and the polarizing direction ofthe I IO cosZ 9 Re ection and Refraction When a light ray encounters a boundan between two transparent media a reflecte ray and a r racted TLZ sin 92 n1 sin 91 39 me index ifthe angle ofincidence exceeds a critical angle 19 where 71 rt1 C 7 sin 2 Interference Diffraction The law of refraction can be derived from Huygens39principle by assumingthatthe ter an edge an obstacle or an aperture the size of index of refraction of any medium Isn 7 617 39 39 oflightinavacuum Index of Refraction The wavelength 1 of light in a medium depends on the index of refraction n ofthe medium In which 17 isthe speed of light in the 71 in In which l is the wavelength in vacuum Becalise ofthis dependency the phase difference between two waves can change ifthey pass through different materials with different indexes of refraction v Voung s 39 39 a single slit falls on two slits in a screen The light leaving these slits flares out by iffraction and in erference o rs in the region eyon the screen A fringe pattern ue to t e interference forms on a viewing screen sin 9 mil for m012 maximabright fringes d n 21 llform 012darkfringes Where 9 isthe angle the light path makes with a central axis and d is the slit se aration Coherence lftwo light wavesthat meet at a point are to interfere perceptible the p ase difference between them must remain constant with time that is the waves must be co erent Intensity in T 39 39 wo wa s each with intensity ID yield a resultant wave of intensity I at the viewing screen with Z 1 27rd 43 where qt 75m 9 I 410 cos Z o interference This is called diffractlon 39 39 width L1 produce ral maximum on a viewing screen and other maxima separated by minima located at angles 9 to the central axis that satisfy 123mini CircularAperture Diffraction Diffraction by a circular aperture or a lens with diameter d produces a central maximum and concentric maxima and minima with the first minimum at angle 9 given by sin 9 122 First minimumcircular aperture Rayleigh39s Crlterion Rayleigh s criterion suggests that two objects are on the verge of resolvability if the central diffraction m x39mum one is at first minimum ofthe ot er Their angular separation must then be at least ll 9 122 Diffraction Gratings A di raction grating is a series of quotslitsquot used to separate an incident wave into its component wavelengths b se arating and displaying their Y diffraction maxima Diffraction by N multiple slits results In maxima lines at angle 9 such th t dsin ml Rela g The Postulates Einstein s special theory of relativity is based on two postulates 1 39 for obseners in all inertial reference r 2 The speed oflight in vacuum hasthe same value c in all direction and in all inertial reference rames The speed of light c in vacuum is an ultimate speed that cannot be exceeded by any entity carrying ener y or in ormation Time Dilation Iftwo successive events occur atthe same place in an inertial reference the time intenal tD between them me sured on a single clock where he roper time betw ents Obseners 39 ame will measure a larger value for this interval For an obsener moving w t relative speed 17 the measure time intenal is AtO At 1 a 52 Length Contraction The length L0 of an object measured by an obsener in an inertial reference frame in which the object is at rest is called its proper length Obseners in frames moving r luuv w mm length 39 sho length For an obsener moving with relative speed 17 the measured length is L Lu 1 r 175Z Relativity ofVelocities When a particle is moving with speed u1 in the positive x1 direction in an inertial reference frame 511 that itself is moving with speed 17 parallel tothe x direction of the a second inertial frame 5 the speed u ofthe particle as measured inS 39s u 1u1v to s and Matter Waves Light QuantaPhotons An electromagnetic wave light is quantized and its quanta are called photons For a light wave of frequencyf and wavelength l the energy E momentum magnitude p of a p f and p E E c Photoelectric EffectWhen light of high enough frequency falls on a clean metal surface electrons are emitted from the surface by photonelectron interactions within the metal The governing relation is hf Kmax P in which hf is the photon energy Km is the kinetic energy ofthe most energetic emitted electron an I is the work function of the target material that is the minimum energy an electron must have if it is to emerge rom the surface ofthe target IF hf is less than 1D e ectrons are n emi ed Compton ShiftWhen x rays are scattered by loosely bound electrons in a target some the scattered x rays ave a longerwavelength than do the incident x rays This Compton shift in wavelength is given All 1 7 cos r In which 1 is the angle at which the x rays are scattered LightWaves and Photon When light interacts with matter energy and momentum are transferred via photons hen li t is in transit however we interpretthe light quot 39 39 ta p oton t e amplitude of the oscillating movin particle ora proton can be described as matter wave its wavelength called the de Broglie wavelength is given byl h p where p isthe magnitu e of the partic e s momentum Heisenberg s Uncertainty Principle The probabilistic nature of quantum physics 39 ortant limitation on de ecting a particle s position and momentum possible to me sure the position and momentum of particle simultaneousy with unlimited precision The uncertainties in the components of Y these quantities are given b Ax Apx 2 FL More About Matter Waves Conduction of Elect g olids An Electron in an in nite Potential Well An infinite potential well is a device for Metals Semiconductors and Insulators Three electrical properties that can be used h u confining an electron From t at the matter wave representing a trap ed e ect one dimensional infinite potential well the energies associated with the quantum states are e con inement principle we ex e c z n nr m 5 m n c 1 5 nr m m c E m n E n m m n nr n m lquot n nr 2 E1 L2 n m In which L isthe width of the wall and n is a quantum number Ifthe electron is to change from one state to another its energy must change bythe amount AE Ehigh Elaw39 If the change is done by photon absorption or emission the energy ofthe photon hf AE Ehigh Elaw39 The wave functions associated with the quantum states are 11x Asin X The probability density 1120 for an allowed state has the physical meaning that 111200 isthe probability that the electron will be detected in the intenal between x andx dx F r an electron in an in inite well the pro ability densities are 111200 AZ sinZ AZ can be found from 39 quot39 quot Z x Ti l39 1 dx 1 w 71 y Which asserts that the electron must be somewhere within the well because the probability 1 implies certainty Electron in a Finite Potential Well A finite potential well is one for which the potential energy ofan electron inside the well is less than that for one outside the well by a finite amount e wave function for an electron trapped in such a well extends in tothe walls of the well to distinguis re etals low p positive and low a large n p negative and high a sma n and Gaps in a Crystalline Solid An isolated atom can exist in only a ms come to et erto form a so 39 orm the discrete energy ban s ofthe solid These ds are separated by energy gaps each of which correspondsto a range of d number density of charge carriersn Solids can be broadly divided and ss Insulators In an insulator the highest band containing electrons is completely filled e vacant band a ove it by an ener gap so arge that electrons can essentially never become thermally agitated enough to jump across e gap Metals the highest band that contains any electrons is only partially filled The energy of the highest filled level at a temperature of0 K is called the Fermi energy ns in the partially filled band are the conduction electrons and their number of conduction electrons in sample number of atoms in sam le number 0 Valence electrons per atom The number densityn o the con uction electrons is n auction electron in sample sample Volume V 39 The number of atoms in a sample is given by sam le mass M number of atoms in sample at Electric Charge The coulomb and Ampere The SI unit of charge isthe coulomb C It is defined by the followmg 1C 1A1s Coulom b s Law Coulomb s Law describes the electrostatic force between small point electric L11 and qz at rest and separated by a distance r 1 2 A r 4n 0rz The two shell theorems for electrostatics are 39 e attracts or repels a charges particle that is outside the shell r e were concentrated at its center lfa char ed particle is located inside a shell of uniform charge there is no net electrostaticforce on the particle from the she o 39 r to Electric Fields f E r 7 go 4n 0rz The direction of E is away from the point charge if the charge is positive and toward it if the charge is negative 39c lJ39nes provide eansforvisualizing the direction and magnitude of ectric ields The electric field vector at any poin is tangent to a field line through that point The density ofthe field lines in any region is proportional to t e magnitude ofthe electric field in the region Field lines originate on positive charges d terminate on Continu us bution The electricfield due to a continuous is charge distribution is found by treating charge elements as point charges and then summing via in egration the electric field vectors produced by all the charge elements to find the net vector Gauss Law qenc 7 E M 50 Applications of Gauss Law Using Gauss39 law and in some cases symmetry arguments we can derive severa 39 39 39 39 39 39 L An exce s charge on an isolated conductor is located entirely on the outer surface of the conduc or 2 The Electric field at any point due to an infinite line ofcharge with uniform linear charge densityl is perpendicularto the line of charge and has ma nitude DE 2 E e e p 4 The electric field outside asphericalshellofcharge with radius R and total charge q is directed radially and had magnitu e The electric field inside a uniform sphere ofcharge is directed radially and has magnitu e 1 E 1 Conducting surface 2 E A line or charge 3 E Lsheet of charge en 271597 2 a 47193 r 7 Li 7 4 E 7 MHZ Spherical shell for r2 R 5 E 7 Electric Potential AVVf 17 7 Ll re 7 VfV7 Eds V Mrsnr 1 da inn7 Calculating E from V The component of E in any direction is the negative of the rate at which the potential changes with distance in that direction 6V EI 5 Electric Potential Energy of a System of Point Charges U W 1 9172 47159 7 Capacitance q CVC ZUSD ijCylindricalMLength and 2 radii n C 41150 14 4L 39 radiiC 4 Dquot 39 t A R CW Z L n capacitors in parallel i Xi n capacitors in series Cu Cr 2 u zc U CVZpotential energy sDEZenergy density 0 3 KE q Gauss Law w a dielectric Here q is the free charge any induced surface charge is accounted for by including the dielectric constant K inside the integral Current and Resistance 7 Hcurrent density ne drift speed V R i 7 i p i presitivityoconductivity L R p 7 p0 puma 7 T0Change ofp with Temp 2 P iV iZR Circ 39ts Loop Rule The algebraic sum of the changes in potential encountered in a complete traversal of any loop ofa circuit must be zero Junction Rule The sum of the currents entering anyjunction must be equal to the sum ofthe currents leaving that junction 7 s 7 msingle loop circuits Reg 7 Z R series M RC I is the capacitive time constant i sitRC charging a capacitor inuaaraiiei I C 1 7 sitRC charginga capacitor i d q q que MRC discharging a capacitor Magnetic Fields 51 n 7 Vl Where n is the number density ofthe charge carriersetheir number per unit volume Where I Ad isthe thickness of the strip em em 717 IqIVB7 7 r7 q 5 7 2m FE lLXB Torque on a CurrentCarrying Coil A coil of area A and N turn carwing current i in a uniform magnetic fieldE will experience a torque f given by Here f isthe magnetic dipole moment ofthe coil with magnitude 1 NiA and direction given bythe righthand rule glh The line integral i B uuin5olenoidri isthe number ofturns per unit Ien L 7 9 7lRC dz RC 9 discharging a capacitor Magne c 39elds Due to Currents Faraday s Law dig a quot152 i a ulll euui 39 L DE dl where the integral is overthe area 7 m 7 s B ong Straight Wireperpendicular distance R fromthe wire amf T F El E 39 dsr Faraday 5 Law Of INdUCtlonl B mrwlarmcx radius R and central angle Min radians If the loop is replaced by a closely packed coil 32A turns the induced emf is 7 7 5 Fb ibLBa sin90 anigll orce between Parallel currents attract in same direction I Smff a 39 Where d isthe wire separation and in an vb are the currents in the wires Lenz s LawAn induced currenthas a direction such that the magnetic field due to the 7 7 V I current opposes the change in the magnetic flux that in ces the current The Bl d5 7 0197 Ampere 5 Law induced emf has the same direction as the induced current n this equation is evaluated around a closed loop called an Amperran loop B Toroid Whereristhe distance L ofthe toroid to the point Nd L lquotdquot 13quot55 LC Circuits and Transformers 7 E 2 2 2 L 7 l Infuctance 011 Ainductance per unit length UECand Us Z U HE Us emf 7L lSelf7nductiondirectionfound using Lenz s Law dzq 1 Emf 3913 L 2 q 0 q Qcoswt vcurrent oscillations i 71 7 93927LSeries RLcircuils1LLR is called the inductive time constant d C 7 L 39 39 39 39 39 39 the system i log 272 current decay U5 LiZ Magnetic Energyu5 ZB energy density N N d d RLC and AC Circuits 2 LZTZ R 39 q 0Damped Oscillation q Qe RZZquot cosw t d3 Where ml in 39 39 with em 39 by i Isinwd 7 d3 Resonance When the driving angularfrequency wd equals the natural angular frequency w ofthe circuit that is at resonance Thean XL 1 0 and the current is in phase with the emf For a capacitorl IXC in which XE lwdC is the capa ve reactance the current here leads the potential difference by 90 790 7 7Zrad For an inductonVL IXL in which XL de is the inductive reactance the current here lags the potential difference by 90 90 grad Series RLC Circuits For a series RLC circuit with an alternating external emfand a resulting alternating current 5 X 7X tan v LR E 7 Rlterzgt quotd Power and Maxwell39s Eguations PM ImZR sm mmlm cos t The term cos iscalled the power factor id so displacement current g E d5 uuidyem uuim Maxwell s Equations 55 111 LIME 0 Gauss s law forelectricity g 111 0 Gauss law for magnetism ti d 7 5 a N dz 53 d uDSDTE u0iem AmpereMaxwell law RC Circuits q Cs 1 7 eTZRC charging a capacitor n which Cs qu isthe equilibrium final charge and RC I is the capacitive time constant ofthe circuit During the charging the current is 7 L 7 dz q que ZRC discharging a capacitor i L dz Faraday s law 1 eTZRC charging a capacitor 7 I7n 7zRC M e dIschargIng a capacItor Oscillations and Waves Simple Harmonic motionsxt xm coswt 13 In which xm is the amplitude ofthe displacement in and f is t e phase constant Th frequency ofthe motio f is the phase of the motion e angular frequency is related to the period and Sinusoidal Waves A sinusoidal wave moving t emathematicform m Where mt e amplitude ofthe wave angular frequency and kx 7 39 sin is is the angular wave number a is the wt Is the phase The wave ength l is related to k by llfwave speed 2111 in the positive direction of an x axis has wt 2739 m klv7 1 k Electromagnetic Waves E 5l sinkx 7 wt and B 7 8l sinkx 7 wt 7E 1 c 7 B mwhere Eand B are the slmultaneous magnltudes ofthe flelds t s gtlt Poyntingvector 1 z a P 17 Emsl7m Polar39zation iftheir elellttiit I single plane called the plane of oscillation lfthe original light is initially unpolarizedI lfthe original light is initially polarized I 10 cosZ 9 Re ection and Refrac Ion ITLZ smaz 1 sm 91 Totallnternal Re ection QC sin 1 1 Interference Diffraction n 7 6171 asin9 ml folrm 123 minima singleslit n a sine 122 First minimumcircular aperture Voung s 39 V 39 single slit falls on two slits in a screen The light leaving these slits flares out by 39 raction and interference o rs in the region eyon the screen A fringe pattern due to the interference forms on a viewing screen sin 9 ml for m012 maximabright fringes dsin m Z lform 012 I 410 cosZ i where 1 gsin 9lntensity in Twoslit interference da rk fringes 9R 122Rayleigh s criterion Rel The Postulates Einstein s special theory of relativity is based on two postulates 1 The laws o physics are the sa or obseners in all inertial reference rames No one rame is referred over any ot er 2 The speed oflight in vacuum hasthe same value c in all direction and in all inertial reference frames Am TIme DIlatIon At 17ltEgt2 Length Contraction L LD 1 7 175Z Relativity ofVelocities When a particle is moving with speed u1 in the positive x1 direction in an inertial reference frame 511 that itself is moving with speed 17 parallel tothe x direction of the a second inertial frame 5 the speed u ofthe particle as measured inS 39s Photons and Matter Waves Light QuantaPhotons hf and p g A Photoelectric Effect hf Km D This Compton shift in wavelength is given by All i 1 7 605 qt is the angle at which the x rays are scattered e Broglie wavelength is given byl hp Heisenberg s Uncertainty Principle Ax Apx 2 ll h In which d W39 More About Matter Waves An Electron in an in nite Potential Well El nZ In which L is the width of the wall and n is a quantum number The wave functions associated with the quantum states areilJl x A sin 11Zx AZ sinZ Xprobability density 1an xdx 1Normalization Conduct of ct Solids Metals the highest band that contains any electrons is only partially filled The energy of the highest filled level at a temperature of0 is called the Fermi energy The electrons in the partially filled band are the conduction electrons and their n u m er Is number of conduction electrons in sample number of atoms in sam e number 0 Valence electrons per atom The number densityn oft e condlliction electrons is 7 duczm electron 171 may n sample Volume V 39 The number of atoms in a sample is given by M sample mass mm number of atoms in sample atomic mass sample mass Mmm 7 molar mass MNA material 5 densitysample Volume V 7 molar mass MNA The number density n of the conduction electrons is n 7 numb f c duczm electron 111 may ampl olume V The Fermi energy for a metal can be found bythe following equation 23 h2 u 1212 23 23 E 1N7 m n n The number of electrons in the conductio m n band ofsilicon can be increases greatly I p lIUlUS us 39 g t 39 39 enumber of holes in the valence band can be greatly increases by doping with aluminum thus forming ptype ma erial e pn Jun A pn 39unction is a single semiconducting crystal with one end doped to form ptype material and the other end doped to form ntype material the two types meeting at a junction plane Physics 2 Exam 1 Crib Sheet Electric Charge Conductors are materials in which a significant number ofcharged particles electrons in metals are free to move The charged particles in nonconductors or insulators are not free to move The Coulomb and Ampere The SI unit ofcharge is the coulomb C It is defined in terms of the unit of current the ampere A as the charge passing a particular point in 1 second when there is a current atthat point 1C 1A 15 This is based on the relation between current i and the rate Z jat which charge passes a point i 4 electric current Coulomb s Law Coulomb s Law describes the elech39ostatic force between small point electric L11 and qz at rest and separated by a distance r a thz f 41r 0rz The two shell theorems for electrostatics are A shell of uniform charge attracts or repels a charges particle that is outside the shell as ifall the shell s charge were concentrated at its cen er lfa charged particle is located inside a shell ofuniform charge there is no net electrostatic force on the particle from the shell The Elementary Charge Electric charge is quantized any charge can be written as ne where n is a positive or negative integer and e is a constant of nature called the elementary charge 1602 X10719 C Electric charge is conserved the net charge of any isolated system cannot change Elech39ic Fields Electric Field TL quot39 quot IquotEatanyr 39 39 39 in terms of 39 39t quot force 1 that would be expected on a positive test charge qu placed there E 10 Electric Field lJ39nes provide a meansforvisualizing the direction and magnitude of electric fields The electric field vector at any point is tangent to a field line through that point The density ofthe field lines in any region is proportional to the magnitude ofthe electric field in the region Field lines originate on positive charges and terminate on negative charges Field Due to a Point Charge The magnitude of the electric field if set up by a point charge q at a distance rfrom the charge is39 E f The direction of E is away from the point charge if the charge is positive and toward it if the charge is negative Field Due to an Electric Dipole An electric dipole consists of two particles with charges of equal magnitude C but opposite sign separated by a small distance d Their electric dipole moment has a magnitude gd and points from the negative charge tothe positive charge The magnitude of the electric field set up by the dipole at a distant point on the dipole axis which runs through both charges is 1 p L 41r 0rz Where 2 is the distance between the point and the centerof the dipole Field Due to a Continuous Charge Distribution The electricfield due to a continuous charge distribution is found by treating charge elements as point charges and then ming via integration the electric field vectors produced by all the charge elements to find the net vector Dipole in and Elech39ic Field When an electric dipole moment 13 is placed in an electric field E the field exerts a torque f on the dipole f 13 x Equot The dipole has a potential energy U associated with its orientation in the field P This quot 39 is defined quot quot LU Eand greatest UpE when 13 is directed oppositeE Gauss Law Gauss Law DE 7 qenc 50 In which q is the net charge inside an imaginary closed surface a Gaussian surface and PE is the netflux of the electric field through the surface cpE E M Applications of Gauss Law Using Gauss law and in some cases symmetry arguments we can derive several important results in electric situations Among these are 1 An excess charge on an isolated conductor is located entirely on the outersurface of the conductor 2 The external electricfield nearthe surface ofa charged conductor is perpendicular to the surface and has magnitude E conducting surface 5n 3 The Electric field at any point due to an infinite line ofcharge with uniform linear charge density Ii is perpendicularto the line ofcharge and has magnitude E ersnr Where r is the perpendicular distance from the line ofcharge to the point 4 The electric field due to an infinite shell ofcharge with uniform surface charge density Jis perpendiculartothe plane ofthe sheet and has magnitude E sheet ofcharge n 5 The electric field outside a sphericalshellofcharge with radius R and total charge q is directed radially and had magni u e 1 I2 Spherlcal shell forrZ R line or charge 111 H 4 6 The electric field inside a uniform sphere ofcharge is directed radially and has magnitude Ll E 41T 0R3 r Electric Potential Electric Potential EnergyThe change AU in the electric potential energy U of a point charge as the charge moves from an initial pointito final pointfin an electric field is 1 UL 7 Where W is the work done by the electrostatic force due to the external electric fieldO on the point charge during the move from itof Electric Potential Difference and Elech39ic Potential We define the potential difference AV between two points i andfin an electric field as AV Vf 7 V 73 Potential and potential difference can also be written in terms ofthe electric potential energy U of a particle of charge q in an electric field H V Ll Finding Vfrom EThe electric potential difference between two points i andfis M a E E 1 Potential Due to Point Charges The electric potential due to a single charge at a distance rfrom that point charge is 1 Where V hasthe same sign has q Potential Due to an Electric Dipole At a distance rfrom an electric dipole moment magnitude p qd the electric potential ofthe dipole is p c059 V 41150 Potential Due to a Continuous Charge Dishibu on Fora continuous distribution of charge 1 d V Elf In which the integral is taken overthe entire distribution Calculating ifrom VThe component ofi in any direction is the negative ofthe rate at which the potential changes with distance in that direction E 7 6V Physics 2 Exam 1 Crib Sheet 9172 Uw Ca nce Capacitor Capacitance A capacitor consists of two isolated conductors the plates with charge q and q Its capacitance C is defined from Ll I Where V is the potential difference be ween the plates 39 39 g 39 ene detenniue 39 a particular capacitor configuration by 1 assuming a charge qto have been placed on the plates 2 finding the electric field E due to this charge 3 evaluating the potential difference V and 4 calculating Cfrom q CV Some specific results are the followin A parallelplate capacitor with flat parallel plates of area A and spacing d has capacitance A d A cylindricalcapacitor two long coaxial cylinders of length L and radii a and b has capaci ance L C 7 21150 A spherical capacitor with concentric spherical plates of radii a and b has capacitance a bra39 C 41150 An isolatedsphere of radius R has capacitance C 41TSDR Capacitors in Parallel and in Series The equivalent capacitance C29 of combinations of individual capacitors connected in parallel and in series can be found from eq Z Ln capacitors in parallel 1 n capacitors in series c Potential Energy and Energy Denslty The electi39ic potential energy U of a charged cap citor 2 U Z C cvz Is equal tothe work required to charge the capacitor This energy can be associated with the capacitor39s electric field E By extension we can associate stored energy with any electric field In vacuum t e energy densi u or potential energy per unit volume within an electric field of magnitude E is given by ZSDEZ Gauss Law with a Dielectric When a dielectric is present Gauss39 law may be generalized to Here q is the free charge any induced surface charge is accounted for by including the dielectric constant K inside the integral Current and Resistance Current An electi39ic current i in a conductor is defined y do i Here dq is the amount of positive charge that passes in time dtthrough a hypothetical surface that cuts across the conductor Current Density Current a scalar is related to current density a vector by l Where H is a vector perpendicularto a surface element of area dA and the integral istake u era 39 Qquot quot 39 39 the velocity of they are negati a Drift Speed ofthe Charge Carriers When an electric field E is established in a conductor the charge carriers assumed positive acquire a drift speed 17d in the direction of E the velocity 17d is related tothe current density by 1 y I as he moving charges ifthey are positive and the opposite direction if ve quot9178 Resistance ofa Conductor The resistance R ofa conductor is defined as R 1 Where Vis the potential difference acrossthe conductor and i is the current Similar equations define the resi typ and conductivity I or a material E 7 7 Where E isthe magnitude ofthe applied electric field The resistance R of a conducting wire of length L and uniform cross section is L R p A Change ofpwith Temperature The resistivityp for most materials changes with temperature I coefficient of resistivity forthe materia Ohm39s Law A given device obe s 0hm s Law if its resistance R defined by Vi is independent ofthe applied potential difference V Resistivityo 39 m eroffree electrons per unitvolume r is the mean time between the collisions of an electron with the atoms ofthe atom m P Pedal To Here TD 39s a reference temperature puis the resistivity at Tuand aisthe temperature I P 2 39 e 717 Power The power P or rate ofenergytransfer in an electrical device across which a potential difference V is maintained is 2 P iV iZR V R Circuits Emf An emf device due 39 quot g 39 39 39 39439 between its output terminals If dW is the workthe device does to force positive dq from the negative tothe positive terminal then the emfwork per unit charge ofthe device is dw g dq Loop Rule The algebraic sum of the changes in potential encountered in a complete traversal of any loop ofa circuit must be zero tion Rule The sum ofthe currents entering anyjunction must be equal to the sum urrents leaving thatjunction SingleLoop Circuits The current in a singleloop circuit containing a single resistance R and an emf device with emf s and internal resistance r is s l n7 Series Resistances When resistances are in series they have the same current The 39 39 LhaLcan 39 39 quot39 39 re istance is Ree Z R Parallel Resistances When resistances are in parallel they have the same potential difference The equivalent resistance that can replace a parallel combination of resistance is given by 1 1 2 J RC Circuits When an emf s is applied to a resistance R and capacitance C in seriesthe charge on the capacitor increases accordingto q Cs 1 7 sitRC charging a capacitor In which Cs qu is the equilibrium final charge an C L39 is the capacitive time constant ofthe circuit During the charging the current is L i ftRC charging a capacitor 1 dz When a capacitordischarges through a resistance R the charge on the capacitor decays accordingto q que ZRC discharging a capacitor Duringthe discharging the current is id 7 erzRc Constants and Extra Information so 885 x 10 12 Fm 89875518 x 10 NmZC Z 41150 e 16 x 10 c 91 X 10 31 kg 39 materia mama efew conductors when they are doped with other material all e become atoms that contributed free electrons e 39 resistance at IUW Recent research quot 39 high temperatures Physics 2 Exam 1 Crib Sheet Physics 2 Exam 2 Crib Sheet Ma n 5 Magnetic eld A magne cfield E is defIned In terms of the force acting on a test particle with charge q movingthrough the field with velocity 1 E 111 X The SI unit for E is the tesla T 1 T 1 N Am 104 gauss Crossed Fields Both an electricfield E and a magnetic field B can produce a force on a charged particle When the two fields are perpendicularto each other they are said to be crossed fields The Hall EffectWhen a conducting strip carwing a current i is placed in a uniform magnetic field B some charge carriers with charge e buildup on one side ofthe doctor creating a potential difference Vacross the strip The polarities ofthe sides indicate the sig of the charge carriers 11 VE ZL39 Where n isthe number density of the charge carriers their number per unit volume Where i Ad is the thickness of the stri A Charged Particle Circu ating in a Magnetic Field A charged particle with mass m And charge magnitudequ moving with velocity 1 perpendicularto a uniform magnetic field B will travel in a circle Applying Newton s second law to the circular motion yields mvz IqlvB From which we find the radius r of the circle to be my r l lE The frequency of revolution f the angularfrequency w and the period ofthe motion T are given 7 1 7 1 7 7 z 7 T 7 2m39 Magnetic Force on a CurrentCarryingWIre A straight wire carwing a current i in a uniform magnetic field experience a sideways force FE 1392 x E idZin a 39 39 is M 139de E The direction of the length vector Z or dz is that ofthe current 139 Torque on a CurrentCarrying Coil A coil of area A and N turn carwing current i in a uniform magnetic fieldE will experience a torque f given by x E The s B I Here f is the magnetic dipole moment of the coil with magnitude u NiA and direction given by the righthand rule Magnet elds Due to Currents The BiotSawrt Law The magnetic eld set up by a currentrcarwing conductor can be found from the BiotrSoVort low This law asserts that he contribution dB to the field produced by a currentrlength element 139 d at a point P located a distance rfrom the current element is 7 WE X f 13 2 A 411 r Here r Is a unIt vector that points from the element toward o o 139 o i he BiotrSavart law gives forthe magnitude ofthe magnetic field at a perpendiculardistance rom the wire 3 2 1339 Magnetic Field of a Circular Arc The magnitudelrofthe magnetic field at the center of a circular arc of radius and central angle 1 in radians carning current i is uid 411R DaralleIWIres ran attract each other 39 39 39 W 39 repel each other The magnitude ofthe force on a length L ofeitherwire is F ibLBa sin 90 ZL 39 Where d isthe wire separation and in and ib are the currents in the wires Ampere s Law Ampere s law states that B39d oim The lIne Integral In thIs equation Is evaluated around a closed loop called an Ampenon loop The current i ont e right side isthe net current encircled bythe loop Field ofa Solenoid and a Toroid Inside a long solenoid carningcurrent i at points not near its ends the magnitude B ofthe magnetic field is Hum Where TL isthe numberofturns per unit length At a point inside a toroid the magnitude Bofthe magnetic field is M1 27 7 Where r isthe distance fromthe centerofthe toroid to the point Faraday s Law Magnetic Flux The mognetic uxllE through an area A in a magnetic fieldB is defined as an I B d Where the integral is taken overthe area The SI unit of magnetic flux is the weber where 1Wb 1 Tmz Faraday s Law of Induction If the magnetic flux DE through an area bounded by a closed conducting loop changes with time a current and an emf are produced in the loop this process is called induction The induced emf 39s d g emf dz lfthe loop is replaced by a closely packed coil of N turns the induced emf is d SW 7 T39 Lenz s LawAn induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current The induced emf has the same direction as the induced current Emf and the Induced Elech39ic Field An emf is induc even if the loop through which the flux is changing is not a physicalconducto b imaginary line The changing magnetic field induces an electric field E by ed by a changing magnetic flux r ut an emf E d Where the integration is taken around the loop From this equation we can write Faraday s law in its most general form d g T39 7 lnduct Inductors An inductor is a device that can be used to produce a known magnetic field in a specified region Ifa currenti is established through each ofthe Nwindings ofan inductor a magnetic flux DB linksthose windingsThe inductanceL ofthe inductor is 7 5 I The SI unit of inductance isthe henryH where 1 henw 1 T mZA The inductance per unit length nearthe middle ofa long solenoid ofcrossrsectional areaA and TLturns er units length as 7 z a 071 SelfInduction Ifa currenti in a coil chahgeswith time an emf is induced in the coil This selfrinduced emf is 7 L d1 5mm g The direction of emf is found from Lenz s law the selfrinduced emf acts to oppose the change that produces 39t Series RL Circuits If a constant emf is introduced into a singlerloop circuit containing a resistance R and an inductance L the current risesto an equilibrium value of emfR accordingto i 7 in e eat7L Here 1LLR governs the rate of rise ofthe current and is called the inductive time constant ofthe circuit When the source of constant emf is re moved the current decays from a value to according to l e L Magnetic Energy If an inductor L carries a current i the inductor s magnetic field stores an e nergy give n by US lLiZ IfB isthe magnitude ofa magnetic field at any pointin an inductor oranywhere elsethe density o Physics 2 Exam 2 Crib Sheet LC Circuits and Transformers LC EnergyTransfers In an oscillating LC circuit energy is shuttled periodically between the electric field of the capacitorand the magnetic field ofthe inductor instantaneous values of the two orms of energy HE Cand Us Where qis the instantaneous charge on the capacitorand i is the instantaneous current through the inductor The total energy U U5 U5 remains constant LC Charge and Current Oscillations The principle of consenation of energy leadsto 2 LZTZ g 0 As the differential equation of LC oscillations with no resistanceThe solution to the equation 395 L12 q Qcoswt In which Q isthe charge amplitude maximum charge on the capacitor and the angularfrequency w ofthe oscillations is w J The phase constant qt is determined by the initial conditions t0 of the system Transformers A transformer assumed to be ideal is an iron core on which are wound a primary coil of N17 turns and a secondaw coil of NI turns If the primaw coil is connected across an alternatingcurrent generator the primaw and secondaw voltages are related by N K Vim The currents through the coils are related by I I TVS and the equivalent resistance of the secondary circuit as seen by the generator is N lt gt R where R is the resistive load in the secondaw circuit The ratioNFNJ is called the tra nsformer39s turns ratio RLC and AC Circuits I i u n when a dissipative r n 3 element R is also present in the circuit Then Liz 397 Rd q 1 a 0 dzz dz cq 7 39 The solution ofthis differential equation is Qe RZZL coswlt 43 where We consider only situations with small R and thus small damping then m m w Alternating Currents 39 39 e i RIC circuit may quot 39 39 39 J39 39 a a jizquany wd byan 39 39 39 emf mmmsinwdt The current driven in the circuit is i Isinwd Where qt is the phase constant ofthe current Resonance The current amplitude I in a series RLC circuit driven by a sinusoidal external emf is a maximum I Milly when the driving angularfrequency wd equals the uequencyw of quot Thean XL 1 0 and the current is in phase with the emf Sin le 39 39 Tquot 39 39 39 39439 resistor has amplitudeVR IR the current is in phase with the potential difference For a capacitorl IXC in which XE 1wdC is the capacitive reactance the current here leads the potential difference by 90 790 izrad f 739 mm For an inductorVL IXL in which XL de is the inductive reactance the current here lags the potential difference by 90 90 grad Series RLC Circuits For a series RLC circuit with an alternating external emfand a resulting alternating current Emma xRZ XL X02 and tanq Power and Maxwell39s Eguations Power In a series RLC circuit the average power PM of the generator is equal to the production rate of thermal energy in the resistor Pm 7m R Smfrm Im 50511 Here nu Lanus for q he 39 39 related tothe maximum quantities byIrm Ixi Vrm Vxi and sm rm smfm7 The term cos p is called the power factor of the circuit Displacement Current We define the fictitious displacement current due to a changing electricfield as an id so 7 Maxwell s extension ofAmpere39s Law then becomes 3 Maia uuim where idle is the displacement current encircled by the integration loop The idea of a displacement current allows us to retain the notion of continuity of currentthrough a capacitor However displacement current is not a transferof charge Maxwell39s Equatio s Gauss39s law for electricity E 39dA LiensSoy Relates net electric flux to net enclosed electric charge au 5 aw for magnetism E Lil o Relates net magnetic flux to net enclosed magnetic charge Faraday s law 515 1 1 Relates induced electric field to changing magnetic flux AmpereMaxwell law gag 39 d5 M050 Muienm Relates induced magnetic field to changing electric flux and to current Extra Information and Constants so 885 x 1012 Fm no 1257 x 10 6 Hm e 16 x 10 C m 167 x 10 Z7kg me 91 x 10 31 kg RC Circuits When an emfs is ap lied to a resistance R and capacitance C in series the charge on the capacitor increases accor i g to q C 1 RC charging a capacitor In which Cs qu isthe equilibrium final charge and RC I is the capacitive time constant ofthe circuit During the charging the current is dq l 3 ftRC charging a capacitor When a capacitor discharges through a resistance R the charge on the capacitor decays according to q que ZRC discharging a capacitor During the discharging the current is d LT 7 errRC Physics 2 Exam 2 Crib Sheet
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