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# Phys 2213 Prelim 1 Studyguide Phys 2213

Cornell

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This 6 page Study Guide was uploaded by FM on Tuesday March 8, 2016. The Study Guide belongs to Phys 2213 at Cornell University taught by in Spring 2016. Since its upload, it has received 57 views. For similar materials see Physics II: Electromagnetism in Physics 2 at Cornell University.

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Date Created: 03/08/16

Cornell Phys 2213 Topic Syllabus 1 Static Electricity as in lab a Charging of objects polarization motion of charges from one object to another etc 2 Coulomb39s Law a Electrostatic force and applications trajectory of charged particles i F2 kc g Q27922 rlt 1 kc g Q2 rz39r1r239r13 b Electric eld and relation to electric force Fqq Er c Superposition and calculation of electric elds Via sums and integrals e g rings lines sheets discs spheres etc may be nite or in nite i Eltrgtkc 2 q rrngtrr3 ii ErkCl rrnrrn3 dq dqt dx 0 dA p dV iii Point Er kc qrZ rA iV Line Er 2 kCld rA V Sheet Er 27 kCa rA d Superposition to get elds from composite objects e g cylinder with a hole drilled out halfpositivehalfnegative line of charge etc Et0tr E ArEBr 3 Gauss39s Law a Fluid analogy and mapping between uid ows and electric elds 139 qn lt gt In ii Er lt gt Vr iii 80 lt gt p0 b FluX integrals uid and electric computing them with integrals solVing ux problems without complicated integrals using ux principle 115101 i Current I mass or charge per time ii J p0 V gives direction of ow and current density current per area iii patch J 39 dA iV I 9cl3J39dA tot c Application of Gauss39s Law to compute electric elds in high symmetry situations in nite plates cylinders spheres could be solid shells or thin wires i QemlsflgtgOE 39 dA ii 395 E dA Q d Differential form of Gauss39s Law and computing densities from elds V 39E pea 4 Field Lines FLs a Basic FL rules 80 encl b Knowing the difference between eld lines and particle trajectories c FL sketches for given charge distributions xed distributions not those near conductors 5 Conductors a Basic familiarity with polarization phenomena of conductors in electric elds b Knowledge of the basic fact that EO inside a conductor c Field Lines FLs and Conductors d Field line sketches and elds near conductors conductors may be neutral or charged external charges may be nearby or embedded in internal cavities e Computing elds near conductors e g coaxial cables concentric spheres plates 6 Voltage a De nition Vba nga J Er 39 dr and path independence of V nga Vy V c Vaa O 95 Er dr path integral around a closed loop always gives zero same thing as Kirchhoff s voltage law 7 Electric potential a VX E VXHef Jbref Er 39 dr typically refoo b V VbVa c Conservation of energy KE q VX const d ErVVr 8 Current and ow of charge a I dQdt J39 dA Jn q v 9 Resistors a vdqb E Jo E onq2b VE L b VIR RLo ApLA p1o c Power dissipated into heatlight etc PIV12RV2R d Ability to calculate resistance in nonstandard geometries like spherical resistors blt a b Path composition of voltages V blt a blt z 39 blt a 10 Capacitors a E 680 oQA VE d CVQ CeOAd b Enhancement with dielectric material GM XM E C eAd KeoAd c Energy storage UQV2CV22Q22C d Ability to calculate resistance in nonstandard geometries like spherical resistors 11 Batteries a Ideal battery i Voltage across terminals V8 ii Power put into circuit P8 I b Realistic battery includes ideal battery and an internal resistance in series 12 Circuit Theory a Kirchhoff s Laws 1 Indicate presumed directions of current ow in each branch use current law 2 1111 2 Iout to introduce minimal needed number of unknown current values ii Write one loop law 0 Vaga ngachbVd Vagd current doing all fundamental inner loops b ResistorsV c Batteries i Ideal battery nga terminal if going from longer line to for each unknown bgaIR if upstream if downstream 8 if going from shorter line terminal to shorter line terminal ii Realistic batteries also include an internal resistance which you treat like any resistor d Capacitors i VQC if going from presumed terminal to presumed terminal if going from presumed terminal to presumed terminal ii Also need to include one equation IdQdt for each capacitor Sign must be consistent with presumed direction of current ow and presumed terminals e Reduction by equivalent circuit elements seriesparallel combinations i Meaning of series combination share same I V s add ii meaning of parallel combination share same V I s add iii Resistors 1 Series R equiv 2 Parallel Requiv391R1391R2391 basically makes a wider resistor with greater crosssectional area R1R2 basically makes a longer resistor iv Capacitors 1 Ser1esCequiv separation 2 Parallel CequiVC1C2 basically makes a capacitor with wider plates with more area 391C1391C2391 basically makes a capactor with greater plate f RC circuits i Exponential approach to steadystate with time constant tRC ii At start before much charge ows onoff capacitors act like ideal batteries of VQC iii At end steadystate capacitors act like open broken circuits with no current passing through 13 Energy storage in electric field a U802 ll E2 dV b Matches exactly UCV22 for a capacitor 14 Magnetic forces on moving charges a Origin ultimately from effects of relativity b Parallel wires with currents in same direction attract parallel wires with currents in opposite directions repel c Lorentz force law Fq E v gtlt B and applications to circular orbits mass spectrometers Hall effect etc d 21 qivi gtlt gt 1 1 dB gtlt dB in presumed direction of current ow e F l 1 dB gtlt B and applications to forces on wires wire loops etc 15 Computation of magnetic fields a Field above infinite wire carrying current Ito the left has magnitude B pol2751quot with direction into the page where r is the distance from the wire b BiotSavart law and applications to magnetic fields from wires and loops and sheets variants below based on the same idea from computing magnetic forces that 2 qivi gtlt 11th JJdvx i Br 21 qui X rl il l i3 ii Br 21 qui X r7r2 iii Br I 1 dt gtlt rfrz iv Br I J dV gtlt rfrz 16 Gauss s law for magnetic eld a V 39 Br O b V 39 Br 0 implies equivalent uid model for magnetic eld of ow of a uid stirred up by corresponding set of rotating wires spinning in direction given by right hand rule 2 e g thumb and curling ngers version c Ability to sketch magnetic eld lines for various arrangements of wirescurrents 17 Ampere s Law a V gtlt Br uOJ uOSO 6E6t Equivalently d5 Br 39 d uolemu080dCDEdt where DEE If dE 39 dA where the area element dA is de ned with a direction consistent with RHR2 right hand rule 2 according to the direction of the line integral for d5 Br 39 dB b Choice of Amperian loop i Passes along a eld line through point of interest ii Always parallel or perpendicular to the magnetic eld lines often you can just use the magnetic eld line going through the point of interest iii Encloses lassos some amount of current c Ability to calculate magnetic elds using Ampere s law for various high symmetry cases such as wires in nitely thin or of nite thickness current sheets in nitely thin or of nite thickness solenoids straight or torroidal thin or nite walls etc 18 Faraday s Law a Motional EMF i Electric eld inside an ideal conductor loop moving through a magnetic eld is no longer zero but EvgtltB ii Motional EMF is 8 5 E 39 d8 S15 V X B 39 d 15 B 39vxd dCIBdt iii Power out from motional EMF equals power needed to be put in to keep loop moving iv Be able to compute EMF for all sorts of moving loops translating rectangles spinning loops etc b Induced EMF i By relativity doesn t matter if conductor loop is moving through a magnetic eld or if the magnetic eld is moved past the loop or if the magnetic eld is changing for any other reason there is still an induced EMF giving by Faraday s Law ii Faraday s Law V gtltEr8B8t Equivalently d5 E 39d dCIBdt with DB If B 39 dA with direction of dA determined by RHR2 relative to direction of loop used for d5 E 39d iii Lenz s Law No free lunch law Induced EMF generates an effect to oppose the change For motional EMF induced current creates a force which opposes the motion and also induced current generates a uX through the loop in the direction countering the change For induction induced EMF due to a changing magnetic eld generates a current ow in a direction generating a ux opposing the external change in ux If no wires happen to be in the given region E still points in the direction that would drive such currents if a wire happened to be there 19 Induction circuits a Mutualinduction M is the induced EMF from the magnetic eld due to changing currents in a different loop or solenoid etc Selfinduction L is the induced EMF from changing currents in the same loop Mutual inductance 1331 E MIL2 dlzdt 1332 E M291 dlldt Always will nd MlezzMzei Self inductance DB E LdIdt Circuit analysis with inductors i Kirchhoff s junctioncurrent law Kl stays the same ii Kirchhoff s loopvoltage law K2 gets generalized to a statement about effective voltages iii K2 02100p V where all voltages for previous circuit element stay the same and the effects voltage across an inductor is L dIdt with the Sign convention that the upstream side according to the presumed direction of the current I of the inductor is at higher potential and the downstream side is at lower potential iv Immediately after any change in the circuit ipping a switch etc the current through any inductor is exactly the same immediately after the change as before the change Currents in other parts of the circuit may change instantly but not the currents through the inductors v After things settle for a long time into a steady state an inductor acts just like a wire in any circuit vi LR circuits The change between the state immediately after any change iv and the steady state v is generally exponential If there are just inductors resistors and EMFs in the circuit LR circuit generally the exponential time constant obeys tLeffReff vii Energy stored in an inductor Umdl2L12 viii LC circuits Show oscillations with angular frequency 0lsqrtLC 20 Energy storage and flow in the electromagnetic field a b c Storage UE l uE dV uE802E2 UB uB dV uBl2u0 B2 Energy inout of the elds from moving charges PQW m J 39 E dV Conservation of energy ddtUt tPQ 0P OW t P OW tJT S 39 dA where SEgtltBu0 is the Poynting vector Poynting vector SEgtltBu0 is for energy what J is for charge S gives the ow of energy Senergytimeareapowerarea dir of S gives direction of energy ow Note From this we see energy ow is always perpendicular to both E and B Be able to use above to evaluate energy stored in different regions of space under different circumstances and to compute the ow of energy from one part of space into another Planar electromagnetic pulses light i E L dir prop B L dir prop E L B E X B along dir prop exactly like expected for the associated ow of energy ii B1vE iii V lsqrt80u0 30 gtlt 108 ms iv Be able to use the above to gure out directions of various vectors E B dir prop based on information about the others be able to gure out E or B in a given electromagnetic pulse given B or E

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