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# Physics II Elec & Magnetism CPHY 122

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This 15 page Class Notes was uploaded by Alexandria Gutkowski on Monday October 5, 2015. The Class Notes belongs to CPHY 122 at Clark Atlanta University taught by Staff in Fall. Since its upload, it has received 30 views. For similar materials see /class/219535/cphy-122-clark-atlanta-university in Physics 2 at Clark Atlanta University.

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Date Created: 10/05/15

CPHY 122 Class Notes 13 Instructor H L Neal 1 Faraday s Law Any change in the magnetic environment of a coil of wire will cause a voltage emf to be induced in the coil No matter how the change is produced the voltage will be generated The change could be produced by changing the magnetic eld strength moving a magnet toward or away from the coil moving the coil into or out of the magnetic eld rotating the coil relative to the magnet etc ds dA Faraday s law is a fundamental relationship which comes from one of Maxwell s four equa tions fE ds dltIgt 39 if B 0 dt where 3 fig01A S d 77 dt 3 The quantity 5 is called the electromotive force emf Faraday s law is a succinct summary of the ways a voltage or emf may be generated by a changing magnetic environment The induced emf is equal to the negative of the rate of change of magnetic ux It involves the interaction of charge with magnetic eld 2 Lenz s Law When an emf is generated by a change in magnetic ux according to Faraday s Law the polarity of the induced emf is such that it produces a current whose magnetic eld opposes the change which produces it The induced magnetic eld inside any loop of wire always acts to keep the magnetic ux in the loop constant In the examples below if the B eld is increasing the induced eld acts in opposition to it If it is decreasing the induced eld acts in the direction of the applied eld to try to keep it constant 3 Eddy Currents Eu lllr nml h 53 as 393 0 saa 6 R 54 An eddy current is caused by a moving magnetic eld intersecting a conductor or vice versa The relative motion causes a circulating ow of electrons or current within the conductor These circulating eddies of current create electromagnets with magnetic elds that oppose the change in the external magnetic eld see Lenz s law above The stronger the magnetic eld or greater the electrical conductivity of the conductor the greater the currents developed and the greater the opposing force 4 Maxwell s Equations I began this course with Maxwell s Equations fE dA QEO Gauss law for E 1 f B ds 01 Ampere s Law 2 fB dA 0 Gauss law for B 3 fE ds 7 Faraday s Law 4 CPHY 122 Class Notes 17 Instructor H L Neal 1 Impedance in AC Circuits The impedance is de ned as 2 Power in AC Circuits The instantaneous power is Where the period of one cycle is T 21 w We may Write 24 paw t Vmaxlmaxi sin wt sin wt 7 qb dt 27139 0 24 melmaX 21 sin wt sin wt cos ab 7 cos wt sin ab dt 77 0 We have 1quot 1 0 sin2wtdt a 24 i w 39 wtcoswtdt 0 2W 0 sm so that paw t max me COS 5 lt15 tan 1 wL 7 3071 21 Series RLC Circuit From the previous Class Notes R cosqb 2 R2 wL 7 marl 71 Sing 7 R2 wL 7 marl max max 7 Z where 2 Z 7 R2 wL 7 marl The average power is PM 7 Imaxvmcosaa 7 Imxfzcosaa 7 INVE 22 Parallel RLC Circuit From the Quiz 8 solution tanq 71120 R7 wL 1 1 2 1m 7 14m2 7 77 C lt gt lt gtR2M w 1 R a5 2 2 1R Hi we 7 Z 7 R7 where 12 1 1 2 7 Z 7 7 7 C R2 wL w gt The average power is pant mamexCOSa5 V 2 1 Vmax2 7 R 3 Resonance in AC Circuits The average power is 1 0 7 V4010 VmwlmaX sin wt sin wt 7 15 CPHY 122 Class Notes 15 Instructor H L Neal 1 Energy of Electric and Magnetic Fields In this section we calculate the energy stored by a capacitor and an inductor It is most pro table to think of the energy in these cases as being stored in the electric and magnetic elds produced respectively in the capacitor and the inductor From these calculations we compute the energy per unit volume in electric and magnetic elds These results turn out to be valid for any electric and magnetic elds 7 not just those inside parallel plate capacitors and inductors Let us rst consider a capacitor Recall that the energy stored is 12 UE 7 E Assuming that we have a parallel plate capacitor let s insert the formula for the capacitance of such a device 6014 7 Let us further recall that the electric eld in a parallel plate capacitor is E 760 qEOA C so that q EOEA and 7 E60142 7 60E2Ad 260Ad 2 The combination Ad is just the volume between the capacitor plates The energy density in the capacitor is therefore i UE i 60E2 7 E 7 2 This formula for the energy density in the electric eld is speci c to a parallel plate capacitor However it turns out to be valid for any electric eld UE A similar analysis of a current increasing from zero in an inductor yields the energy density in a magnetic eld The work done by the generator in time dt is dW qu SIdt so that the power is This implies that W 7 71L 2 But AU 7W 1 7L 2 Or 1 U ELI2 constant The constant term is usually ignored Now recall that for a solenoid N B WY N2 L 0714 Putting this int0 to the equation for U gives 1 N2 z 2 U3 5 OT4 MB 2 Since AZ is the volume of the solenoid the magnetic energy density is 71UB 112 B m 2 LC Circuits switch We know that Applying Kirchoff s rules after closing the switch give 5L V or since d 1 I i dt we have 2 dt 039 Taking the derivative of both sides with respect to t gives d2 7 1 dt2 C Rearranging gives d2 1 I i 0 dt2 LC The solution of this equation is of the form I t max cos out where M 3 RLC Circuits Once the switch is closed we must have 1 5L 7 6 R or dt2 L dt LC This equation has three types of solutions 1 underdamped 2 overdamped and 3 critically damped For underdamping N S N B 9 gt4 CD gt4 U A IJI For overdamping I Ale Alt A2e2t 2 1 A1 A1 E i 2L 2L LO CPHY 122 Class Notes 16 Instructor H L Neal 1 Alternating Current Sources An alternating current may be generated by a time varying voltage source of the form VS t Vmax sin wt TM The application of Ohm s law to the circuit above implies that Clearly V W Not that VS t and I t have minima and maxima at the same values of the time t They are said to be in phase The instantaneous power is P z 10 R The average power is de ned as Where the period T is 27139 T 7 i w Thus 24 t R7 I t 2d PM 2 n IWVR Where 24 w w Tm32 Imax2 s1n2wtdt 1 max 2lt gt Therefore 1 ms Elmax Similarly 1 VW 7 vm 2 Inductor in AC circuit L 11m We must have Vs i 5L 07 Where d E 7L7 L dt This means that Lg 7 V 39 t dz max 5111 W 7 so that me t sinwtdt L V 7 cos wt Vmax L s1nwt 7 772 There are two things to notice about this result 1 The current lags behind VS t by 772 2 The quantity wL71 inductive reactance has the same units as a resistor 3 Capacitor in an AC circuit We must have Where so that Then Tim C 40ch7 VC q t CVmax sin wt I t g uvamax cos wt uvamax sin wt 772 Again7 there are two things to notice 1 The current leads VS t by 772 2 The quantity wC capacitive reactance has the same units as a resistor 4 The RLC Series AC Circuit Applying Kirchoff s rules gives 1 q Vmax 39 t 7 L7 R 7 s1nw dt C or 1 VW sin m LE R Differentiating both sides With respect to t gives 12 d 1 Vmax cos wt L RE 5 We may assume that I t max sin wt 7 Then 1 E wmax cos wt 7 b 7 2 7w21max sin wt 7 b Inserting this into the differential equation gives vmx wt 7 7min sin wt 7 a5 BMW wt 7 a5 1 Elmax sin wt 7 b or Vmax cos wt max 7 Lw2gt sin wt 7 l5 RwlmaX cos wt 7 5 Recall that sin wt 7 b sin wt cos 7 cos wt sin 7 cos wt 7 b cos wt cos sin wt sin ab Inserting this gives me21X cos wt C RwIm21X cos wt cos sin wt sin 11m 7 Lw2gt cos ab Rwsin m sin w W l 7 M sin wt w 7 wt sin am 11m Rw cos ab 7 7 Lw2gt sin m cos m Equating the coef cients of cos wt and sin wt on both sides of the equation gives lt 7 Lw2gt cos Rw sin 07 max chos 7 7 Lw2gt sin meax From the rst equation we get Therefore Now recall that L 2 7 i tanq 7w 0 7 wL 7 wC71 R 7 wL 7 wC71 t 1 ab an R 1 cos tanilx V902 1 sin tanilx 35 V902 1 Class Notes 8 Instructor H L Neal 1 Electromotive Force Electromotive force emf is de ned as the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric eld produced by that device It is measured in volts Sources of emf include electric generators both alternating current and direct current types and batteries Electromotive force is often denoted by E In the circuit Fig 17 the emf prodiced by the battery pushes electric charge aroubd the circuit to produce the current I The electric potential difference accross the resistor is VIRE 2 Resistors in Series and Parallel R1 Rz IRl IRz R2 The resistors add and resistors in parallel divide is a good way to remember how to determine the equivalent resistance for any combination Note that this es exactly opposite to the rule for capacitor combinations

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