PHYSICS - CLUTCH 1E CH 28: INDUCTION AND INDUCTANCE.

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2 CONCEPT: ELECTROMAGNETIC INDUCTION A col of wre wth a VOLTAGE across each end wll have a current n t - Wre doesn t HAVE to have voltage source, voltage can be INDUCED V Common ways to INDUCE a voltage on a col of wre: - Move a bar magnet - Vary current n electromagnet - Turn electromagnet on and off Induced voltage known as INDUCED EMF INDUCED CURRENT - Process known as ELECTROMAGNETIC INDUCTION EXAMPLE of current nducton: vሬԧ vሬԧ = 0 vሬԧ CURRENT INDUCED NO CURRENT INDUCED CURRENT INDUCED CERTAIN changes wll nduce a current, and the magntude of the current depends on the rate of these changes - Bar magnet movng nto col Faster t goes, larger the nduced current - Current changng n electromagnet near a col Faster the current changes, larger the nduced current Page 2

3 CONCEPT: MAGNETIC FLUX Remember! Electrc flux s just the amount of passng through a surface - MAGNETIC FLUX s just the amount of passng through a surface MAGNETIC FLUX s gven by A θ Normal Φ B = (UNITS 1 Wb = 1 ) B Magnetc flux changes wth,, and - IMPORTANT to remember ths changes n magnetc flux wll be mportant later! EXAMPLE: What s the magnetc flux through the square surface depcted n the followng fgure, f B = 0.05 T? Assume the sde length of the square s 5 cm. Surface B 30 o Page 3

4 PRACTICE: MAGNETIC FLUX THROUGH A RING A rng of radus 0.5m les n the xy-plane. If a magnetc feld of magntude 2 T ponts at an angle of 22 o above the x-axs, what s the magnetc flux through the rng? EXAMPLE: ROTATING RING A rng of radus 2 cm s n the presence of a 0.6 T magnetc feld. If the rng begns wth ts plane parallel to the magnetc feld, and ends wth the plane of the rng perpendcular to the magnetc feld, what s the change n the magnetc flux? Page 4

5 CONCEPT: MAGNETIC FLUX WITH CALCULUS Magnetc flux when the magnetc feld DOESN T CHANGE over the surface Φ B = BAcosθ - For changng felds Φ B = EXAMPLE: What s the flux through the loop shown n the followng fgure? d w L Page 5

6 CONCEPT: FARADAY S LAW Changng magnetc feld through conductng loops - Ths s actually due to a changng MAGNETIC FLUX A changng MAGNETIC FLUX leads to an nduced EMF: Ɛ nd = - Ths s known as Faraday s Law Remember! Φ = BA cos θ - So, magnetc flux changes wth,, and EXAMPLE 1: A square conductng wre of sde length 4 cm s n a 2 T magnetc feld. It rotates such that the angle of the magnetc feld to the normal of the square ncreases from 30 o to 60 o n 2 s. What s the nduced current on the wre f ts resstance s 5 Ω? Page 6

7 PRACTICE: FARADAY S LAW AND TWO SOLENOIDS Two solenods are placed end to end, wth one solenod connected to a varable power source, and the other solenod connected to a 10 Ω resstor. The frst solenod has 10 turns per cm and has as an ntal current of 2 A, and the second solenod has 5 turns and a radus of 2 cm. a) What s the change n magnetc feld emtted by the frst solenod f the current ncreases from 2 A to 5 A n 1 s? b) What s the change n the magnetc flux through the other solenod durng ths 1 s? c) What s the nduced EMF on the second solenod? d) What s the nduced current on the second solenod? EXAMPLE: CURRENT IN A CIRCUIT WITH A CHANGING, EXTERNAL MAGNETIC FIELD What current does the ammeter read f the followng crcut, wth an area of 50 cm 2, s placed n a magnetc feld that s changng at 0.05 T/s? Note that the resstor has a resstance of 2 Ω. BሬሬԦ A Page 7

8 CONCEPT: FARADAY S LAW WITH CALCULUS FARADAY S LAW states, more precsely, - Ɛ = EXAMPLE: Fnd the current n the loop shown n the followng fgure f the current n the straght wre ncreases at d/dt. d w L b Remnder! Between any two ponts, a and b, V ab = E dl a - For a col whch has an nduced EMF, the ntegral has to be a CLOSED ntegral over the loop E dl = dφ dt Page 8

9 EXAMPLE: RAIL GUN A ral gun s composed of a moveable conductng rod on a U-shaped crcut, as shown below. What s the next force on the conductng rod n ths case? Page 9

10 CONCEPT: LENZ S LAW Faraday s Law tells us the magntude of the nduced EMF magntude of nduced current - To fnd DIRECTION of nduced current, we use Lenz s Law LENZ s LAW states: A conductor wll nduce a magnetc feld on tself to changes n ts magnetc flux B v v B nd v B B nd v B B Once the drecton of the nduced magnetc feld s known, rght hand rule gves drecton of nduced current B B v B nd v nd EXAMPLE: In the followng scenaros, fnd the drecton of the current nduced on the conductors. v v v Page 10

11 PRACTICE: DIRECTION OF INDUCED CURRENT IN A RING What s the drecton of the nduced current n the nner rng shown n the followng fgure? For ths problem, consder the battery s voltage as contnuously INCREASING. Note: the arrow strkng through the battery n the crcut dagram ndcates that the voltage of the battery s varable (.e. t can be changed). EXAMPLE: BAR MAGNET VS CURRENT-CARRYING WIRE A bar magnet moves relatve to a col of wre as ndcated n the fgure below and nduces a current n the col. A current carryng wre carres a current relatve to a col as shown n the second fgure. Would you need to ncrease or decrease the magntude of the current n the wre to nduce a current n the col that moves n the SAME drecton as the current nduced by the bar magnet? Scenaro 1 v Scenaro 2 Page 11

12 CONCEPT: MOTIONAL EMF Remember! A changng magnetc feld can produce an EMF - BUT so can moton. Ths s referred to as a MOTIONAL EMF. If a conductor moves through a magnetc feld, charges feel a Postve charges feel the force [ UPWARD / DOWNWARD] Separaton of charges L FሬԦ B BሬሬԦ vሬԧ Separaton of charges E feld Electrc force that magnetc force - To balance, E = vb Induced EMF Ɛ = EL = EXAMPLE: If a conductor of length 10 cm moves wth a velocty of 20 m/s n a magnetc feld of 0.05 T, what s the current through the conductor f ts resstance s 15 Ω If a conductor moves along U-shaped wre, MAGNETIC FLUX changes - Change n Change n magnetc flux Producton of L BሬሬԦ vሬԧ Change n area of Change n magnetc flux of Induced EMF Ɛ = ΔΦ B Δt = EXAMPLE: In the crcut below, f the wre has a resstance of 10 Ω, what s the current nduced f the length of the bar s 10 cm, the speed of the bar s 25 cm/s, and the magnetc feld s 0.02 T? What about the power generated by the crcut? a BሬሬԦ vሬԧ b Page 12

13 PRACTICE: BAR MOVING IN UNKNOWN MAGNETIC FIELD A thn rod moves n a perpendcular, unknown magnetc feld. If the length of the rod s 10 cm and the nduced EMF s 1 V when t moves at 5 m/s, what s the magntude of the magnetc feld? Page 13

14 CONCEPT: TRANSFORMERS Power n North Amerca s delvered to outlets n homes at 120 V. - Ths s too large to operate many delcate electroncs, such as computers. Remember! A col wth a changng magnetc feld can nduce an EMF on a second col - Ths nduced EMF can be as small as needed. A TRANSFORMER does exactly ths t uses Faraday s law to convert a large voltage to a small EMF: V 1 V 2 The rato of the VOLTAGES n a transformer depends upon the rato of the TURNS: V 2 V 1 = N 2 N 1 EXAMPLE: You need to buld a transformer that drops the 120 V of a regular North Amercan outlet to a much safer 15 V. You already have a solenod wth 50 turns made, but you need to make a second solenod to complete your transformer. What s the least number of turns the second solenod could have? Page 14

15 PRACTICE: OPERATING A LAPTOP An outlet n North Amerca outputs electrcty at 120 V, but a typcal laptop needs to operate at around 20 V. In order to do so, a transformer s placed n a laptop s power supply. If the col n the crcut connected to the laptop has 20 turns, how many turns must the col n the crcut wth the outlet have? Page 15

16 CONCEPT: MUTUAL INDUCTANCE If two cols are brought close, and the current changes through one col, an EMF s nduced on the other col - Ths s known as MUTUAL INDUCTANCE - The col wth the changng current s known as the, the other the Magnetc feld B depends on Flux through 2 depends on Total flux through 2 s to current through 1 - We call the proportonalty constant MUTUAL INDUCTANCE, M Col 1 N 1 1 Col 2 N 2 The MUTUAL INDUCTANCE between two cols s M = (UNITS: Henry, 1 H = 1 / ) EXAMPLE: What s the mutual nductance of two solenods of length L and area A, one wth N1 turns and the other wth N2? L N 1 N 2 Induced EMF on second col s dependent upon the mutual nductance: - Ɛ 2 = M Δ 1 Δt EXAMPLE: Whch of the followng prmary cols would nduce the largest EMF on a secondary col of wre f they all had ther current changng at the same rate. 10 turns 10 cm 20 turns 2.5 cm Page 16

17 PRACTICE: MUTUAL INDUCTION BETWEEN TWO SOLENOIDS A solenod of 25 turns, wth an area of m 2 s wound around a 10.0 cm solenod wth 50 turns, as shown n the fgure below. If, at some nstant n tme, the current through the 10.0 cm solenod s 0.5 A and changng at 50 ma/s, what s the nduced EMF on the 25 turn solenod? L N 1 N 2 Page 17

18 CONCEPT: SELF INDUCTANCE When consderng two cols, one acted as source of magnetc feld and one felt a changng magnetc flux - Sngle col wll also experence an nduced EMF due to changng ts own magnetc flux SELF-INDUCED EMF We defne SELF INDUCTANCE, L, dentcally to mutual nductance L = (UNITS are also H) Self-nduced EMF s then - Ɛ = L Δ Δt EXAMPLE: What s the self-nductance of a col of wre, wth N turns and a radus R? EXAMPLE: A solenod has 500 turns, wth a current of 0.5 A producng an average flux through col of Wb. If the selfnduced EMF on ths solenod s 10 mv, how quckly must the current be changng? Page 18

19 PRACTICE: SELF-INDUCED EMF IN A COIL OF WIRE A crcular col of wre wth 20 turns has a current changng at a rate of 0.12 A/s. If the radus of the col s 20 cm, what s the nduced EMF on the col? Page 19

20 CONCEPT: INDUCTORS A col of wre placed n a crcut s known as an INDUCTOR or In order to use nductors n crcuts, we need to know how to apply KIRCHHOFF S LOOP RULE to them: a b a b Remember! By Faraday s law, the nduced EMF on an nductor s Ɛ = EXAMPLE: Wrte out Krchhoff s loop rule for the followng crcut. Treat the capactor as ntally charged. +q -q Snce nductors oppose changes n current (by Lenz law), they re use n crcuts to do just that. - For nstance, they re used n power transmsson lnes, n the event of lghtnng strkes Page 20

21 CONCEPT: LR CIRCUITS LR Crcuts are, as the name mples, crcuts contanng and S 2 L R S 1 V There are two steps needed to analyze ths crcut: - CURRENT GROWTH: When S1 s closed, but S2 s open, the battery produces a current n the crcut - CURRENT DECAY: When S1 s open and S2 s closed, the current decays because the battery s removed CURRENT GROWTH n an LR crcut does not occur nstantly the nductor ressts changes to currents (t) = V R (1 e t/τ ) L R V CURRENT DECAY n an LR crcut does not occur nstantly the nductor ressts changes to currents (t) = V R e t/τ L R The TIME CONSTANT, τ = L, determnes the how quckly growth and decay occurs R Page 21

22 EXAMPLE: UNKOWN RESISTANCE IN LR CIRCUIT An LR crcut has a tme constant of s and s ntally connected to a 10 V battery. If after s of beng dsconnected from the battery, the current s 0.5 A, what s the resstance of the crcut? PRACTICE: TIME TO HALF MAXIMUM CURRENT An LR crcut wth L = 0.1 H and R = 10 Ω are connected to a battery wth the crcut ntally broken. When the crcut s closed, how much tme passes untl the current reaches half of ts maxmum value? Page 22

23 PRACTICE: UNKNOWN CURRENT IN AN LR CIRCUIT Consder the LR crcut shown below. Intally, both swtches are open. Frst, swtch 1 s closed, and current s allowed to grow to ts maxmum value. Then, swtch two s closed and swtch 1 s open, and current s allowed to decay for 0.05 s. What s the maxmum current n the crcut? What s the fnal current n the crcut f V = 10 V, L = 0.02 H, and R = 5 Ω? S 2 L R S 1 V Page 23

24 CONCEPT: LC CIRCUITS LC Crcuts are made up of and, as ther name mples The current n ths crcut OSCILLATES: +q -q q = 0 -q +q q = 0 +q -q MATHEMATICALLY, the current and charge are represented by - (t) = (ω = 1/LC s the angular frequency of oscllaton) - q(t) = φ s known as the PHASE ANGLE, and t determnes what part of the oscllaton you begn at EXAMPLE: An LC crcut wth an nductor of 0.05 H and a capactor of 35 µf begns wth the current at half ts maxmum value. What s the phase angle of ths oscllaton? Page 24

25 EXAMPLE: OSCILLATIONS IN AN LC CIRCUIT An LC crcuts, wth L = 0.05 H and C = 50 mf, begns wth the capactor fully charged. After 0.1 s, the current s 0.2 A. Under these condtons, how many seconds does t take for a fully charged plate to transfer all of ts charge to the other plate? PRACTICE: MAXIMUM CURRENT IN LR CIRCUIT An LR crcut has a 0.5 mf capactor ntally charged to 1 mc. If t s connected to a 0.04 H nductor, what s the maxmum current n the crcut? Page 25

26 CONCEPT: ENERGY IN AN LC CIRCUIT An nductor s just a col of wre we don t assume t has any resstance So when the capactor loses ts charge, and therefore ts energy, where does t go? An nductor can store MAGNETC ENERGY - U = So long as the wres don t have any resstance, energy s conserved n an LC crcut: +q -q q = 0 -q +q q = 0 +q -q E U C U L E U C U L E U C U L E U C U L E U C U L EXAMPLE: An LC crcut has an 0.1 H nductor and a 15 nf capactor, and begns wth the capactor maxmally charged. After 0.1 s, how much energy s stored by the nductor? If the ntal charge on the capactor were 50 mc, what s the maxmum current n the crcut? Page 26

27 PRACTICE: ENERGY LOSS DUE TO RESISTANCE Let s say an LC crcut begns wth the capactors carryng a maxmum charge of 10 mc. After the capactor has lost half of ts charge, what s the current n the crcut f L = 0.01 H and C = 50 mf? If durng the tme for the capactor to lose half ts charge, resstance wthn the crcut dsspated 0.2 mj, what then would the current n the crcut be? Page 27

28 CONCEPT: LRC CIRCUITS As the name mples, an LRC crcut contans,, and In and LRC crcut, wth the capactor ntally charged, we have: - ΣV = = 0 = 0 = 0 +q -q There are 3 solutons to the equaton above: the UNDERDAMPED, CRITICALLY DAMPED, and OVERDRAMPED UNDERDAMPED CRITICALLY DAMPED OVERDAMPED - q(t) = Qe (R/2L)t cos (ω t + φ) - q(t) = Qe (R/2L)t - No smple equaton q t - Occurs for small R - Looks almost lke an LC crcut - But R s sappng energy - Occurs when R 2 = 4L/C - Looks lke an RC Crcut - Occurs for large R - Looks lke an RC Crcut The new angular frequency s - ω = 1 LC R2 4L 2 Page 28

29 EXAMPLE: AMPLITUDE DECAY IN AN LRC CIRCUIT An LRC crcut has an nductance of 10 mh, a capactance of 100 µf, and a resstance of 20 Ω. What type of LRC crcut s ths? How long wll t take for the maxmum charge stored on the capactor to drop by half? Page 29