Induction and Inductance

Transcription

1 Induction and Inductance How we generate E by B, and the passive component inductor in a circuit. 1. A review of emf and the magnetic fux. 2. Faraday s Law of Induction 3. Lentz Law 4. Inductance and inductor 5. Sef-inductance and L in circuits (RL and LC) 6. Energy considerations.

2 Eectricity and Magnetism Charge q as source Current I as source Gauss s Law Eectric fied E! F = q! E Faraday s Law Ampere-Maxwe Law Magnetic fied B Ampere s Law! F = q! v! B Force on q in the fied Force on q v! or I in the fied Summarized in Maxwe equations

3 Eectricity and Magnetism Faraday s Law Eectric fied E Magnetic fied B

4 Chemica emf: A dry-ce battery Chemica reactions in the battery ce transport charge carriers (eectrons) from one termina to the other to create the needed eectric potentia (emf) which drives the current through the outside oad, a ight bub here.

5 Motiona emf, the concept! F B = q v! B! With We can group charges by moving them in a magnetic fied à motiona emf. A motiona emf is the emf induced in a conductor moving through a magnetic fied The eectrons in the! conductor experience a force, F B = q v! B! that is directed aong Charges are accumuated at the ends of the conductor to create an eectric fied inside the conductor to stop further charge transportation. When in equiibrium: F B = qvb = F E = qe

6 Motiona emf, the cacuation Start from the equiibrium condition One has F B = qvb = F E = qe E = vb Or the emf, potentia difference: emf = ΔV = E = vb As ong as the bar is kept moving with a veocity v, the motiona emf is maintained to be vb.

7 Motiona emf, put in use to power a resistor Condition: A bar moves on two rais. The bar and the rais have negigibe resistance. A resistor of R is connected to the end of the two rais. Resut: The emf = vb, so the current I = vb/r Bar moved by Two issues need attention: 1. The moving bar carrying current I, inside the magnetic fied, experiences a force from the fied is F B =IB 2. The magnetic fux in the encosed area (bar, rais and resistor) is Φ B =xb, and it is changing with time as dφ b d dx = ( xb ) = B = vb = emf dt dt dt I! F app Equivaent circuit diagram dφ b dt = emf

8 Exampe, what is the termina veocity? A bar of mass m sides on two vertica rais. A resistor is connected to the end of the rais. When the bar is reeased at t = t 0, (a) cacuate the veocity of the bar at time t, (b) what is the termina veocity? Assuming that the rais and the magnetic fied is ong and arge enough. Once the bar starts to move, acceerated by the gravitationa force, there is: emf = vb And there is current as we: ( ) R I = vb And there is magnetic force on the bar, pointing opposite to the gravitationa force: ( ) 2! F B = I B ˆx = v B R ˆx! F G = mgˆx I m

9 Exampe, what is the termina veocity? Construct the equation of veocity v:! F G + F! " B = mg v B $ # R mg v ( B ) 2 R Sove this equation ( ) 2 = ma = m dv dt dv v τ g = dt τ, τ mr B # v = τ g 1 e t & τ % ( $ ' ( ) 2 % ' ˆx = maˆx & This is the answer to (a). For (b), the termina veocity is when τ g t! F G = mgˆx I m

10 Faraday s Law of induction In the siding bar experiment, we proved that: emf = dφ B dt We aso know that the magnetic fux is defined as Φ B =! B d! A or Φ B = BAcosθ In the siding bar experiment, we changed A by moving the bar. More practicay peope change B or the ange θ to achieve a changing fux. Changing B Changing θ

11 Faraday s Law of induction In any case, the induced emf foows the Faraday s Law of induction Yes, I seeked in the in front of emf = dφ B dt dφ B dt Because Mr. Lenz tod me so in order to answer the question that in which direction shoud the induced current fow. Or more fundamentay because of energy conservation.

12 Faraday s Law, in Engish and in Math Faraday s aw of induction states that the emf induced in a circuit is directy proportiona to the time rate of change of the magnetic fux through the circuit Mathematicay, emf = dφ B dt Lenz s Law

13 Lenz s Law, the direction of the induced emf Lenz s aw: the induced current in a oop is in the direction that creates a magnetic fied that opposes the change in magnetic fux through the area encosed by the oop. The induced current tends to keep the origina magnetic fux through the circuit from changing. The mathematica expression of this Law is the minus sign in Faraday s Law of induction.

14 Lenz s Law tes the induced current direction source B increases, so induced current generates a secondary B that opposes the increase. source B decreases, so induced current generates a secondary B that opposes/compensates the decrease.

15 Lenz s Law, the concept of change

16 Induced emf and Eectric Fieds An eectric fied is created in the conductor as a resut of the changing magnetic fux Even in the absence of a conducting oop, a changing magnetic fied wi generate an eectric fied in empty space I = dφ B dt emf ( = V ), R E V = E d = emf = V = E ds! circe

17 Induced emf and Eectric Fieds The emf for any cosed path can be expressed as the ine integra of E ds the path over Faraday s aw can be written in a genera form:! " E d s! = dφ B dt circe

18 Eddy Currents Circuating currents caed eddy currents are induced in buk pieces of meta moving through a magnetic fied The eddy currents are in opposite directions as the pate enters or eaves the fied Eddy currents are often undesirabe because they represent a transformation of mechanica energy into interna energy. But there are appications of Eddy currents as we.

19 Eddy Currents The magnetic fied is directed into the page The induced eddy current is countercockwise as the pate enters the fied It is opposite when the pate eaves the fied The induced eddy currents produce a magnetic retarding force and the swinging pate eventuay comes to rest

20 Eddy Currents To reduce energy oses by the eddy currents, the conducting parts can Be buit up in thin ayers separated by a nonconducting materia Have sots cut in the conducting pate Both prevent arge current oops and increase the efficiency of the device

21 (Review on) Resistance, Capacitance and (introduce) Inductance Ohm s Law defines resistance: R ΔV I Resistors do not store energy, instead they transform eectrica energy into thermo energy at a rate of: ΔV R 2 2 P= ΔV I = = I R

22 (Review on) Resistance, Capacitance and (introduce) Inductance Ohm s Law defines resistance: R ΔV I Resistors do not store energy, instead they transform eectrica energy into thermo energy at a rate of: ΔV R 2 2 P= ΔV I = = I R Capacitance, the abiity to hod charge: C ΔV Capacitors store eectric energy once charged: 2 1 Q 1 ( Δ ) ΔUE = = C V 2 C 2 2 Q

23 (Review on) Resistance, Capacitance and (introduce) Inductance Ohm s Law defines resistance: R ΔV I Resistors do not store energy, instead they transform eectrica energy into thermo energy at a rate of: ΔV R 2 2 P= ΔV I = = I R Capacitance, the abiity to hod charge: C ΔV Capacitors store eectric energy once charged: 2 1 Q 1 ( Δ ) ΔUE = = C V 2 C 2 Inductance, the abiity to hod current (moving charge). Inductors store magnetic energy once charged with current, i.e., current fows through it. 2 Q

24 Inductance, the definition When a current fows through a coi, there is magnetic fied estabished. If we take the soenoid assumption for the coi: B= µ ni When this magnetic fied fux changes, it induces an emf, E L, caed sef-induction: Φ d( NAB) d( NAµ ni) E d di di L dt dt dt dt dt B 0 2 L = = = = µ 0nV or: E L di L dt 0 For a soenoid: This defines the inductance L, which is constant reated ony to the coi. The sefinduced emf is generated by current fowing though a coi. According to Lenz Law, the emf generated inside this coi is aways opposing the change of the current which is deivered by the origina emf. L = µ E! 2 0n V I + E L Where n: # of turns per unit ength. N: # of turns in ength. A: cross section area V: Voume for ength.

25 Inductor We used a coi and the soenoid assumption to introduce the inductance. But the definition L L E di dt hods for a types of inductance, incuding a straight wire. Any conductor has capacitance and inductance. But as in the capacitor case, an inductor is a device made to have a sizabe inductance. An inductor is made of a coi. The symbo is Once the coi is made, its inductance L is defined. The sef-induced emf over this inductor under a changing current I is given by: di EL = L dt

26 Unit for Inductance The SI unit for inductance is the henry (H) Named for Joseph Henry: V s 1H = 1 A American physicist First director of the Smithsonian Improved design of eectromagnet Constructed one of the first motors Discovered sef-inductance

27 Discussion about Some Terminoogy Use emf and current when they are caused by batteries or other sources Use induced emf and induced current when they are caused by changing magnetic fieds When deaing with probems in eectromagnetism, it is important to distinguish between the two situations

28 Exampe: Inductance of a coaxia cabe Start from the definition: emf = dφ B dt We have dφ B = LdI, or Φ B = LI Φ B = BdA = = µ 0 I 2π n b a b a So the inductance is L = Φ B I = µ 0 2π n b a µ 0 I dr 2πr = L di dt

29 Put inductor L to use: the RL Circuit An RL circuit contains a resistor R and an inductor L. There are two cases as in an RC circuit: charging and discharging. There are aso two case in an RL circuit: except that here one charges with current, not eectric charge. Charging: When S 2 is connected to position a and when switch S 1 is cosed (at time t = 0), the current begins to increase Discharging: When S 2 is connected to position b.

30 RL Circuit, charging Appying Kirchhoff s oop rue to the circuit in the cockwise direction gives Here because the current is increasing, the induced emf has a direction that shoud oppose this increase. Sove for the current I, with initia condition that I (t = 0) = 0, we find I = E R E IR L di dt = 0 ( L 1 ) e Rt E R ( τ 1 ) e t Where the time constant is defined as: τ = L R

31 RL Circuit, discharging When switch S 2 is moved to position b, the origina current disappears. The sef-induced emf wi try to prevent that change, and this determines the emf direction (Lenz Law). Appying Kirchhoff s oop rue to the previous circuit in the cockwise direction gives IR + L di dt = 0 Sove for the current I, with initia condition that I ( t = 0) = E we find R I = E R e Rt L E R e t τ

32 Energy stored in an inductor In the charging case, the current I from the battery suppies power not ony to the resistor, but aso to the inductor. From Kirchhoff s oop rue, we have Mutipy both sides with I: This equation reads: power battery = power R +power L So we have the energy increase in the inductor as: Sove for U L : E = IR + L di dt E I = I 2 R + LI di dt du L dt = LI di dt U L = I LI di = 1 2 LI 2 0

33 Stored energy type and the Energy Density of a Magnetic Fied Given and assume (for simpicity) a soenoid with U L = 1 LI 2 2 L = µ 0 n 2 V U L = 1 2 µ! 0n 2 V B $ # & " µ 0 n % 2 = B2 2µ 0 V So the energy stored in the soenoid voume V is magnetic (B) energy. Since V is the voume of the soenoid, the magnetic energy density, u B is u B = U L V = B2 2µ 0 And the energy density is proportiona to B 2. This appies to any region in which a magnetic fied exists (not just the soenoid)

34 RL and RC circuits comparison RL RC Charging Discharging Energy Energy density I ε = R I = UL Rt L ( 1 e ) ε e R = Rt L 1 2 LI Magnetic fied u B = B 2µ 2 2 o ( ) I t ( ) I t = = ε e R t RC Q e RC t RC 2 Q 1 ( ) 2 UC = = C ΔV 2C 2 Eectric fied u E = 1 2 εe o 2

35 Energy Storage Summary Inductor and capacitor store energy through different mechanisms Charged capacitor Stores energy as eectric potentia energy When current fows through an inductor, it Stores energy as magnetic potentia energy A resistor does not store energy Energy deivered is transformed into thermo energy

36 Exampe: EMF produced by a changing A oop of wire is connected to a sensitive ammeter Determine the current in the oop and the magnet is being Moved into the oop Moved out of the oop Hed sti inside the oop magnetic fied

37 Exampe: a transformer A primary coi is connected to a switch and a battery The wire is wrapped around an iron ring A secondary coi is aso wrapped around the iron ring There is no battery present in the secondary coi The secondary coi is not directy connected to the primary coi Cose the switch and observe the current readings given by the ammeter

38 Appications of Faraday s Law GFI A GFI (ground faut indicator) protects users of eectrica appiances against eectric shock When the currents in the wires are in opposite directions, the fux is zero When the return current in wire 2 changes, the fux is no onger zero The resuting induced emf can be used to trigger a circuit breaker

39 Appications of Faraday s Law Pickup Coi The pickup coi of an eectric guitar uses Faraday s aw The coi is paced near the vibrating string and causes a portion of the string to become magnetized When the string vibrates at some frequency, the magnetized segment produces a changing fux through the coi The induced emf is fed to an ampifier

40 Rotating Loop Assume a oop with N turns, a of the same area rotating in a magnetic fied The fux through the oop at any time t is Φ B = BAcosθ So = BAcosωt emf = N dφ B dt = NBAω sinωt The emf is a sin wave: AC.

41 Generators Eectric generators take in energy by work and transfer it out by eectrica transmission The AC generator consists of a oop of wire rotated by some externa means in a magnetic fied

42 DC Generators The DC (direct current) generator has essentiay the same components as the AC generator The main difference is that the contacts to the rotating oop are made using a spit ring caed a commutator

43 Motors Motors are devices into which energy is transferred by eectrica transmission whie energy is transferred out by work A motor is a generator operating in reverse A current is suppied to the coi by a battery and the torque acting on the current-carrying coi causes it to rotate

44 Motors Usefu mechanica work can be done by attaching the rotating coi to some externa device However, as the coi rotates in a magnetic fied, an emf is induced This induced emf aways acts to reduce the current in the coi The back emf increases in magnitude as the rotationa speed of the coi increases

45 Motors The current in the rotating coi is imited by the back emf The term back emf is commony used to indicate an emf that tends to reduce the suppied current The induced emf expains why the power requirements for starting a motor and for running it are greater for heavy oads than for ight ones

46 Reading materia and Homework assignment Pease watch this video (about 50 minutes each): and Pease check wieypus webpage for homework assignment.