Physics 11b Lecture #13

Transcription

1 Physics 11b Lecture #13 Faraday s Law S&J Chapter 31

2 Midterm #2 Midterm #2 will be on April 7th by popular vote Covers lectures #8 through #14 inclusive Textbook chapters from 27 up to 32.4 There will be 5 problems again One is multiple-choice One is from the homework Other rules are same as #1

3 What We Did Last Time Applied iot-savart law to linear current = field rotates around the current Parallel wires attract each other by F = Definition of Ampere Ampère s Law Applies to any boundary that encircle current I Examples: thick wire, infinite current sheet Solenoid field Gauss s Law d s =µ 0I = µ ni 0 Φ = d A = 0 µ I I π a µ I 0 2π r

4 Today s Goals Induction: changing magnetic field generates electricity asis of much of the electric industry Faraday s Law and Lenz s Law Describes the induction Completes the four basic laws of E&M Practical examples AC and DC electric generators Eddy currents

5 Induction Current is generated in a loop of wire when Magnet approaches Current flows in a nearby wire A S N A Current is induced by the changing magnetic fields It flows only when the magnet is moving, or when the switch is thrown on/off The induced emf must be related to d/dt

6 Faraday s Law The emf E induced in a circuit equals to the rate of change of the magnetic flux through the circuit E = dφ dt where Φ = d A da E looks like

7 Magnetic Flux Magnetic flux Φ depends on the field, the size, shape and angle of the loop A Simple case: a flat loop of area A in a uniform field θ Φ = d A = Acosθ dφ d E = = ( A cos θ ) dt dt Induction emf may appear because of Changing field Changing area A of the loop Changing angle θ between and the loop

8 Practical Examples To measure magnetic field, we can rotate a small loop and measure the induced emf Called a flip-coil Attach a very light coil on a thin membrane in a magnetic field V Sound waves make the membrane vibrate induce emf Called a moving-coil microphone sound S N Can you imagine how a movingmagnet microphone works? V

9 Moving-Rod Circuit A circuit is made of a movable metal rod on two rails Rod moves with v Area of the circuit A = x Consider Lorentz force x on a charge q in the rod F = qv Charge moves up Upper rail is +, lower rail is da v dt = This creates E field pointing down In the steady state da dφ V = E = v = = dt dt R qv + qe= 0 E = v F q F E v Faraday s law except for the sign

10 Moving-Rod Circuit Induced emf causes current I I V = = R v R Lorentz force on this current is F = IL F = I = v R 2 2 How much work per unit time do we have to do? v P = Fv = = I V R R V = v x To keep the bar moving, we must pull the bar with this much force to the right F Exactly the power dissipated in the resistor I v

11 Faraday and Lorentz In the case of moving-rod circuit, Faraday s law can be derived from Lorentz force on the charges in the bar The two laws are equivalent This doesn t work if the circuit is not moving and the magnetic field is changing Faraday s law predicts an emf No Lorentz force unless charges are moving Faraday s law is a part of the four fundamental laws that connects magnetic field to electric field efore getting there, let s deal with the minus sign

12 Lenz s Law Induced current creates magnetic field that opposes the change in the magnetic flux If the flux in a loop is increasing, the magnetic field due to the induced current is opposite to the original field Vice versa Nature opposes changes The minus sign in Faraday s Law symbolizes this Actual direction of the induced emf and current can be found by Lenz s law E ext = dφ dt ind d dt ext Induced current > 0 < 0 ind

13 Lenz s Law A magnet approaches a conductive ring Which way is the induced current? S N N S What if the magnet is moving away? A coil and a ring is wrapped around an iron bar Which way is the induced current when the switch is closed? and opened?

14 Induced Electric Field A conductive ring is in an increasing field Induced current is counter-clockwise Charges in the ring feels force, as if there is an E field Call it an induced electric field d dt > 0 Integrating the induced E field around the ring, we get the induced emf + E = E d s Faraday s Law can be rewritten as E d s = dφ dt Point your right thumb along the field the other fingers curl in the opposite direction from the loop integral sign

15 Four E&M Equations We now have four fundamental E&M equations Gauss s law for E Gauss s law for Faraday s law Ampère s law E d A = S d A = S E d s = d s =µ 0I q ε 0 0 dφ dt Surface integral is taken over a closed surface S Line integral is taken around a closed loop One of them is still incomplete Next lecture

16 AC Generators Alternate Current (AC) generators are very simple For a loop area A rotating with angular velocity ω Φ = Acosθ = Acosωt dφ E = = Aω sinωt dt If the loop has N turns E = NAω sinωt For commercial 60Hz power generator ω = 2 π (50 sec) = 314.2radian sec

17 DC Generators Direct Current (DC) generators use a split ring + brush (commutator) to keep the output flowing in one way E AC generator t E DC generator t

18 AC Generator and Power While generating electricity the generator needs power θ = ωt input (e.g. steam turbine) Current flowing the loop receives torque τ = IA Use E = Aω sinωt and assume a resistor R as the load 2 2 Aωsinωt A ω 2 τ = Asinωt = sin ωt R R 2 2 A ω Average torque is τ = 2R Power required is Aω max τ ω = = E 2R 2R

19 Eddy Currents Faraday s Law works in conductor of any shape Consider a simple plate Increase field Rotating current Move the plate into a field Ditto Rotating current in a continuous body of conductor due to changing field is called the eddy current Direction is given by Lenz s Law Eddy currents always slow down the change Used for braking systems of various machines

20 Summary Changing magnetic field induces emf dφ Faraday s law: E = where dt Lenz s law: induced current reduces the change in Φ Moving-rod example connection with Lorentz force Faradays law in terms of E field One of the four basic laws of E&M Others: Gauss s law for E/, Ampère s law AC generator Eddy current E = NAω sinωt Φ = d A E d s = dφ dt