Chapter 27 Magnetic Induction
Motional EMF Consider a conductor in a B-field moving to the right. In which direction will an electron in the bar experience a magnetic force? V
e - V The electrons in the bar will move toward the bottom of the bar. F This charge separation creates an electric field in the bar and results in a potential difference between the top and bottom of the bar. What is the electric field? The motional EMF is where L is the length of the bar.
What if the bar were placed across conducting rails (in red) so that there is a closed loop for the electrons to follow? V L In this circuit, what direction is the current? a) clockwise b) counterclockwise
Now the rod has a current through it. What is the direction of the magnetic force on the rod due to the external magnetic field? 6 Using the right hand rule, the force on the bar is directed to the left. The magnitude of the magnetic force on the rod is:
7 To maintain a constant EMF, the rod must be towed to the right with constant speed. An external agent must do work on the bar. Power: Where does the energy go? Ohmic dissipation:
Induced EMF Motion of wire in B field induces an emf (thus a current if circuit is closed) Are there other ways?
Magnetic flux Φ B = B da
Faraday s Law of Induction The induced emf in a closed loop equals the negative of the rate of change of the magnetic flux through the loop: V emf = dφ B dt Valid regardless of the reason for the change in magnetic flux (could be a motional emf, a changing magnetic field, changing circuit geometry, etc
CT 33.3 A loop of wire is moving rapidly through a uniform magnetic field as shown. Is a nonzero EMF induced in the loop? A: Yes, there is B: No, there is not
Lenz s Law What is with the minus sign in Faraday s Law? The direction of any magnetic induction effect is such as to oppose the cause of the effect. If current in loop is solely due to induction, direction of current is such that the induced magnetic field causes a reduction in the rate of change of the magnetic flux. Due to conservation of energy
Recall the sliding bar: Induced emf oriented such that induced current produces a magnetic field directed out of page. This reduces the rate of change of the magnetic flux. Note: Induced current tries to cause magnetic flux to not change. It always fails.
CT 33.6 A current-carrying wire is pulled away from a conducting loop. As the wire moves, is there a current induced around the loop? A: Yes, CW B: Yes, CCW C: No
CT 33.6c A loop of wire is near a long straight wire that is carrying a large current I, which is decreasing with time. The loop and wire are in the same plane. The current induced in the loop is I to the right, but decre loop A: CW B: CCW C: No current
Example A long straight wire has a current I(t) changing in time by the equation I(t) = d t. A current loop of length l and width w is situated such that the nearest end is a distance a from the straight wire. What is the induced current if the resistance of the loop is R?
Applications of Induction Magnetic recording and playback: hard drives, tape, credit cards, answering machines, etc Recording: varying current in solenoid produced varying magnetic field, which aligns magnetic dipoles in material. Playback: Varying B in time induces emf in solenoid which produces varying current and voltage.
18 Generators (discussed soon) Alternators (B produced by solenoid, current loop rotated by alternator belt) Spark Plugs (quick I(t) produces huge EMF) Dynamic Microphones (sound waves move diaphragm and coil in magnetic field, inducing current) Transformers and inductors (next chapter) Metal Detectors Electric guitar pickup
Electric Generators 20 A coil of wire is spun in a magnetic field. This produces an EMF and also a current; both vary with time. (AC-alternating current) Power plant Alternator Portable home generator Real generators use more than one coil!
CT 33.7c An electric motor consists of a coil, free to turn on an axis, in an magnetic field created by permanent magnets. Which way will the coil shown rotate? B I A) CW CCW B) C) It won t rotate in this configuration B
http://www.sciencejoywagon.com/physicszone/otherpub/wfendt/ generatorengl.htm
CT 33.8b What can you say about the current generated by the loop at this moment shown? A) Maximum B) Zero C) Nonzero, and changing
The EMF produced by an AC generator is: = dφ B = d dt dt (BAcos(ωt)) = ωbasin(ωt) = 0 sin(ωt) In the United States and Canada V emf = 170 Volts and f = ω/2π = 60 Hz (for home outlets). (V rms = 120 Volts) An energy source is needed to turn the wire coil. Examples include burning coal or natural gas to produce steam (which drives pistons); falling water; engine in hybrid car
Example An electric generator consists of a 100-turn circular coil 50 cm in diameter. Its rotated at f=60 Hz inside a solenoid of radius 75 cm and winding density n = 50 cm -1. What DC current in the solenoid is needed for the the maximum emf of the generator to be 170 V?
Eddy Currents 26 Eddy currents are induced currents in a large (2D or 3D) chunk of a conductor. 26 currents cause conductor to heat up (ohmic dissipation) http://www.youtube.com/watch?v=jdcgxz87onc&nr=1 Application: Roller coaster car breaking, induction stoves, magnetic levitation (trains), metal detectors, etc
Induced electric fields The windings of a long solenoid carrying a current I Where does the emf in the wire loop come from? What actually drives the current? How does wire know B field changed? E dr = dφ B dt E(r, t) = R2 db 2r dt The electric field does not have a radial component (due to Gauss s law). This field is not conservative! (It has curl)
28 Electric field is generated (induced) by the changing B field (regardless of whether a wire is there or not). This induced electric field is what produces the EMF. Note: this electric field has curl (non-conservative)!
Mutual Inductance Consider two stationary circuits in the vicinity of each other. Current in one circuit produces a magnetic field, and thus a magnetic flux through the other circuit. Mutual inductance is the constant of proportion between I 1 and the magnetic flux in circuit #2 Φ B2 = M 21 I 1 If I 1 is changing, an emf is induced in circuit #2 2 = dφ B2 dt = M 21 di 1 dt Mutual inductance is the basis of transformers (covered next chapter)
CT 33.18 Which arrangement of two coils has the larger mutual inductance? Pink A Yellow B
Self-Inductance and Inductors Self inductance determines the magnetic flux in a single circuit due to the circuit s own current. Φ B = LI L = L di dt Inductor circuit device designed to have a particularly high inductance (typically a solenoid)
CT 33.23 Two long solenoids, each of inductance L, are connected together to form a single very long solenoid of inductance L total. What is L total? L(total)=? A: L B: 2L C: 4L D: 8L E: other L + L L = L total =?
LR circuit with constant voltage source: Modified Kirchoff s Loop law: E dl = L di dt = V i i L di dt = IR V emf Note: not proper to think of a voltage difference across inductor (as many books do) (Electric field isn t conservative any more!) always integrate in direction of current
CT 33.28 The switch is closed at t=0. What is the current through the resistor, at t=0+? R = 20Ω V = 10V L = 10H A) 0 A B) 0.5 A/s C) 1 A/s D) 10 A/s E) other
CT 33.29 The switch is closed at t=0. What is the current through the resistor after a very long time? R = 20Ω A) 0 A B) 0.5 A/s C) 1 A/s D) 10 A/s E) other V = 10V L = 10H
E d = IR ξ 0 = L di dt I 0 di = 1 IR ξ 0 L I(t) = ξ 0 R 1 e Rt/L t 0 dt timescale to reach steady state: τ = L/R
Suppose the switch has been closed at B for a very long time. At t=0, the switch is flipped to position B. Immediately afterwards, the current through the resistor is 1. 1 A 2. 0.5 A 3. zero L =1V R = 1 Ohm L = 1 H
Back emf in motor circuit Back EMF in motors causes current to be much smaller than if motor didn t rotate: V emf IR = L di dt = L If resistance is very small, back emf is approximately equal to driving voltage. I(t) = V emf L R overloading the motor or jamming it causes back emf to go away, causing large current and can fry the solenoid