27 Essential University Physics Richard Wolfson Electromagnetic Induction
In Chapter 27 you learnt • To explain the phenomenon of electromagnetic induction • To calculate induced emfs and currents • To use energy conservation to find the direction of induced effects • To describe important technological applications of induction • To explain inductance • And describe the role of inductance in simple circuits • That magnetic fields store energy • And how to calculate that energy • To recognize Faraday’s law as one of the four fundamental laws of electromagnetism • And to calculate induced electric fields
Electromagnetic induction • Electromagnetic induction involves electrical effects resulting from changing magnetic fields. • Simple experiments (next slide) show that it doesn’t matter how the magnetic field changes: Induced electrical effects occur in all cases of changing magnetic fields. • (1) Move a magnet near a circuit; an induced current results. • (2) Move the circuit near a magnet; an induced current results. • (3) Energize one coil to make it an electromagnet; move it near a circuit and an induced current results. • (4) Energize one coil to make it an electromagnet; hold it stationary and move a circuit near it—an induced current results. • (5) Change the current in one circuit, and thus the magnetic field it produces; an induced current results in a nearby circuit.
Five simple experiments • Experiment 1:moving magnet • Experiment 2:moving circuit/coil • Experiments 3 and 4:two circuits; either one moving • Experiment 5: changing field/current; no motion
Faraday’s law • Faraday’s law, in its simplest form, describes induction by relating the emf induced in a circuit to the rate of change of magnetic flux through the circuit: • Here the magnetic flux is given by • With a flat area and uniform field,this becomes • The flux can change because ofchanging field B, changing area A,or changing orientation . Moving a magnet near a wire loop increases the flux through the loop. The result is an induced emf given by Faraday’s law. The induced emf drives an induced current in the loop.
Two examples: • A changing field • The loop has radius r, resistance R, and is in a magnetic field changing at the rate dB/dt. The induced emf is = πr2dB/dt and the induced current is I = (πr2/R) dB/dt. • A changing area • The bar slides on the conducting rails, increasing the circuit area at the rate l dx/dt = lv. The induced emf is = Blv, and the induced current is I = Blv/R.
Clicker question • What will be the direction of the current in the loop when it first enters the field shown, coming into the field from the left side? • clockwise • counterclockwise
Clicker question • What will be the direction of the current in the loop when it first enters the field shown, coming into the field from the left side? • clockwise • counterclockwise
Direction of the induced current: Lenz’s law • The direction of the induced emf and current is described by the minus sign in Faraday’s law. • But it’s easier to get the direction from conservation of energy. • The direction of the induced current must be such as to oppose that change that gives rise to it. • This is known as Lenz’s law. • Otherwise we could produce energy without doing any work! • Here the north pole of the magnet approaches the loop. So the induced current makes the loop a bar magnet with north to the left, opposing the approaching magnet.
It’s the change that matters • Lenz’s law says that induced effects oppose the changes that give rise to them. • Now the induced current flows the opposite way, making the loop’s south pole to the left and opposing the withdrawal of the magnet.
Electric generators • Electric generators use a rotating coil in a magnetic field to convert mechanical to electrical energy. • Here it’s the orientation that’s changing to produce the changing magnetic flux. • Lenz’s law makes it hard to turn a generator that’s supplying electrical energy. • That’s why we have to burn fuels or use the energies of water or wind to generate electricity.
Clicker question • A copper penny falls on a path that takes it between the poles of a magnet. Does the penny hit the ground going faster or slower than if the magnet were not present? • The penny will hit the ground going faster. • The penny will hit the ground going slower. • The penny will hit the ground going the same speed either way.
Clicker question • A copper penny falls on a path that takes it between the poles of a magnet. Does the penny hit the ground going faster or slower than if the magnet were not present? • The penny will hit the ground going faster. • The penny will hit the ground going slower. • The penny will hit the ground going the same speed either way.
Inductance • Mutual inductance occurs when a changing current in one circuit results, via changing magnetic flux, in an induced emf and thus a current in an adjacent circuit. • Mutual inductance occurs because some of the magnetic flux produced by one circuit passes through the other circuit. • Self-inductance occurs when a changing current in a circuit results in an induced emf that opposes the change in the circuit itself. • Self-inductance occurs because some of the magnetic flux produced in a circuit passes through that same circuit.
Clicker question • A long wire carries a current I as shown. What is the direction of the current in the circular conducting loop when I is decreasing? • The current flows counterclockwise. • The current flows clockwise.
Clicker question • A long wire carries a current I as shown. What is the direction of the current in the circular conducting loop when I is decreasing? • The current flows counterclockwise. • The current flows clockwise.
Self-inductance • The self-inductance L of a circuit is defined as the ratio of the magnetic flux through the circuit to the current in the circuit: • In differential form: • Therefore, by Faraday’s law,the emf across an inductor is • The minus sign shows that the directionof the inductor emf is such as to opposethe change in the inductor current.
Clicker question • Current flows from left to right through the inductor as shown. A voltmeter connected across the inductor gives a constant reading, and shows that the left end is positive. Is the current in the inductor changing, and if so, how? • The current is increasing. • The current is decreasing. • The current is constant.
Clicker question • Current flows from left to right through the inductor as shown. A voltmeter connected across the inductor gives a constant reading, and shows that the left end is positive. Is the current in the inductor changing, and if so, how? • The current is increasing. • The current is decreasing. • The current is constant.
Magnetic energy • As current builds up in an inductor, the inductor absorbs energy from the circuit. • That energy is stored in the inductor’s magnetic field. • For an inductor, the stored energy is • Considering the uniform magnetic field inside a solenoid shows that the magnetic energy density is • This is a universal expression: wherever there’s a magnetic field, there’s energy with density B2/20.
Induced electric fields • The induced emf in a circuit subject to changing magnetic flux results from an induced electric field. • Induced electric fields result from changing magnetic flux. • This is described by the full form of Faraday’s law, one of the four fundamental laws of electromagnetism:where the integral is taken around any closed loop, and where the flux is through any area bounded by the loop. • Loosely, Faraday’s law states that a changing magnetic field produces an electric field. • Thus not only charges but also changing magnetic fields are sources of electric field. • Unlike the electric field of a static charge distribution, the induced electric field is not conservative. O
Static and induced electric fields • Static electric fields begin and end on charges. • But induced electric fields generally form closed loops.
Summary • Faraday’s law describes electromagnetic induction, most fundamentally the phenomenon whereby a changing magnetic field produces an electric field: • This induced electric field is nonconservative and its field lines have no beginnings or endings. • In the presence of a circuit, the induced electric field gives rise to an induced emf and an induced current. • Lenz’s law states that the direction of the induced current is such that the magnetic field it produces acts to oppose the change that gives rise to it. • Self-inductance is a circuit property whereby changing current in a circuit results in an induced emf that opposes the change. • Consideration of current buildup in an inductor shows that all magnetic fields store energy, with energy density B2/20. O
Magnetic induction and Faraday’s law • Ch 27 Problem 40 • A square wire loop with sides of 3.0 m is perpendicular to a uniform magnetic field of 2.0 T. A 6.0 V light bulb is in series with the loop. The magnetic field is reduced steadily to zero over time ∆t. • Find ∆t such that the light will shine at full brightness during this time. • Which way will the loop current flow?
(a) State Faraday’s law and Lenz’s law using words as well as symbols. Explain all symbols that you use. [6] • (b) A small circular copper ring is inside a larger loop that is connected to a battery and a switch as shown in the figure. • Use Lenz’s law to find the direction of the current induced in the small ring • just after switch S is closed, • after S has been closed a long time and • (iii) just after S has been reopened after being closed a long time. [4]
Magnetic induction and Faraday’s law Ch 27 Exercise 18 A conducting loop of area 240 cm2 and resistance 12 Ω lies at right angles to a spatially uniform magnetic field. The loop carries an induced current of 320 mA. At what rate is the magnetic field changing? Ch 27 Exercise 19 The magnetic field inside a 20 cm diameter solenoid is increasing at a rate of 2.4 T/s. How many turns should a coil wrapped around the outside of the solenoid have so that the emf induced in the coil is 15 V? Both on Sheet 10
