# Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 20: Electromagnetic Induction.

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Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 20: Electromagnetic Induction Motional EMF Electric Generators Faraday’s Law Lenz’s Law Transformers Eddy Currents Induced Electric Fields Mutual- and Self-Inductance LR Circuits

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 § 20.1 Motional EMF Consider a conductor in a B-field moving to the right. V

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 An electron in the conductor experiences a force downward. V e-e- F The electrons in the bar will move toward the bottom of the bar. This creates an electric field in the bar and results in a potential difference between the top and bottom of the bar.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 What if the bar were placed across conducting rails (in red) so that there is a closed loop for the electrons to follow? In this circuit, the electrons flow clockwise; the current is counterclockwise. V L

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 The motional EMF is where L is the separation between the rails. The current in the rod is where R is the resistance in the “wires”.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 The magnitude of the magnetic force on the rod is: The rod has a current through it. What is the direction of the magnetic force on the rod due to the external magnetic field? Using the right hand rule, the force on the bar is directed to the left.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 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. (Energy conservation)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 §20.2 Electric Generators 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) An energy source is needed to turn the wire coil. Examples include burning coal or natural gas to produce steam; falling water.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 The EMF produced by an AC generator is: In the United States and Canada  0 = 170 volts and f =  /2  = 60 Hz.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 §20.3 Faraday’s Law Moving a conductor through a B-field will generate an EMF. Another way to generate an EMF is to place a stationary conductor in a B-field that varies with time.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 The magnetic flux is proportional to the number of B-field lines that cross a given area. Loop of wire with area A The unit of magnetic flux is the weber: 1 Wb = 1 Tm 2

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Faraday’s Law: An induced EMF in a “coil” of N loops is due to a changing magnetic flux. Ways to induce an EMF: 1.Vary the magnetic field. 2.Vary the area of the coil. 3.Change the angle between B and A.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Example: If the magnetic field in a region varies with time according to the graph shown below, find the magnitude of the induced EMF in a single loop of wire during the following time intervals: (a) 0-2.0 ms, (b) 2.0-4.0 ms, and (c) 4.0-8.0 ms. The loop has area 0.500 m 2 and the plane of the loop is perpendicular to the B-field. 0.50 T 2 4 8 B (T) t (ms)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 Using Faraday’s Law: This is the slope of the given B versus time graph. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 Example continued: (a) In the interval 0.0-2.0 ms, (b) In the interval 2.0-4.0 ms,

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 (c) In the interval 4.0-8.0 ms, Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 §20.4 Lenz’s Law The direction of induced EMFs and currents always oppose the change in flux that produced them. That is, the induced I (and thus induced B) tries to keep the total flux through the loop constant.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 Example: Towing the bar to the right produced an induced current that was CCW. What is the direction of the induced magnetic field? The induced B is out of the page to maintain the flux originally through the loop before the bar started to move to the right (the area of the loop is increasing). V L

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 Example (text problem 20.12): A long straight wire carrying a steady current is in the plane of a circular loop of wire. (a) If the loop of wire is moved closer to the wire, what is the direction of the induced current in the wire loop? I Wire loop There is a magnetic field into the page at the location of the loop. As the loop gets closer to the wire there is an increase in flux. To negate this increase in flux, the induced B-field must point out of the page. This requires a CCW current.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 (b) At one instant, the induced EMF in the loop is 3.5 mV. What is the rate of change of the magnetic flux through the loop in that instant? Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 §20.5 Transformers Wrap an iron core with wire. Primary coil Secondary coil Apply a varying voltage to the primary coil. This causes a changing magnetic flux in the secondary coil.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 Since the flux through the coils is the same The “turns ratio” gives the ratio of the EMFs. Depending on the turns ratio, a transformer can be used to step-up or step-down a voltage. The rate that power is supplied to both coils is the same

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Example (text problem 20.25): A step-down transformer has a turns ratio of 1/100. An AC voltage of amplitude 170 V is applied to the primary. If the primary current is 1.0 mA, what is the secondary current?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Example (text problem 20.27): The primary coil of a transformer has 250 turns and the secondary coil has 1000 turns. An AC voltage is sent through the primary. The EMF of the primary is 16.0 V. What is the EMF in the secondary?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 §20.6 Eddy Currents If a conductor is subjected to a changing magnetic flux, a current will flow. (This includes sheets of metal, etc.)

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Consider a metal plate that swings through a magnetic field. An external magnetic field into the page created by a magnet. X pivot

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 As the plate swings through the region of magnetic field, some regions of the plate are entering the B-field (increasing flux), and other regions of the plate are leaving the B-field (decreasing flux). There will be induced currents in the conductor called eddy currents. The eddy currents dissipate energy (according to I 2 R); this results in the damping of the amplitude of the metal sheet.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 § 20.7 Induced Electric Fields When a stationary conductor sits in a changing magnetic field it is an induced electric field that causes the charges in the conductor to move.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 § 20.8 Mutual- and Self-Inductance Coil 1 Coil 2 A variable current I 1 flows in coil 1. I 1 then induces a current in coil 2. The flux (  21 ) through coil 2 due to coil 1 is

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Writing this as an equality, Where M is the mutual inductance. It depends only on constants and geometrical factors. The unit of inductance is the Henry (1H = 1Vs/A). The induced EMF in the coils will be:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 Self-inductance occurs when a current carrying coil induces an EMF in itself. The definition of self-inductance (L) is

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 Example (text problem 20.41): The current in a 0.080 Henry solenoid increases from 20.0 mA to 160.0 mA in 7.0 s. Find the average EMF in the solenoid during that time interval.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 An inductor stores energy in its magnetic field according to: The energy density in a magnetic field is:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 § 20.9 LR Circuits An inductor and resistor are connected in series to a battery. As with an RC circuit, the current in the circuit varies with time.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 The voltage drop across an inductor is given by When an inductor is “charging” (the energy stored is increasing) the current in the circuit is: Where  = L/R is the time constant for the circuit and I f =  b /R maximum current in the circuit.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 Applying Kirchhoff’s loop rule to the circuit gives the EMF in the inductor as:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 Plots of  L (t) and I(t) for this LR circuit:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 For a “discharging” inductor, The LR circuit time constant  plays the same role as in an RC circuit. where I 0 is the current in the inductor when t=0.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40 Example: A coil has an inductance of 0.15 H and a resistance of 33.0 . The coil is connected to a 6.0 V ideal battery. When the current reaches one-half the maximum value: (a) At what rate is the magnetic energy being stored in the inductor? V max = emf of the battery (  b ) = 6.0 Volts

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 41 (b) At what rate is energy being dissipated? Energy is dissipated in the resistor at a rate Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 42 (c) What is the total power the battery supplies? The battery must supply energy to the inductor and the resistor. Part a and b calculate the rate at which energy is delivered to the inductor and resistor respectively; the battery must supply the sum of these: P battery = 0.54 Watts. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 43 Summary Motional EMF Faraday’s Law Lenz’s Law Transformers Eddy Currents Inductance and Inductors LR Circuits

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