1. Chapter 21
Magnetic Induction

2. Magnetic Induction
Electric and magnetic forces both act only on particles carrying an electric charge
Moving electric charges create a magnetic field
A changing magnetic field creates an electric field
This effect is called magnetic induction
This links electricity and magnetism in a fundamental way
Magnetic induction is also the key to many practical applications
Introduction

3. Electromagnetism
Electric and magnetic phenomena were connected by Ørsted in 1820
He discovered an electric current in a wire can exert a force on a compass needle
Indicated a electric field can lead to a force on a magnet
He concluded an electric field can produce a magnetic field
Did a magnetic field produce an electric field?
Experiments were done by Michael Faraday
Section 21.1

4. Faraday’s Experiment
Faraday attempted to observe an induced electric field
He used an ammeter instead of a light bulb
If the bar magnet was in motion, a current was observed
If the magnet is stationary, the current and the electric field are both zero
Section 21.1

5. Another Faraday Experiment
A solenoid is positioned near a loop of wire with the light bulb
He passed a current through the solenoid by connecting it to a battery
When the current through the solenoid is constant, there is no current in the wire
When the switch is opened or closed, the bulb does light up
Section 21.1

6. Conclusions from Experiments
An electric current is produced during those instances when the current through the solenoid is changing
Faraday’s experiments show that an electric current is produced in the wire loop only when the magnetic field at the loop is changing
A changing magnetic field produces an electric field
An electric field produced in this way is called an induced electric field
The phenomena is called electromagnetic induction
Section 21.1

7. Magnetic Flux
Faraday developed a quantitative theory of induction now called Faraday’s Law
The law shows how to calculate the induced electric field in different situations
Faraday’s Law uses the concept of magnetic flux
Magnetic flux is similar to the concept of electric flux
Let A be an area of a surface with a magnetic field passing through it
The flux is ΦB = B A cos θ
Section 21.2

8. Magnetic Flux, cont.
If the field is perpendicular to the surface, ΦB = B A
If the field makes an angle θ with the normal to the surface, ΦB = B A cos θ
If the field is parallel to the surface, ΦB = 0
Section 21.2

9. Magnetic Flux, final The magnetic flux can be defined for any surface
A complicated surface can be broken into small regions and the definition of flux applied
The total flux is the sum of the fluxes through all the individual pieces of the surface
The surfaces of interest are open surfaces
With electric flux, closed surfaces were used
The SI unit of magnetic flux is the Weber (Wb)
1 Wb = 1 T . m2
Section 21.2

10. Faraday’s Law
Faraday’s Law indicates how to calculate the potential difference that produces the induced current
Written in terms of the electromotive force induced in the wire loop
The magnitude of the induced emf equals the rate of change of the magnetic flux
The negative sign is Lenz’s Law
Section 21.2

11. Applying Faraday’s Law
The ε is the induced emf in the wire loop
Its value will be indicated on the voltmeter
It is related to the electric field directly along and inside the wire loop
The induced potential difference produces the current

12. Applying Faraday’s Law, cont.
The emf is produced by changes in the magnetic flux through the circuit
A constant flux does not produce an induced voltage
The flux can change due to
Changes in the magnetic field
Changes in the area
Changes in the angle
The voltmeter will indicate the direction of the induced emf and induced current and electric field
Section 21.2

13. Faraday’s Law, Summary Only changes in the magnetic flux matter
Rapid changes in the flux produce larger values of emf than do slow changes
This dependency on frequency means the induced emf plays an important role in AC circuits
The magnitude of the emf is proportional to the rate of change of the flux
If the rate is constant, then the emf is constant
In most cases, this isn’t possible and AC currents result
The induced emf is present even if there is no current in the path enclosing an area of changing magnetic flux
Section 21.2

14. Flux Though a Changing Area
A magnetic field is constant and in a direction perpendicular to the plane of the rails and the bar
Assume the bar moves at a constant speed
The magnitude of the induced emf is ε = B L v
The current leads to power dissipation in the circuit
Section 21.2

15. Conservation of Energy
The mechanical power put into the bar by the external agent is equal to the electrical power delivered to the resistor
Energy is converted from mechanical to electrical, but the total energy remains the same
Conservation of energy is obeyed by electromagnetic phenomena
Section 21.2

16. Electrical Generator
Need to make the rate of change of the flux large enough to give a useful emf
Use rotational motion instead of linear motion
A permanent magnet produces a constant magnetic field in the region between its poles
Section 21.2

17. Generator, cont. A wire loop is located in the region of the field
The loop has a fixed area, but is mounted on a rotating shaft
The angle between the field and the plane of the loop changes as the loop rotates
If the shaft rotates with a constant angular velocity, the flux varies sinusoidally with time
This basic design could generate about 70 V so it is a practical design
Section 21.2

18. Changing a Magnetic Flux, Summary
A change in magnetic flux and therefore an induced current can be produced in four ways
If the magnitude of the magnetic field changes with time
If the area changes with time
If the loop rotates so that the angle changes with time
If the loop moves from one region to another and the magnitude of the field is different in the two regions
Section 21.2

19. Changing a Magnetic Flux, Summary, cont.
Section 21.2

20. Lenz’s Law
Lenz’s Law gives an easy way to determine the sign of the induced emf
Lenz’s Law states the magnetic field produced by an induced current always opposes any changes in the magnetic flux
Section 21.3

21. Lenz’s Law, Example 1
Assume a metal loop in which the magnetic field passes upward through it
Assume the magnetic flux increases with time
The magnetic field produced by the induced emf must oppose the change in flux
Therefore, the induced magnetic field must be downward and the induced current will be clockwise
Section 21.3

22. Lenz’s Law, Example 2
Assume a metal loop in which the magnetic field passes upward through it
Assume the magnetic flux decreases with time
The magnetic field produced by the induced emf must oppose the change in flux
Therefore, the induced magnetic field must be upward and the induced current will be counterclockwise
Section 21.3

23. Problem Solving Strategy
Recognize the principle
The induced emf always opposes changes in flux through the Lenz’s Law loop or path
Sketch the problem
Show the closed path that runs along the perimeter of a surface crossed by the magnetic field lines
Identify
Is the magnetic flux increasing or decreasing with time?
Section 21.3

24. Problem Solving Strategy, cont.
Solve
Treat the perimeter of the surface as a wire loop
Suppose there is a current in the loop
Determine the direction of the resulting magnetic field
Find the current direction for which this induced magnetic field opposes the change in the magnetic flux
This current direction gives the sign (direction) of the induced emf
Check
Consider what your answer means
Check that your answer makes sense
Section 21.3

25. Lenz’s Law and Conservation of Energy
Mathematically, Lenz’s Law is just the negative sign in Faraday’s Law
It is actually a consequence of conservation of energy
Therefore, conservation of energy is contained in Faraday’s Law
Nowhere in the laws of electricity and magnetism is there any explicit mention of energy or conservation of energy
Physicists believe all laws of physics must satisfy the principle of conservation of energy
Section 21.3

26. Inductance In some cases, you must include the induced flux
When the switch is closed, a sudden change in current occurs in the coil
This current produces a magnetic field
An emf and current are induced in the coil
Section 21.4

27. Inductor A coil is type of circuit element called an inductor
Many inductors are constructed as small solenoids
Almost any coil or loop will act as an inductor
Whenever the current through an inductor changes, a voltage is induced in the inductor that opposes this change
This phenomenon is called self-inductance
The current changing through a coil induces a current in the same coil
The induced current opposes the original applied current, from Lenz’s Law
Section 21.4

28. Inductance of a Solenoid
Faraday’s Law can be used to find the inductance of a solenoid
L is the symbol for inductance
The voltage across the solenoid can be expressed in terms of the inductance
Section 21.4

29. Inductance, final The results apply to all coils or loops of wire
The value of L depends on the physical size and shape of the circuit element
The voltage drop across an inductor is
The unit of inductance is the Henry
1 H = 1 V . s / A
Section 21.4

30. Mutual Inductance
It is possible for the magnetic field of one coil to produce an induced current in a second coil
The coils are connected indirectly through the magnetic flux
The effect is called mutual inductance
Section 21.4

31. RL Circuit
DC circuits may contain resistors, inductors, and capacitors
The voltage source is a battery or some other source that provides a constant voltage across its output terminals
Behavior of DC circuits with inductors
Immediately after any switch is closed or opened, the induced emfs keep the current through all inductors equal to the values they had the instant before the switch was thrown
After a switch has been closed or opened for a very long time, the induced emfs are zero
Section 21.5

32. RL Circuit Example
Section 21.5

33. RL Circuit Example, Analysis
The presence of resistors and an inductor make the circuit an RL circuit
The current starts at zero since the switch has been open for a very long time
At t = 0, the switch is closed, inducing a potential across the inductor
Just after t = 0, the current in the second loop is zero
After the switch has been closed for a long time, the voltage across the inductor is zero
Section 21.5

34. Time Constant for RL Circuit
The current at time t is found by
τ is called the time constant of the circuit
For a single resistor in series with a single inductor, τ = L / R
The voltage is given by VL = V e-t/τ
Section 21.5

35. Real Inductors
Most practical inductors are constructed by wrapping a wire coil around a magnetic material
Filling a coil with magnetic material greatly increases the magnetic flux through the coil and therefore increases the induced emf
The presence of magnetic material increases the inductance
Most inductors contain a magnetic material inside which produces a larger value of L in a smaller package
Section 21.5

36. Energy in an Inductor
Energy is stored in the magnetic field of an inductor
The energy stored in an inductor is PEind = ½ L I2
Very similar in form to the energy stored in the electric field of a capacitor
The expression for energy can also be stated as
In terms of the magnetic field,
Section 21.6

37. Energy in an Inductor, cont.
Energy contained in the magnetic field actually exists anywhere there is a magnetic field, not just in a solenoid
Can exist in “empty” space
The potential energy can also be expressed in terms of the energy density in the magnetic field
This expression is similar to the energy density contained in an electric field
Section 21.6

38. Bicycle Odometers An odometer control unit is shown
A permanent magnet is attached to a wheel
A pickup coil is mounted on the axle support
When the magnet passes over the pickup coil, a pulse is generated
A computer keep tracks of the number of pulses
Section 21.7

39. Ground Fault Interrupters
A ground fault interrupter (GFI) is a safety device used in many household circuits
It uses Faraday’s Law along with an electromechanical relay
The relay uses the current through a coil to exert a force on a magnetic metal bar in a switch
Section 21.7

40. GFI, cont.
During normal operation, there is zero magnetic field in the relay
If the current in the return coil is smaller, a non-zero magnetic field opens the relay switch and the current turns off

41. Electric Guitars
An electric guitar uses Faraday’s Law to sense the motion of the strings
The metal string passes near a pickup coil wound around a permanent magnet
As the string vibrates, it produces a changing magnetic flux
The resulting emf is sent to an amplifier and the signal can be played through speakers
Section 21.7

42. Generators, Motors and Cars
Motors and generators provide examples of conservation of energy and the conversion of energy from one type to another
A hybrid car contains two motors and a generator
The hybrid car “recaptures” some of the energy normally converted to heat when braking and stores it in batteries
A hybrid car is a practical example of the conversion between mechanical and electrical energy
Section 21.7

43. Induction from a Distance
Assume a very long solenoid is inserted at the center of a single loop of wire
The field from the solenoid at the outer loop is essentially zero
Section 21.8

44. Induction from a Distance, cont.
The field inside the solenoid at the center of the loop still produces a magnetic flux through the inner portion of the loop
Energy is transferred across the empty space between the two conductors
The energy is carried from the solenoid to the outer loop by an electromagnetic wave
Section 21.8