1. Induced Voltages and Inductance
Chapter 20
Induced Voltages and Inductance

2. Michael Faraday 1791 – 1867 Great experimental scientist
Invented electric motor, generator and transformers
Discovered electromagnetic induction
Discovered laws of electrolysis

3. Faraday’s Experiment – Set Up
A current can be produced by a changing magnetic field
First shown in an experiment by Michael Faraday
A primary coil is connected to a battery
A secondary coil is connected to an ammeter

4. Faraday’s Experiment
The purpose of the secondary circuit is to detect current that might be produced by the magnetic field
When the switch is closed, the ammeter reads a current and then returns to zero
When the switch is opened, the ammeter reads a current in the opposite direction and then returns to zero
When there is a steady current in the primary circuit, the ammeter reads zero

5. Faraday’s Conclusions
An electrical current is produced by a changing magnetic field
The secondary circuit acts as if a source of emf were connected to it for a short time
It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field

6. Magnetic Flux
The emf is actually induced by a change in the quantity called the magnetic flux rather than simply by a change in the magnetic field
Magnetic flux is defined in a manner similar to that of electrical flux
Magnetic flux is proportional to both the strength of the magnetic field passing through the plane of a loop of wire and the area of the loop

7. Magnetic Flux, 2 You are given a loop of wire
The wire is in a uniform magnetic field
The loop has an area A
The flux is defined as
ΦB = BA = B A cos θ
θ is the angle between B and the normal to the plane

8. Magnetic Flux, 3
When the field is perpendicular to the plane of the loop, as in a, θ = 0 and ΦB = ΦB, max = BA
When the field is parallel to the plane of the loop, as in b, θ = 90° and ΦB = 0
The flux can be negative, for example if θ = 180°
SI units of flux are T. m² = Wb (Weber)

9. Magnetic Flux, final
The flux can be visualized with respect to magnetic field lines
The value of the magnetic flux is proportional to the total number of lines passing through the loop
When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum
When the area is parallel to the lines, no lines pass through the area and the flux is 0

10. Example 1
A square loop 2.00 m on a side is placed in a magnetic field of magnitude T. If the field makes an angle of 50.0° with the normal to the plane of the loop, find the magnetic flux through the loop.

11. Example 2
A solenoid 4.00 cm in diameter and 20.0 cm long has 250 turns and carries a current of 15.0 A. Calculate the magnetic flux through the circular cross-sectional area of the solenoid.

12. Practice 1
Find the flux of the Earth’s magnetic field of magnitude 5.00 × 10−5 T through a square loop of area 20.0 cm2 (a) when the field is perpendicular to the plane of the loop, (b) when the field makes a 40.0° angle with the normal to the plane of the loop, and (c) when the field makes a 90.0° angle with the normal to the plane.

13. Electromagnetic Induction – An Experiment
When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a)
When the magnet is held stationary, there is no current (b)
When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c)
If the loop is moved instead of the magnet, a current is also detected

14. Electromagnetic Induction – Results of the Experiment
A current is set up in the circuit as long as there is relative motion between the magnet and the loop
The same experimental results are found whether the loop moves or the magnet moves
The current is called an induced current because is it produced by an induced emf

15. Faraday’s Law and Electromagnetic Induction
The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit
If a circuit contains N tightly wound loops and the flux changes by ΔΦB during a time interval Δt, the average emf induced is given by Faraday’s Law:

16. Faraday’s Law and Lenz’ Law
The change in the flux, ΔΦB, can be produced by a change in B, A or θ
Since ΦB = B A cos θ
The negative sign in Faraday’s Law is included to indicate the polarity of the induced emf, which is found by Lenz’ Law
The current caused by the induced emf travels in the direction that creates a magnetic field with flux opposing the change in the original flux through the circuit

17. Example 3
A 300-turn solenoid with a length of 20 cm and a radius of 1.5 cm carries a current of 2.0 A. A second coil of four turns is wrapped tightly about this solenoid so that it can be considered to have the same radius as the solenoid. Find (a) the change in the magnetic flux through the coil and (b) the magnitude of the average induced emf in the coil when the current in the solenoid increases to 5.0 A in a period of 0.90 s.

18. Example 4
A circular coil enclosing an area of 100 cm2 is made of 200 turns of copper wire. The wire making up the coil has resistance of 5.0 Ω, and the ends of the wire are connected to form a closed circuit. Initially, a 1.1-T uniform magnetic field points perpendicularly upward through the plane of the coil. The direction of the field then reverses so that the final magnetic field has a magnitude of 1.1 T and points downward through the coil. If the time required for the field to reverse directions is 0.10 s, what average current flows through the coil during that time?

19. Application of Faraday’s Law – Motional emf
A straight conductor of length ℓ moves perpendicularly with constant velocity through a uniform field
The electrons in the conductor experience a magnetic force
F = q v B
The electrons tend to move to the lower end of the conductor

20. Motional emf
As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor
As a result of this charge separation, an electric field is produced in the conductor
Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force
There is a potential difference between the upper and lower ends of the conductor

21. Motional emf, cont
The potential difference between the ends of the conductor can be found by
ΔV = B ℓ v
The upper end is at a higher potential than the lower end
A potential difference is maintained across the conductor as long as there is motion through the field
If the motion is reversed, the polarity of the potential difference is also reversed

22. Motional emf in a Circuit
Assume the moving bar has zero resistance
As the bar is pulled to the right with a given velocity under the influence of an applied force, the free charges experience a magnetic force along the length of the bar
This force sets up an induced current because the charges are free to move in the closed path

23. Motional emf in a Circuit, cont
The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop
The induced, motional emf, acts like a battery in the circuit

24. Example 5
A conducting rod of length ℓ moves on two horizontal frictionless rails, as in Figure P A constant force of magnitude 1.00 N moves the bar at a uniform speed of 2.00 m/s through a magnetic field that is directed into the page. (a) What is the current in an 8.00-Ω resistor R? (b) What is the rate of energy dissipation in the resistor? (c) What is the mechanical power delivered by the constant force?

25. Example 6
A helicopter has blades of length 3.0 m, rotating at 2.0 rev/s about a central hub. If the vertical component of Earth’s magnetic field is 5.0 × 10−5 T, what is the emf induced between the blade tip and the central hub?

26. Practice 2
A 12.0-m-long steel beam is accidentally dropped by a construction crane from a height of 9.00 m. The horizontal component of the Earth’s magnetic field over the region is 18.0 μT. What is the induced emf in the beam just before impact with the Earth? Assume the long dimension of the beam remains in a horizontal plane, oriented perpendicular to the horizontal component of the Earth’s magnetic field.

27. Lenz’ Law – Moving Magnet Example
A bar magnet is moved to the right toward a stationary loop of wire (a)
As the magnet moves, the magnetic flux increases with time
The induced current produces a flux to the left, so the current is in the direction shown (b)

28. Lenz’ Law, Final Note
When applying Lenz’ Law, there are two magnetic fields to consider
The external changing magnetic field that induces the current in the loop
The magnetic field produced by the current in the loop

29. Example 7
A copper bar is moved to the right while its axis is maintained in a direction perpendicular to a magnetic field, as shown in Figure P If the top of the bar becomes positive relative to the bottom, what is the direction of the magnetic field?

30. Generators Alternating Current (AC) generator
Converts mechanical energy to electrical energy
Consists of a wire loop rotated by some external means
There are a variety of sources that can supply the energy to rotate the loop
These may include falling water, heat by burning coal to produce steam

31. AC Generators, cont Basic operation of the generator
As the loop rotates, the magnetic flux through it changes with time
This induces an emf and a current in the external circuit
The ends of the loop are connected to slip rings that rotate with the loop
Connections to the external circuit are made by stationary brushes in contact with the slip rings

32. AC Generators, final
The emf generated by the rotating loop can be found by
ε =2 B ℓ v=2 B ℓ sin θ
If the loop rotates with a constant angular speed, ω, and N turns
ε = N B A ω sin ω t
ε = εmax when loop is parallel to the field
ε = 0 when when the loop is perpendicular to the field

33. DC Generators
Components are essentially the same as that of an ac generator
The major difference is the contacts to the rotating loop are made by a split ring, or commutator

34. DC Generators, cont The output voltage always has the same polarity
The current is a pulsing current
To produce a steady current, many loops and commutators around the axis of rotation are used
The multiple outputs are superimposed and the output is almost free of fluctuations

35. Motors
Motors are devices that convert electrical energy into mechanical energy
A motor is a generator run in reverse
A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

36. Motors and Back emf
The phrase back emf is used for an emf that tends to reduce the applied current
When a motor is turned on, there is no back emf initially
The current is very large because it is limited only by the resistance of the coil

37. Motors and Back emf, cont
As the coil begins to rotate, the induced back emf opposes the applied voltage
The current in the coil is reduced
The power requirements for starting a motor and for running it under heavy loads are greater than those for running the motor under average loads

38. Example 8
A 100-turn square wire coil of area m2 rotates about a vertical axis at 1 500 rev/min, as indicated in Figure P The horizontal component of the Earth’s magnetic field at the location of the loop is 2.0 × 10−5 T. Calculate the maximum emf induced in the coil by the Earth’s field.

39. Example 9
A motor has coils with a resistance of 30 Ω and operates from a voltage of 240 V. When the motor is operating at its maximum speed, the back emf is 145 V. Find the current in the coils (a) when the motor is first turned on and (b) when the motor has reached maximum speed. (c) If the current in the motor is 6.0 A at some instant, what is the back emf at that time?

40. Self-inductance
Self-inductance occurs when the changing flux through a circuit arises from the circuit itself
As the current increases, the magnetic flux through a loop due to this current also increases
The increasing flux induces an emf that opposes the change in magnetic flux
As the magnitude of the current increases, the rate of increase lessens and the induced emf decreases
This opposing emf results in a gradual increase of the current

41. Self-inductance cont
The self-induced emf must be proportional to the time rate of change of the current
L is a proportionality constant called the inductance of the device
The negative sign indicates that a changing current induces an emf in opposition to that change

42. Self-inductance, final
The inductance of a coil depends on geometric factors
The SI unit of self-inductance is the Henry
1 H = 1 (V · s) / A
You can determine an expression for L

43. Example 10
A coiled telephone cord forms a spiral with 70.0 turns, a diameter of 1.30 cm, and an unstretched length of 60.0 cm. Determine the self-inductance of one conductor in the unstretched cord.

44. Joseph Henry 1797 – 1878 First director of the Smithsonian
First president of the Academy of Natural Science
First to produce an electric current with a magnetic field
Improved the design of the electro-magnetic and constructed a motor
Discovered self-inductance

45. Inductor in a Circuit
Inductance can be interpreted as a measure of opposition to the rate of change in the current
Remember resistance R is a measure of opposition to the current
As a circuit is completed, the current begins to increase, but the inductor produces an emf that opposes the increasing current
Therefore, the current doesn’t change from 0 to its maximum instantaneously

46. RL Circuit
When the current reaches its maximum, the rate of change and the back emf are zero
The time constant, , for an RL circuit is the time required for the current in the circuit to reach 63.2% of its final value

47. RL Circuit, cont The time constant depends on R and L
The current at any time can be found by

48. Example 11
Consider the circuit shown in Figure P Take ε = 6.00 V, L = 8.00 mH, and R = 4.00 Ω. (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250 μs after the switch is closed. (c) What is the value of the final steady-state current? (d) How long does it take the current to reach 80.0% of its maximum value?

49. Energy Stored in a Magnetic Field
The emf induced by an inductor prevents a battery from establishing an instantaneous current in a circuit
The battery has to do work to produce a current
This work can be thought of as energy stored by the inductor in its magnetic field
PEL = ½ L I2

50. Example 12
A 300-turn solenoid has a radius of 5.00 cm and a length of 20.0 cm. Find (a) the inductance of the solenoid and (b) the energy stored in the solenoid when the current in its windings is A.

51. Practice 3
How much energy is stored in a 70.0-mH inductor at an instant when the current is 2.00 A?