Chapter 20

Induced Voltages and Inductance

Faraday’s Experiment• A primary coil is connected to a battery and a

secondary coil is connected to an ammeter

• The purpose of the secondary circuit is to detect current that might be produced by a (changing) magnetic field

• When there is a steady current in the primary circuit, the ammeter reads zero

Faraday’s Experiment• When the switch is opened, the ammeter reads a

current and then returns to zero

• When the switch is closed, the ammeter reads a current in the opposite direction and then returns to zero

• An induced emf is produced in the secondary circuit by the changing magnetic field

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 (defined similar to that of electrical 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

• For a loop of wire with an area A in a uniform magnetic field, the flux is (θ is the angle between B and the normal to the plane):

ΦB = BA = B A cos θ

Magnetic Flux• When the field is perpendicular to the plane of the

loop, θ = 0 and ΦB = ΦB, max = BA

• When the field is parallel to the plane of the loop, θ = 90° and ΦB = 0

• The flux can be negative, for example if θ = 180°

• SI unit of flux: Weber

• Wb = T. m²

Wilhelm Eduard Weber1804 – 1891

Magnetic Flux• The value of the magnetic flux is proportional to the

total number of magnetic field 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

Electromagnetic Induction

• When a magnet moves toward a loop of wire, the ammeter shows the presence of a current

• When the magnet moves away from the loop, the ammeter shows a current in the opposite direction

• When the magnet is held stationary, there is no current

• If the loop is moved instead of the magnet, a current is also detected

Electromagnetic Induction

• A current is set up in the circuit as long as there is relative motion between the magnet and the loop

• The current is called an induced current because is it produced by an induced emf

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:

tN B

Faraday’s Law and Lenz’ Law• Since ΦB = B A cos θ, the change in the flux, ΔΦB, can

be produced by a change in B, A or 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

tN B

Heinrich FriedrichEmil Lenz

1804 – 1865

Faraday’s Law and Lenz’ Law• Example

• The magnetic field, B, becomes smaller with time and this reduces the flux

• The induced current will produce an induced field, Bind, in the same direction as the original field

tN B

Chapter 20Problem 14

A square, single-turn wire loop 1.00 cm on a side is placed inside a solenoid that has a circular cross section of radius 3.00 cm, as shown in Figure P20.14. The solenoid is 20.0 cm long and wound with 100 turns of wire. (a) If the current in the solenoid is 3.00 A, find the flux through the loop. (b) If the current in the solenoid is reduced to zero in 3.00 s, find the magnitude of the average induced emf in the loop.

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

• As the negative charges accumulate at the base, a net positive charge exists at the upper end of the conductor

Motional emf• 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

q E = q v B; E = v B;

• There is a potential difference between the upper and lower ends of the conductor

Motional emf• The potential difference between the

ends of the conductor (the upper end is at a higher potential than the lower end):

ΔV = E ℓ = B ℓ v• 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

Motional emf in a Circuit• As the bar (with zero resistance) is

pulled to the right with a constant 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

• The changing magnetic flux through the loop and the corresponding induced emf in the bar result from the change in area of the loop

Motional emf in a Circuit• The induced, motional emf, acts like a

battery in the circuit

• As the bar moves to the right, the magnetic flux through the circuit increases with time because the area of the loop increases

• The induced current must be in a direction such that it opposes the change in the external magnetic flux (Lenz’ Law)

B vB v and I

R

Motional emf in a Circuit• The flux due to the external field is

increasing into the page

• The flux due to the induced current must be out of the page

• Therefore the current must be counterclockwise when the bar moves to the right

• If the bar is moving toward the left, the magnetic flux through the loop is decreasing with time – the induced current must be clockwise to produce its own flux into the page

Chapter 20Problem 57

A conducting rod of length ℓ moves on two horizontal frictionless rails. 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?

Lenz’ Law – Moving Magnet Example• As the bar magnet is moved to the right toward a

stationary loop of wire, the magnetic flux increases with time

• The induced current produces a flux to the left, so the current is in the direction shown

• When applying Lenz’ Law, there are two magnetic fields to consider: changing external and induced

AC Generators

• Alternating Current (AC) generators convert mechanical energy to electrical energy

• Consist of a wire loop rotated by some external means (falling water, heat by burning coal to produce steam, etc.)

• As the loop rotates, the magnetic flux through it changes with time inducing an emf and a current in the external circuit

AC Generators

• 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

• The emf generated by the rotating loop can be found by ε = 2 B ℓ v= 2 B ℓ v sin θ

• If the loop rotates with a constant angular speed, ω,

and N turns ε = N B A ω sin ω t

AC Generators• The magnetic force on the charges in the wires AB and

CD is perpendicular to the length of the wires

• An emf is generated in wires BC and AD

• The emf produced in each of these wires is ε = B ℓ v= = B ℓ v sin θ

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

• The output voltage always has the same polarity

• The current is a pulsing current

DC Generators

• 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

Motors

• Motors are devices that convert electrical energy into mechanical energy (generators run in reverse)

• A motor can perform useful mechanical work when a shaft connected to its rotating coil is attached to some external device

• As the coil begins to rotate, the induced back emf opposes the applied voltage and the current in the coil is reduced

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 inducing 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

Self-inductance• The self-induced emf must be proportional to the time

rate of change of the current

• L: inductance of a coil (depends on geometric factors)

• The negative sign indicates that a changing current induces an emf in opposition to that change

• The SI unit of self-inductance: Henry

• 1 H = 1 (V · s) / A

IL

t

tN B

t

I

IN B

I

NL B

Joseph Henry1797 – 1878

Chapter 20Problem 40

An emf of 24.0 mV is induced in a 500-turn coil when the current is changing at a rate of 10.0 A/s. What is the magnetic flux through each turn of the coil at an instant when the current is 4.00 A?

Inductor in a Circuit

• Inductance can be interpreted as a measure of opposition to the rate of change in the current (while resistance 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

• As a result, the current doesn’t change from 0 to its maximum instantaneously

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:

• The current can be found by:

LR

/1 tI eR

Chapter 20Problem 46

Consider the circuit shown in the figure. 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?

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

Answers to Even Numbered Problems

Chapter 20:

Problem 10

34 mV

Answers to Even Numbered Problems

Chapter 20:

Problem 18

1.00 ms

Answers to Even Numbered Problems

Chapter 20:

Problem 28

(a) left to right(b) no induced current(c) right to left

Answers to Even Numbered Problems

Chapter 20:

Problem 30

13 mV

Answers to Even Numbered Problems

Chapter 20:

Problem 42

(a) 1.00 kΩ(b) 3.00 ms