Chapter 20

Induced Voltages and Inductance

Michael Faraday 1791 – 1867 Great experimental

scientist Invented electric

motor, generator and transformers

Discovered electromagnetic induction

Discovered laws of electrolysis

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

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

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

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

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 = BA = B A cos θ

θ is the angle between B and the normal to the plane

B

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)

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

Example 1

A square loop 2.00 m on a side is placed in a magnetic field of magnitude 0.300 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.

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.

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.

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

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

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 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

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.

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?

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

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

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

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

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

B vB v and I

R

Example 5A conducting rod of length ℓ moves on two horizontal frictionless rails, as in Figure P20.18. 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?

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?

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.

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)

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

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 P20.27. If the top of the bar becomes positive relative to the bottom, what is the direction of the magnetic field?

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

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

AC Generators, final The emf generated by

the rotating loop can be found byε =2 B ℓ v=2 B ℓ sin θ

If the loop rotates with a constant angular speed, ω, and N turnsε = 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

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

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

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

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

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

Example 8

A 100-turn square wire coil of area 0.040 m2 rotates about a vertical axis at 1 500 rev/min, as indicated in Figure P20.30. 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.

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?

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

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

IL

t

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

B BNL N

I I

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.

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

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

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

RL Circuit, cont The time constant depends on R

and L

The current at any time can be found by

LR

/1 tI eR

Example 11

Consider the circuit shown in Figure P20.46. 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

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 0.500 A.

Practice 3

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