Chapter 20Chapter 20

Induced Voltages and Inductance

General Physics

Inductors & RL Circuits

Sections 5–8

General Physics

Generators

Alternating Current (AC) and Direct Current (DC) generators– Converts mechanical energy to electrical energy– Consists of a wire loop rotated through a magnetic

field 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

General Physics

AC Generators As the loop rotates (θ changes), the

magnetic flux through the loop changes with time

This induces an emf and a current in the external circuit (toaster)

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 output voltage oscillates between positive and negative polarity

The current is an AC current

General Physics

AC Generators – Rotating Loop Wires BC and AD act as bars moving

vertically through the horizontal magnetic field between the N and S poles.

An emf is generated in wires BC and AD

The total emf produced in these 2 wires is ε = 2 B ℓ v= 2 B ℓ v sin θ

If the loop rotates with a constant angular speed, ω, the emf generated by the rotating loop is ε =2 B ℓ (a / 2) ω sin ωt = B A ω sin ωt

If a coil has N turns, the emf is N times as large ε = N B A ω sin ω t

Active Figure: AC Generator

General Physics

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 DC pulsing current

Active Figure: DC Generator

General Physics

Motors Motors are devices that

convert electrical energy (through magnetic forces) 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

General Physics

Motors and Back emf

As the motor rotates, the magnetic flux through the loop changes with time

This induces a back emf that tends to reduce the current applied to the motor from the external source

When a motor is first turned on, the current is very large because there is no back emf initially

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

General Physics

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

General Physics

Self-inductance Self-inductance occurs when the

changing flux through a circuit arises from the circuit itself– When the switch is closed, the

current increases from zero– 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 rather than a sharp increase

General Physics

Self-inductance, cont The self-induced emf is proportional to

the time rate of change of the current

– L is a proportionality constant called the self-inductance of the circuit or device

– The SI unit of self-inductance is the Henry1 H = 1 (V · s) / A

– The negative sign indicates that a changing current induces an emf in opposition to that change – Lenz’s law

IL

t

General Physics

Self-inductance, cont The inductance of a coil depends on

geometric factors You can determine L from the

expression

For a solenoid the inductance is

I

NL B

l

rNL

220

General Physics

Self-Inductance and Lenz’ Law

Consider an increasing current through the inductor

The self-induced emf has a direction so as to oppose the increase in the current

Consider a decreasing current through the inductor

The self-induced emf has an opposite direction so as to oppose the decrease in the current

General Physics

Inductor in a Circuit – RL Circuit When the switch is closed, the current

in the RL circuit increases from zero The increasing current induces an emf

in the inductor that opposes the change in the current

As the magnitude of the current increases, the rate of increase lessens and the self-induced emf decreases

When the current reaches its maximum, the rate of change and the self-induced emf become 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

General Physics

RL Circuit, cont

The time constant depends on R and L

The current at any time can be found by

LR

/1 tI eR

Active Figure: An RL Circuit

General Physics

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 results in energy being stored by the

inductor in its magnetic field

PEL = ½ L I2

– Note that this result is similar to the expression for the energy stored by a capacitor in its electric field

PEC = ½ C ΔV2