Chapter 20Self-Inductance

LR CircuitsMotional EMF

V R+

_x x x x x xx x x x x x

What happens when the switch is first closed?

Initially, there is no current in the circuit.

We know the final current in thiscircuit (from Ohm’s Law) will be I = V / R

So, for some period of time, the current ischanging from 0 to V / R.

Changing Currents Changing MagneticFields

V R+

_x x x x x xx x x x x x

As a result of the changing current, themagnetic field that the current createschanges as well.

That magnetic field passes through the planeof this circuit.

There is a magnetic flux through the circuit.

The magnetic flux is changing as thecurrent increases.

An EMF is induced in the circuit as thecurrent increases!

And by Lenz’s Law, the induced EMFcreates a current which opposes thechanges in the magnetic flux.

x x x x x xx x x x x x

V R+

_

x x x x x xx x x x x x

V R+

_

Induced emf

The very fact that the circuit is a closedloop made of conducting material hasresulted in an induced EMF in the circuitwhich opposes the current!

Of course, once the current attains its finalvalue, the induced EMF = 0 because themagnetic field is no longer changing.

x x x x x xx x x x x x

V R+

_

For the circuits we deal with in class, wegenerally ignore the self-inductance of thecircuit itself.

However, you might observe its effect on the circuits you construct in the laboratory.

When you first put voltageacross the circuit, the currentshould ramp up to its finalvalue...

We will, however, pay attention to theself-inductance of a solenoid.

When present as a circuit element, we callsolenoids INDUCTORS.

To power supply To power supply

What is an inductor (solenoid)?

A bunch of current loops connected together!

So a magnetic flux exists through the inductor.

As was the case with the electrical circuit of thelast example, an induced EMF will be generatedacross the inductor when the current throughthe inductor changes.

Again, by Lenz’s Law, the induced EMF will generate a current to oppose the changing magnetic flux through the inductor.

What is the magnetic flux through an inductorof length l with N turns and a current I flowing through it?

NBA NN

IAN

AI 0 0

2

l l

What is the induced EMF through such a coil?

t

NA

I

t0

2

l

LI

t

Where L is the inductanceand is defined to be

LN

A0

2

l

More generally, to define theinductance we can use

Nt

AND

LI

t

LN

A0

2

lTRUE ONLY FORA SOLENOID!!!!

That is, equate these two statements of therelationship of EMF to changing fluxes andchanging currents.

L NI

[L] N]I]

Wb

A [

[ ]

[

turns

[L]Tm

A

2

NAmm

A

2

[L]Nm

A

Js

AC

Vs

A

2

General expression forself-inductance:

These are actual physics demonstrationstaken from the closets of Chem/Phys

and are meant for exhibition purposes,not for competition.

NO WAGERING, PLEASE!

Induced current due to changingmagnetic flux.

LENZ’S LAW

Inductors set up a back-EMF in a circuit asthe current in the circuit changes. Themagnitude of that EMF is given by

As a circuit element, therefore, inductorsaffect the rate at which the current in acircuit changes.

In some respects, their effect on a circuitis analogous to the role of a capacitor.

LI

t

Recall, capacitors are circuit elements whichstore energy in an electric field between apositively charged plate and a negativelycharged plate.

Inductors store energy in a magnetic field.

The amount of energy stored in the magneticfield of an inductor is given by

U LI1

22

V R+

_L

I

t

I=V/R

I=0.632V/R

I I( ) ( )/t e t max 1

Imax V R/

= the time constant = L / R

We can use the loop rule and Ohm’s lawto determine the voltage across the inductor.

V = VL + VR

V IR ( ) ( ) ( )/ /t e et t I R RV

R Rmax 1 1

VR ( ) ( )/t e t V 1

VL ( ) ( )/ /t e et t V V V1

Induced Back - EMF in the Inductor

x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x x

Lv

What happens to the charges in this conducting bar as it moves with a constant velocity v through a uniform magnetic field?

The positive charges within the conductor feela magnetic force upwards.

The negative charges feel a downwards force.

x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x x

Lv

++++++

------

F+

F-

How long can charge be separated this way?

Until the electrostatic forcebalances the

magnetic force!

x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x x

Lv

++++++

------

F+

F-

FE = FB

FE = q E

FB = q v B

E = v B

x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x x

Lv

What is the magnitude of the potential difference between the ends of the bar when the electrostatic and magnetic forces are in balance?

|V| = E d = B L v