Lecture 26: WED 18 MARLecture 26: WED 18 MARCh30Ch30..4–6

Induction and Inductance Induction and Inductance

Physics 2102Spring 2007

Jonathan Dowling

A Changing Magnetic Field Produces a Current in a Wire Loop

dA

FaradayFaraday’’s Laws Law• A time varying magnetic FLUX

creates an induced EMF• Definition of magnetic flux is

similar to definition of electricflux

B

�

E = !d"

B

dt

�

!B

=! B " d" A

S

#• Take note of the MINUS sign!!• The induced EMF acts in such away that it OPPOSES thechange in magnetic flux (“Lenz’sLaw”).

�

d! A

Some ApplicationsSome ApplicationsMagLev train relies onFaraday’s Law: currentsinduced in non-magnetic railtracks repel the movingmagnets creating the induction;result: levitation!

ELECTRIC Guitar pickups also use Faraday’sLaw — a vibrating string modulates the fluxthrough a coil hence creating an electricalsignal at the same frequency.

Fender StratocasterSolenoid Pickup

Start Your Engines: The Ignition CoilStart Your Engines: The Ignition Coil• The gap between the spark plug

in a combustion engine needs anelectric field of ~107 V/m inorder to ignite the air-fuelmixture. For a typical spark pluggap, one needs to generate apotential difference > 104 V!

• But, the typical EMF of a carbattery is 12 V. So, how does aspark plug work??

spark

12V

The “ignition coil” is a double layer solenoid:• Primary: small number of turns -- 12 V• Secondary: MANY turns -- spark plug

http://www.familycar.com/Classroom/ignition.htm

•Breaking the circuit changesthe current through “primarycoil”• Result: LARGE change in fluxthru secondary -- largeinduced EMF!

Start Your Engines: The Ignition CoilStart Your Engines: The Ignition Coil

Transformer: P=iV

dA

Changing B-Field Produces E-Field!Changing B-Field Produces E-Field!

• We saw that a time varyingmagnetic FLUX creates aninduced EMF in a wire,exhibited as a current.

• Recall that a current flows ina conductor because ofelectric field.

• Hence, a time varyingmagnetic flux must induce anELECTRIC FIELD!

• A Changing B-Field Producesan E-Field in Empty Space!

B

dt

dsdE

B

C

!"=#$

!!

To decide SIGN of flux, useright hand rule: curl fingersaround loop, +flux -- thumb.

�

d! A

�

E ! i

ExampleExample• The figure shows two circular

regions R1 & R2 with radii r1 =1m & r2 = 2m. In R1, themagnetic field B1 points out ofthe page. In R2, the magneticfield B2 points into the page.

• Both fields are uniform and areDECREASING at the SAMEsteady rate = 1 T/s.

• Calculate the “Faraday”integral for the two pathsshown.

R1 R2

! "C

sdE!!

I

II

VsTrdt

dsdE

B

C

14.3)/1)((2

1 +=!!="

!=#$ %!!

Path I:

[ ] VsTrsTrsdE

C

42.9)/1)(()/1)((2

2

2

1 +=!+!!!="# $$!!

Path II:

B1

B2

Solenoid ExampleSolenoid Example• A long solenoid has a circular cross-

section of radius R.• The magnetic field B through the solenoid

is increasing at a steady rate dB/dt.• Compute the variation of the electric field

as a function of the distance r from theaxis of the solenoid.

First, let’s look at r < R:

dt

dBrrE )()2( 2

!! =

dt

dBrE

2=

Next, let’s look at r > R:

dt

dBRrE )()2( 2

!! =

dt

dB

r

RE

2

2

=

magnetic field lines

electric field lines

B

Solenoid Example Cont.Solenoid Example Cont.

r

E(r)

r = R

Er<R

=r

2

dB

dt

! r

Er>R

=R2

2r

dB

dt

!1 / r

Added Complication: Changing B FieldIs Produced by Changing Current i inthe Loops of Solenoid!

B = µ0ni

dB

dt= µ

0ndi

dt

i

i

E

B

Er<R =

µ0nr

2

di

dtEr>R =

µ0nR

2

2r

di

dt

SummarySummaryTwo versions of Faradays’ law:

– A time varying magnetic flux produces an EMF:

�

E = !d"

B

dt

dt

dsdE

B

C

!"=#$

!!

–A time varying magnetic flux produces anelectric field: