Lecture 26: WED 18 Lecture 26: WED 18 MARMARCh30Ch30..4–6

Induction and InductanceInduction and Inductance

Physics 2102Spring 2007

Jonathan Dowling

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are needed to see this picture.

A Changing Magnetic Field Produces a Current in a Wire Loop

dA

Faraday’s LawFaraday’s Law• A time varying magnetic

FLUX creates an induced EMF

• Definition of magnetic flux is similar to definition of electric flux

B

€

E =−dΦB

dt

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ΦB =r B ⋅d

v A

S∫

• Take note of the MINUS sign!!

• The induced EMF acts in such a way that it OPPOSES the change in magnetic flux (“Lenz’s Law”).

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dv A

Some ApplicationsSome ApplicationsMagLev train relies on Faraday’s Law: currents induced in non-magnetic rail tracks repel the moving magnets creating the induction; result: levitation!

ELECTRIC Guitar pickups also use Faraday’s Law — a vibrating string modulates the flux through a coil hence creating an electrical signal at the same frequency.

QuickTime™ and a decompressor

are needed to see this picture.Fender StratocasterSolenoid Pickup

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Start Your Engines: The Start Your Engines: The Ignition CoilIgnition Coil

• The gap between the spark plug in a combustion engine needs an electric field of ~107 V/m in order to ignite the air-fuel mixture. For a typical spark plug gap, one needs to generate a potential difference > 104 V!

• But, the typical EMF of a car battery is 12 V. So, how does a spark 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 changes the current through “primary coil”• Result: LARGE change in flux thru secondary -- large induced EMF!

Start Your Engines: The Start Your Engines: The Ignition CoilIgnition Coil

Transformer: P=iV

dA

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

• We saw that a time varying magnetic FLUX creates an induced EMF in a wire, exhibited as a current.

• Recall that a current flows in a conductor because of electric field.

• Hence, a time varying magnetic flux must induce an ELECTRIC FIELD!

• A Changing B-Field Produces an E-Field in Empty Space!

B

dtdsdE B

C

Φ−=⋅∫

rr

To decide SIGN of flux, use right hand rule: curl fingers around loop, +flux -- thumb.

€

dv A

€

E ∝ i

ExampleExample • The figure shows two circular regions R1 &

R2 with radii r1 = 1m & r2 = 2m. In R1, the magnetic field B1 points out of the page. In R2, the magnetic field B2 points into the page.

• Both fields are uniform and are DECREASING at the SAME steady rate = 1 T/s.

• Calculate the “Faraday” integral for the two paths shown.

R1 R2

∫ ⋅C

sdE rrI

II

VsTrdt

dsdE B

C

14.3)/1)(( 21 +=−−=

Φ−=⋅∫ π

rrPath I:

[ ] VsTrsTrsdEC

42.9)/1)(()/1)(( 22

21 +=−+−−−=⋅∫ ππ

rrPath II:

B1

B2

Solenoid Example Solenoid 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 the axis of the solenoid.

First, let’s look at r < R:

dtdBrrE )()2( 2ππ =

dtdBrE

2=

Next, let’s look at r > R:

dtdBRrE )()2( 2ππ =

dtdB

rRE2

2

=

magnetic field lines

electric field lines

B

Solenoid Example Cont.Solenoid Example Cont.

r

E(r)

r = R

Er<R =r2dBdt

∝ r

Er>R =R 2

2rdBdt

∝1 / r

Added Complication: Changing B Field Is Produced by Changing Current i in the Loops of Solenoid!

B =μ0nidBdt

=μ0ndidt

i

i

EB

Er<R =μ0nr2

didt

Er>R =μ0nR

2

2rdidt

SummarySummaryTwo versions of Faradays’ law:

– A time varying magnetic flux produces an EMF:

€

E =−dΦB

dt

dtdsdE B

C

Φ−=⋅∫

rr–A time varying magnetic flux produces an electric field: