Faraday’s Law and Inductance

Faraday’s Law• A moving magnet can exert a force on a

stationary charge.

• Faraday’s Law of Induction

• Induced emf is directly proportional to the change in the magnetic flux.

dt

dN m

AdBm

Remember:

Collapsing Field• Ex. What is the induced emf in a 100 turn

coil with an area of 0.035 m² when the magnetic field decreases linearly from 2.5 T to 0 T in 10 seconds?

B

Moving Loop• Ex. For a rectangular loop moving through

a magnetic field, plot the total flux, emf, and force on the loop with respect to time.

w

lv

B l = 20 cmw = 20 cmv = 5 m/sB = 0.75 Td = 60 cmR = 1 d

Motional EMF• For a moving conductor in a magnetic

field

or

With a resistor attached the current is

qvBqE

lvBV

R

lvBI

R vl

Lenz’s Law• Polarity of the induced emf is such that it

produces a current that will create a magnetic flux to oppose the change in magnetic flux through the loop.

vI

As the loop enters the fieldthe downward flux increases.Therefore, the current flowsin the loop to generate anupward flux to cancel thechange.

Induced EMF• A changing magnetic field always generates

an electric field.

• This electric field is non-conservative and time varying.

dt

dsdE m

Generators & Motors

• AC Generator - A spinning loop in a magnetic field generates an AC voltage.

• Eddy Currents - Currents generated within a conductor due to changes in the magnetic field.

tBAm cos tNBA sin

Induced EMF• Self induced emf is due to a

change in current.

• Inductance, L

• Units, henry (H) which is a V·s/A

dt

dIL

dt

dN m

I

NL m

Solenoid• From Ampere’s Law

• If the cross sectional area of the solenoid is A, then

• Inductance is then

dsl

INlBsdB 0

l

IANABm

0

l

AN

I

NL m 0

2

Solenoid (cont.)• Ex. What is the inductance of a solenoid of

500 turns where the length is 0.05 m and the diameter of the coil is 0.04 m?

)05.0(

)1026.1(/104)500( 232720

2

m

mAN

l

ANL

2322 1026.1)02.0( mmrA

HL 3109.7

RL Circuits• After the switch is closed,

Kirchoff’s rules gives

• Solution is0

dt

dILIR

/0

/ 11)( tLRt eIeR

tI

R

L

• If power supply is shorted out, then/

0)( teItI

R

L

RL Circuit Current

• How much current is flowing in a 10 0.1H RL circuit 15 ms after it is hooked up to a 10V power supply?

0

0.5I0

1.0I0

Current Flow

msH

1010

1.0

AeV

tI msms 78.0110

10)( 10/15

Energy in the Magnetic Field

• Power is voltage times current, but also the rate at which work is done.

dt

dIILIP

dt

dU

dt

dWP m

dt

dIIL

dt

dUm 221 LIUm

• Energy Density of a solenoid:

0

2

2B

lA

Uu m

m

Important E & M Equations

• Maxwell’s Equations

• Lorentz Force

0

QAdE

0AdB

dt

dIsdB e 000

dt

dsdE m

BvqEqF