dt

demf mag

Faraday’s law cannot be derived from the other fundamental principles we have studied

Formal version of Faraday’s law:

Sign: given by right hand rule

Faraday’s Law

Differential form ofFaraday’s law:

∮𝐸 ∙𝑑 �⃗�=−𝑑𝑑𝑡 [∫ �⃗� ∙ �̂�𝑑 𝐴 ]

�⃗�× �⃗�=−𝜕 �⃗�𝜕𝑡

curl(

dt

demf mag

Two ways to produce curly electric field:1. Changing B2. Changing A

ABdt

d

dt

d mag

dt

dABA

dt

dB

Two Ways to Produce Changing

Constant voltage – constant I, nocurly electric field.

Increase voltage: dB/dt is notzero emf

d

NIB 0For long solenoid:

Change current at rate dI/dt:

20

1 Rd

NI

dt

d

dt

demf mag

dt

dIR

d

N 20 (one loop)

dt

dIR

d

Nemf 2

20

emfbat

Remfcoil

Inductance

emfbat

Remfcoil

dt

dIR

d

Nemf 2

20

EC

Increasing I increasing B

dt

demf mag

ENC

dt

dILemf ind

emfbat

R

emfind

L – inductance, or self-inductance

22

0 Rd

NL

Inductance

ENC

EC

emfbat

R

emfind

L

dt

dILemf ind

IremfV solindsol

22

0 Rd

NL

Unit of inductance L: Henry = Volt.second/Ampere

Inductance

Increasing the current causes ENC to oppose this increase

EC

dt

demf mag

ENC

emfbat

R

emfind

L

dt

dILemf ind

Conclusion: Inductance resists changes in current

Inductance: Decrease Current

Orientation of emfind depends on sign of dI/dt

202

1E

Volume

energy Electric

)()(dt

dILIemfIVIP

∫∫ f

i

I

I

IdILPdtEnergy

22

2

1

2

1LILIEnergy

f

i

I

I

22

0 Rd

NL

d

NIB 0

2

0

22

0

2

1

N

BdR

d

NEnergy

dR

BEnergy 2

0

2

2

1

VBEnergy 2

0

1

2

1

2

0

1

2

1B

Volume

energy Magnetic

Magnetic Field Energy Density?

L I2

202

1E

Volume

energy Electric

2

0

1

2

1B

Volume

energy Magnetic

2

0

20

1

2

1

2

1BE

Volume

Energy

Electric and magnetic field energy density:

Field Energy Density

0 inductorresistorbattery VVV

0dt

dILRIemfbattery

ctbeatI )(

0 ctctbattery LbceRbeRaemf

R

emfa battery LbcRb

L

Rc

tL

Rbattery beR

emftI

)(

If t is very long:R

emftI battery )(

Current in RL Circuit

tL

Rbattery beR

emftI

)(

If t is zero: 0)0( I

01)0( bR

emfI battery

R

emfb battery

tL

Rbattery eR

emftI 1)(

Current in RL circuit:

Current in RL Circuit

tL

Rbattery eR

emftI 1)(

Current in RL circuit:

Time constant: time in which exponential factor become 1/e

1tL

R

R

L

Time Constant of an RL Circuit

0 inductorcapacitor VV

0dt

dIL

C

Q

dt

dQI

02

2

dt

QdLCQ

ctbaQ cos

0coscos 2 ctbcLCctba

a=0LC

c1

LC

tbQ cos

LC

tQQ cos0

Current in an LC Circuit

LC

tQQ cos0

dt

dQI

LC

t

LC

QI sin0

Current in an LC circuit

Period: LCT 2

Frequency: f 1 / 2 LC

Current in an LC Circuit

0 Rinductorcapacitor VVV

0dt

dILRI

C

Q

Non-ideal LC Circuit

Initial energy stored in a capacitor:C

Q

2

2

At time t=0: Q=Q0C

QUcap 2

20

At time t= : Q=0LC2

2

2

1LIU sol

System oscillates: energy is passed back and forth between electric and magnetic fields.

Energy in an LC Circuit

1/4 of a period

What is maximum current?

At time t=0:

mageltotal UUU C

Q

2

20

At time t= :LC2

mageltotal UUU 2max2

1LI

C

QLI

22

1 202

max LC

QI 0

max

Energy in an LC Circuit

Frequency: f 1 / 2 LC

Radioreceiver:

LC Circuit and Resonance

Varying B is created by AC current in a solenoid

What is the current in this circuit?

tmag sin0

dt

demf

temfemf cos0

tR

emf

R

emfI cos

220

Advantage of using AC: Currents and emf ‘s behave as sine and cosine waves.

Two Bulbs Near a Solenoid

Add a thick wire:

Loop 1

Loop 2

I1

I2

I3

Loop 1: 02211 IRIRemf

Loop 2: 022 IR 02 I

Node: 321 III 31 II

11 R

emfI

Two Bulbs Near a Solenoid

Add a thick wire:

Loop 1

Loop 2

I1

I2

I3

Loop 1: 02211 IRIRemf

Loop 2: 022 IR 02 I

Node: 321 III 31 II

11 R

emfI

Two Bulbs Near a Solenoid

Exercise

Exercise