Lecture 18-Lecture 18-11 Ways to Change Magnetic Flux

• Changing the magnitude of the field within a conducting loop (or coil).

• Changing the area of the loop (or coil) that lies within the magnetic field.

• Changing the relative orientation of the field and the loop.

motor generator

cosB BA

http://www.wvic.com/how-gen-works.htm

Lecture 18-Lecture 18-22

N

1. define the direction of ; can be any of the two normal direction, e.g. point to right

2. determine the sign of Φ. Here Φ>0

3. determine the sign of ∆Φ. Here ∆Φ >0

4. determine the sign of using faraday’s law. Here <0

5. RHR determines the positive direction for EMF • If >0, current follow the direction of the curled

fingers. • If <0, current goes to the opposite direction of

the curled fingers.

n

n

How to use Faraday’s law to determine the induced current direction

Lecture 18-Lecture 18-33Eddy Current

• The presence of eddy current in the objectresults in dissipation of electric energy that is derived from mechanical motion of the object.

• The dissipation of electric energy in turncauses the loss of mechanical energy ofthe object, i.e., the presence of the field damps motion of the object.

A current induced in a solid conducting object, due to motion of the object in an external magnetic field.

Lecture 18-Lecture 18-44 Self-Inductance

• As current i through coil increases, magnetic flux through itself increases.

This in turn induces counter EMF in the coil itself

• When current i is decreasing, EMF is induced again in the coil itself in such a way as to slow the decrease.

Self-induction

BLi

BN

Li

(if flux linked)

Faraday’s Law:Bd dI

dtL

dt

2 / /L T m A A HWb (henry)

Lecture 18-Lecture 18-55

Lecture 18-Lecture 18-66Solenoid: Archetypical Inductor

0

Nr l B i

l

Current i flows through a long solenoid of radius r with N turns in length l

2 20B

NA r BA i r

l For each turn

222 2

0 0BN N N

L r l ri l l

For the solenoid

or2

0L n Al

Inductance, like capacitance, only depends on geometry 0 /H m

Lecture 18-Lecture 18-77Potential Difference Across Inductor

00 dt

dILIR

V I r

V

+

-

I

internal resistance

• Analogous to a battery

• An ideal inductor has r=0

• All dissipative effects are to be included in the internal resistance (i.e., those of the iron core if any)

00 dt

dILIR

Lecture 18-Lecture 18-88RL Circuits – Starting Current

0dI

IR Ldt

2. Loop Rule:

3. Solve this differential equation

/( / ) /( / )1 ,t L R t L RL

dII e V L e

R dt

τ=L/R is the inductive time constant

2 2/ /

//

T m A T m AL R s

V A

1. Switch to e at t=0

As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I.

+

-

Lecture 18-Lecture 18-99

Starting Current through Inductor vs Charging Capacitor

/( / )t L RL

dIV L e

dt

/(1 )t RCq C e

/t RCI eR

/( / )1 t L RIR e

Lecture 18-Lecture 18-1010

R1

R2 LV

Which of the following statement is correct after switch S is closed ?

1. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R1

2. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V.

3. At t = 0, the potential drop across the inductor is 0; When t = ∞, the current through R1 is V/(R1+R2)

4. At t = 0, the potential drop across the inductor is V; When t = ∞, the current through R1 is V/R2

Warm-up

S

Lecture 18-Lecture 18-1111Remove Battery after Steady I already exists in RL Circuits

3. Loop Rule: 0dI

IR Ldt

4. Solve this differential equation

/( / )

/( / )

t L R

t L RL

I eR

dIV L e

dt

I cannot instantly become zero!

Self-induction

like discharging a capacitor

1. Initially steady current Io is flowing: 0 1I R R

-

+

2. Switch to f at t=0, causing back EMF to oppose the change.

Lecture 18-Lecture 18-1212Behavior of Inductors

• Increasing Current

– Initially, the inductor behaves like a battery connected in reverse.

– After a long time, the inductor behaves like a conducting wire.

• Decreasing Current

– Initially, the inductor behaves like a reinforcement battery.

– After a long time, the inductor behaves like a conducting wire.

Lecture 18-Lecture 18-1313Energy Stored By Inductor

1. Switch on at t=0

0dI

IR Ldt

2. Loop Rule:

3. Multiply through by I

As the current tries to begin flowing, self-inductance induces back EMF, thus opposing the increase of I. +

-

2 dII I R LI

dt

Rate at which battery is supplying energy

Rate at which energy is dissipated by the resistor

Rate at which energy is stored in inductor L

mdU dILI

dt dt 21

2mU LI

Lecture 18-Lecture 18-1414Where is the Energy Stored?

• Energy must be stored in the magnetic field!

Energy stored by a capacitor is stored in its electric field

20L n Al0 ,B nI• Consider a long solenoid where

2

2 2 20

0

1 1 1

2 2 2m

BU LI n Al I Al

area A

length l2

0

1

2m

m

U Bu

Al

• So energy density of the magnetic field is

20

1

2Eu E (Energy density of the electric field)

Lecture 18-Lecture 18-1515Physics 241 –Quiz A

The switch in this circuit is initially open for along time, and then closed at t = 0. What is themagnitude of the voltage across the inductorjust after the switch is closed?

a) zero

b) V

c) R / L

d) V / R

e) 2V

Lecture 18-Lecture 18-1616Physics 241 –Quiz B

The switch in this circuit is closed at t = 0. What is the magnitude of the voltage across the resistor a long time after the switch is closed?

a) zero

b) V

c) R / L

d) V / R

e) 2V

Lecture 18-Lecture 18-1717Physics 241 –Quiz C

The switch in this circuit has been open for a long time. Then the switch is closed at t = 0. What is the magnitude of the current through the resistor immediately after the switch is closed?

a) zero

b) V / L

c) R / L

d) V / R

e) 2V / R