Induced Electric Fields and Magnetic Flux: Understanding Faraday's Law and Lenz's Law, Slides of Physics

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27. Electromagnetic Induction

Induced Currents

Faraday’s Law

Induction & Energy

Inductance

Magnetic Energy

Induced Electric Fields

It takes fourteen 110-car trainloads of each week to fuel this power plant.What feature of the equation

ε^

=^

－

d Φ

/ B dt

demands this prodigious fuel consumption?

The minus sign, which denotes energy conservation in electromagnetic induction.

3. An induced current also results when acurrent-carrying circuit replaces the magnet.4. A current is also induced when thecurrent in an adjacent circuit changes.

changing

B

induces currents (electromagnetic induction)

27.2. Faraday’s Law^ •

Magnetic flux

-^

Flux & Induced EMF

Example 27.1. Solenoid

A solenoid of circular cross section has radius

R

consists of

n

turns per unit length, and carries current

I

Find the magnetic flux through each turn of the solenoid.

B

A

2

0

n I

R

B^

out of plane

I^

B

Example 27.2. Nonuniform Field

A long, straight wire carries current

I

A rectangular wire loop of dimensions

l^

by

w

lies in a plane containing

the wire, with its closest edge a distance

a

from the wire, and its

dimension

l^

parallel to the wire.

Find the magnetic flux through the loop.

0 2 a^

w a

I

l dr r

^ 





B^

d

^

B

A

a^

w a

I l

d r r



0

ln

I l

a^

w a

Area element for integration

Example 27.3. Changing

B

A wire loop of radius 10 cm has resistance 2.



.

The plane of the loop is perpendicular to a uniform

B

that’s increasing at 0.10 T/s.

Find the magnitude of the induced current in the loop.

2

B

r 



B^

BA

^

B

d

d^

B

r

d t

d t



3

V



2

d B r^

d t

 

B

d d t

E

^

2

m

T

s

I^

R

E

3

V





3

A



CCW

S

C

I

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Example 27.3. Changing

B

A wire loop of radius 10 cm has resistance 2.



.

The plane of the loop is perpendicular to a uniform

B

that’s increasing at 0.10 T/s.

Find the magnitude of the induced current in the loop.

2

B

r

B^

BA

^

B

d^

d

B

r

d t

d t

3

V



2

d B r^

d t

B

d d t

E

^

2

m

T

s

I^

R

E

3

V





3

A



 CCW

S

C

I

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Example 27.4. Changing Area

Two parallel conducting rails a distance

l^

apart are connected at one end by a resistance

R

.

A conducting bar completes the circuit, joining the two rails electrically but free to slide along.The whole circuit is perpendicular to a uniform

B

, as show in figure.

Find the current when the bar is pulled to the right with constant speed

v

.

B

l x



B^

B A

B

d

 d t

Let E

x

= 0 be at the left end of rail.

B

l v 

I^

R

E

B l v

R

CCW

S

x

C

I

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27.3. Induction & Energy

Direction of emf is to opposemagnet’s motion.

Lenz’s law :Direction of induced emf issuch that

B

created by the

induced current opposes thechanges in

that created

the current.

Magnet moving right

RH rule: thumb //

m

.

Loop ~ magnet with N to left.^ RH rule: thumb //

m

.

Loop ~ magnet with S to left.

Magnet moving left

m

I I

m

Motional EMF & Len’s Law

Motional emf: induced emf due tomotion of conductor in

B

Square loop of sides

L

& resistance

R

pulled with

constant speed

v

out of uniform

B

Force on

e

:^

^

e 

F

v^

B

downward force 

upward

I

(CW)

Force on current carrying wire:

mag

I

F

L

B

, mag net

applied



F

F

B^

B L x

^ B

d^

B L v

 d t

x^

B L v  E P

I

E

2 R

E

I^

R

E

B L v

R

^

B L v

R

F v

F v 

I L B v 

^

B L v

R

Work done is used to heat up circuit ( E conservation ).

S^ C

d x

v

d t

I

CW

B

d

 d t

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GOT IT? 27.

What will be the direction of the current when the loop firstenters the field from the left side?

Current is CCW

S

B

d

 d t

E

B

d

 d t

B 

C

GOT IT? 27.

What will be the direction of the current when the loop firstenters the field from the left side?

Current is CCW

S

B

d

 d t

E

B

d

 d t

B 

C