Physics 1304: Lecture 13, Pg 1 Faraday’s Law and Lenz’s Law ~ B(t) i.

Physics 1304: Lecture 13, Pg 1

Faraday’s Law and Lenz’s Law

~

B(t)

i


Overview of Overview of LectureLecture

Induction Effects Faraday’s Law (Lenz’ Law)

• Energy Conservation with induced currents? Faraday’s Law in terms of Electric Fields

Text Reference: Chapter 31.1-4


Induction Induction EffectsEffects

v

vS N

vN S

N S

S N

Bar magnet moves through coil

Current induced in coil

• Change pole that enters

Induced current changes sign

Bar magnet stationary inside coil

No current induced in coil

Coil moves past fixed bar magnet

Current induced in coil


Induction Induction EffectsEffects

from Currentsfrom Currents

• Switch closed (or opened)

current induced in coil b

• Steady state current in coil a

no current induced in coil b

ab

Conclusion:

A current is induced in a loop when:

• there is a change in magnetic field through it

• loop moves through a magnetic field How can we quantify this?


Faraday's LawFaraday's Law

• Define the flux of the magnetic field through a surface (closed or open) from:

Faraday's Law:

The emf induced in a circuit is determined by the time rate of change of the magnetic flux through that circuit.

The minus sign indicates direction of induced current (given by Lenz's Law).

dS

B B


Lenz's Lenz's LawLaw• Lenz's Law:

The induced current will appear in such a direction that it opposes the change in flux that produced it.

Conservation of energy considerations:

Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated.

» Why??? If current reinforced the change, then the change

would get bigger and that would in turn induce a larger current which would increase the change, etc..

v

BS N

v

BN S


Lecture 18, Lecture 18, CQCQ A conducting rectangular loop moves with

constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown. What is the direction of the induced current in the

loop?

(a) ccw (b) cw (c) no induced current• A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown.

• What is the direction of the induced current in the loop?

1A

X X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X XX X X X X X X X X X X X

v

x

y

(a) ccw (b) cw (c) no induced current

1B v

I

x

y


Lecture 16, ACT 1Lecture 16, ACT 1 A conducting rectangular loop moves with

constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown. What is the direction of the induced current in the

loop?

• There is a non-zero flux B passing through the loop since B is perpendicular to the area of the loop.• Since the velocity of the loop and the magnetic field are CONSTANT, however, this flux DOES NOT CHANGE IN TIME.• Therefore, there is NO emf induced in the loop; NO current will flow!!

(c) no induced current(a) ccw (b) cw

1A


v

x

y


Lecture 16, ACT 1Lecture 16, ACT 1 A conducting rectangular loop moves with constant

velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown. What is the direction of the induced current in the

loop?

y

• The flux through this loop DOES change in time since the loop is moving from a region of higher magnetic field to a region of lower field.• Therefore, by Lenz’ Law, an emf will be induced which will oppose the change of flux.• The current i is induced in the clockwise direction to restore the flux.

(a) ccw (b) cw (c) no induced current• A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown. • What is the direction of the induced current in the loop?


v

x

(a) ccw (b) cw (c) no induced current

v

I

x

y i


DemoDemoE-M E-M

CannonCannonv

Connect solenoid to a source of alternating voltage.

~

side view

F

F

B

B

B

top view

The flux through the area to axis of solenoid therefore changes in time.

A conducting ring placed on top of the solenoid will have a current induced in it opposing this change.

There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field.


Lecture 18, Lecture 18, CQCQ

Let us predict the results of variants of the electromagnetic cannon demo which you just observed.

Suppose two aluminum rings are used in the demo; Ring 2 is identical to Ring 1 except that it has a small slit as shown. Let F1 be the force on Ring 1; F2 be the force on Ring 2.

– Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8 -m); Ring 2 is made of aluminum (resistivity = 2.8X10-8 -m). Let F1 be the force on Ring 1; F2 be the force on Ring 2.

(a) F2 < F1 (b) F2 = F1 (c) F2 > F1

Ring 1

Ring 2

(a) F2 < F1 (b) F2 = F1 (c) F2 > F1



For this ACT, we will predict the results of variants of the electromagnetic cannon demo which you just observed.


(a) F2 < F1 (b) F2 = F1 (c) F2 > F1

Ring 1

Ring 2

• The key here is to realize exactly how the force on the ring is produced.• A force is exerted on the ring because a current is flowing in the ring and the ring is located in a magnetic field with a component perpendicular to the current.• An emf is induced in Ring 2 equal to that of Ring 1, but NO CURRENT is induced in Ring 2 because of the slit!• Therefore, there is NO force on Ring 2!



For this ACT, we will predict the results of variants of the electromagnetic cannon demo which you just observed.


(a) F2 < F1 (b) F2 = F1 (c) F2 > F1

Ring 1

Ring 2

– Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8 -m); Ring 2 is made of aluminum (resistivity = 2.8X10-8 -m). Let F1 be the force on Ring 1; F2 be the force on Ring 2.

(a) F2 < F1 (b) F2 = F1 (c) F2 > F1

• The emf’s induced in each ring are equal.• The currents induced in each ring are NOT equal because of the different resistivities.• The copper ring will have a larger current induced (smaller resistance) and therefore will experience a larger force (F proportional to current).


CalculatioCalculationn

Suppose we pull with velocity v a coil of resistance R through a region of constant magnetic field B. What will be the induced current?

» What direction?

Lenz’ Law clockwise!!

x x x x x x

x x x x x x

x x x x x x

x x x x x x

vw

x

I

What is the magnitude?

» Magnetic Flux:

» Faraday’s Law:


Energy Conservation?Energy Conservation?

x x x x x x

x x x x x x

x x x x x x

x x x x x x

vw

x

I

F

F'

F'

• Energy is dissipated in circuit at rate P'

P = P' !

• The induced current gives rise to a net magnetic force ( ) on the loop which opposes the motion.

F

• Agent must exert equal butopposite force to move the loop with velocity v; agent does work at rate P, where


Faraday's LawFaraday's Law

• Define the flux of the magnetic field through a surface (closed or open) from:

Faraday's Law:

The emf induced in a circuit is determined by the time rate of change of the magnetic flux through that circuit.

The minus sign indicates direction of induced current (given by Lenz's Law).

dS

B B


Conceptual QuestionConceptual Question

In figure (a), a solenoid produces a magnetic field whose strength increases into the plane of the page. An induced emf is established in a conducting loop surrounding the solenoid, and this emf lights bulbs A and B. In figure (b), points P and Q are shorted. After the short is inserted,

1. bulb A goes out; bulb B gets brighter.2. bulb B goes out; bulb A gets brighter.3. bulb A goes out; bulb B gets dimmer.4. bulb B goes out; bulb A gets dimmer.5. both bulbs go out.6. none of the above


B B E E

x x x x x x x x x x

x x x x x x x x x x

x x x x x x x x x x

x x x x x x x x x x

x x x x x x x x x x

r

E

E

E

E

B

• Suppose B is increasing into the screen as shown above. An E field is induced in the direction shown. To move a charge q around the circle would require an amount of work =

• This work can also be calculated from = W/q.

W qE dl

Faraday's law a changing B induces an emf which can produce a current in a loop.

In order for charges to move (i.e., the current) there must be an electric field.

we can state Faraday's law more generally in terms of the E field which is produced by a changing B field.


B B E E

• Putting these 2 eqns together:

• Therefore, Faraday's law can be rewritten in terms of the fields as:

x x x x x x x x x x

x x x x x x x x x x

x x x x x x x x x x

x x x x x x x x x x

x x x x x x x x x x

r

E

E

E

E

B

Note that for E fields generated by charges at rest (electrostatics) since this would correspond to the potential difference between a point and itself. Consequently, there can be no "potential function" corresponding to these induced E fields.



The magnetic field in a region of space of radius 2R is aligned with the -z-direction and changes in time as shown in the plot.What is sign of the induced emf in a ring of

radius R at time t=t1?

– What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R ?

(a) E2R = ER(b) E2R = 2ER (c) E2R = 4ER

z

t

B

t1

X X X X

X X X X X X X X

X X X X X X X X X

X X X X X X X

X X X X X X X X Xx

y

X X X X X X X X X X X X X X X

X X X X

R

(a) < 0( E ccw)

(b) = 0 (c) > 0( E cw)



The magnetic field in a region of space of radius 2R is aligned with the -z-direction and changes in time as shown in the plot.

What is sign of the induced emf in a ring of radius R at time t=t1?

X X X X X X X X X X XX X X X X X X X

X X X X X X X X Xx

y

X X X X X X X X XX X X X X X X X X X X X X X X

X X X X

R

(a) < 0( E ccw)

(b) = 0 (c) > 0( E cw)

• There will be an induced emf at t=t1 because the magnetic field (and therefore the magnetic flux) is changing. It makes NO DIFFERENCE that at t=t1 the magnetic field happens to be equal to ZERO!• The magnetic field is increasing at t=t1 (actually at all times shown!) which induces an emf which opposes the corresponding change in flux. ie electric field must be induced in a counter clockwise sense so that the current it would drive would create a magnetic field in the z direction.

t

B

t1



The magnetic field in a region of space of radius 2R is aligned with the -z-direction and changes in time as shown in the plot.

What is sign of the induced emf in a ring of radius R at time t=t1?

X X X X X X X X X X XX X X X X X X X

X X X X X X X X Xx

y

X X X X X X X X XX X X X X X X X X X X X X X X

X X X X

R

(a) < 0( E ccw)

(b) = 0 (c) > 0( E cw)

t

B

t1 What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R ?

(a) E2R = ER(b) E2R = 2ER (c) E2R = 4ER

• The rate of change of the flux is proportional to the area:• The path integral of the induced electric field is proportional to the radius.

NOTE: This result is important for the operation of the Betatron.


Textbook ProblemsTextbook Problems

Problems 31-33, 31-39, 31-46, 31-67


Induced EMF by motion


Rod Moving in B-FieldRod Moving in B-Field

Consider a metal rod moving in a B-field. The free charges in the rod will experience a force given by,

The work done in separating the charges is:

By definition the EMF is then,

X X X X X X X X X X X X

X X X X X X X X

X X X X X X X X

X X X X X X X X



+++

++

---

-- -

F qv B

the top the force would be upwards and negative charge would thus accumulate on end.

W F l

qvBl

W

q

i.e.

vBl


The previous result is connected with Faraday’s law. To see this we re-interpret Faradays law as the induced EMF along the path of a moving conductor in the presence of a constant of changing B-field. This induced EMF would be equal to the rate at which magnetic flux sweeps across the path.

x x x x x x

x x x x x x

x x x x x x

x x x x x x s

sBLt

vBL

( ) ( )d dsBL BA

dt dt Bd

dt

Bd

dt

: rate at which flux is swept.

sweptA sL


Electric Generators and MotorsElectric Generators and Motors

Consider a loop of wire in a constant magnetic field. If we rotate the loop the flux through the cross sectional area will change.

By Faradays Law an Induced EMF will be generated.

The converse is also true if we run a current through thewire then a torque will be excerted on the loop making itturn.

B

i

i

i

B

B BA

( ) cos ( )B t BA t

n̂


Electric Motors and GeneratorsElectric Motors and Generators

Physics 1304: Lecture 13, Pg 1 Faraday’s Law and Lenz’s Law ~ B(t) i.

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Transcript of Physics 1304: Lecture 13, Pg 1 Faraday’s Law and Lenz’s Law ~ B(t) i.