PHYS 2421 - Fields and Waves

Idea: We have seen: currents can produce B fields

We will now see: B fields can produce currentsFacts: Current is produced in closed loops when the magnetic flux changes

Notice: As the magnet moves in and out, the B-flux

changes creating a current Direction of current changes if B field

increases or decreases Magnitude of current proportional to velocity

of magnet, i.e. to rate of change of B-flux No change of B-flux no current

A similar case

As the coil moves in and out, the B-flux changes creating a current

Direction of current changes if B field increases or decreases

Magnitude of current proportional to velocity of coil, i.e. to rate of change of B-flux

No change of B-flux no current

And another one

No motion of coils As the current in inner

coil sets in, the B-flux changes creating a current

With steady current in inner coil there is no change in B-flux no current

Direction of current changes if current in inner coil changes

More cases

Constant B-flux no current

Decreasing B-flux + current

Increasing B-flux -- current

And yet one more case

Summary of Section 29.1

Bddt

A changing flux produces a current, i.e. it can be taken as a EMF

EMF

Changing flux

Sign to be explained later . . . A curious

application:

Bddt

A changing flux produces a current, i.e. it can be taken as a EMF

cosB B A BA

Consider a B field traversing an area

The magnetic flux is

BddtIf the field (or the area)

is changing, its rate of change is

and the EMF it produces is

Faraday’s law of induction

Michael Faraday

1791-1867

Hmwk: Probls. 29.2, 5, 9 (11th Ed.), 2, 4, 9 (12th or 13th Eds.)

Soln 29.2: a) 1.44x10-5 T, 0; b) 3.6x10-4 V

0.020 0.012 0.242 T/s m mVBd dB Adt dt

B d BAd dB Adt dt dt

B B A BA

Check units:

2 2

/T N N N Jm m m m Vs Ams C s s C C

Summary of Section 29.2

Bddt

Faraday’s law of

induction

Hmwk Sect/ 29.2: Probls. 29.2, 5 and 9 (11th Ed.) or

29.2, 4 and 9 (12th or 13th Eds.)

The induced B field will have the direction needed

to reduce the change in flux

Lenz’s law

Book’s version:“The direction of any magnetic

induction effect is such as to oppose the cause of the effect”

Complicated?Let us see some

examples

Direction of induced EMF

A changing fluxInduces a current in

wirewhich produces a second B

fieldThe induced B field will have the direction needed to reduce the

change in fluxLenz’s

lawMore examples

There are two possible directions for the induced current, which is the correct one?

Homework : Problems 29.16 and 17 (11th Ed.) or

15 and 17 (12th or 13th Eds.)

Which current direction produces this induced B?

Increasing B induced B should oppose original B

Which current direction produces this induced B?

Decreasing B induced B should align original B

Which current direction produces this induced B?

Increasing B induced B should oppose original B

Summary of Section 29.3

The induced B field will have the direction needed

to reduce the change in flux

Lenz’s law

Homework : Problems 29.16 and 17 (11th Ed.) or

15 and 17 (12th and 13th Eds.)

Consider the following example

Moving conducting

barIncreasing

area increasing flux

Induced current on

circuitMagnitude and direction of

EMF?

B d BA d Ad Bdt dt dt

d Lx dxB BL BLvdt dt

Direction? The induced B field will reduce the change in fluxIncreasing area increasing flux

induced B should oppose original B

Counterclockwise direction

Magnitude and direction of electric field induced?

abV vBL

Homework: Problem 29.20 (11th or 12th Eds.), or 28 (13th Ed.)

(Answer: a) 5.6 V; b) counterclockwise; c) 0.22 A)

Magnitude and direction of electric

field induced?

Take field inside wire as uniform

abV EL E vB

v B dl

In general, for any moving

loopBd

dt

Equivalent to

2.5 0.6 0.10

0.15

m/s T m V

vBL

0.15 5.0 V A0.03 I

R

Extra: E field inside rod:

EMF:

Induced current

5 0.10 0.6 0.3 A m T NF ILB Force on rod

2.5 0.6 1.5 1.5 m/s T Tm/s N/CE vB

Homework: Problems 29.24 (11th Ed.), or 25 (12th Ed.)

or 27 (13th Ed.)

Summary of Section 29.4

BLv

E vB

EMF:

Electric field

Homework: Problems 29.20 and 24 (11th Ed.) or

20 and 25 (12th Ed.), or 28 and 27 (13th Ed.)

Idea:A changing

flux

Which induces a current on the circuitWhat is the electric field

induced?

Bddt

Induces an EMF

(voltage)Which induces an E-

field

Remember connections between EMFs and E-fields

Parallel plates:

/V Ed E V d a

a bb

V V E dl dV E dl

Potential gradient (Sect 23.5)

, , x y zV V VE E Ex y z

In general:

Bd E dldt

From which the E-field can be obtained

For a loop, eg.

122

B Bd dE dl Edl E r Edt r dt

The direction is determined

using the right-hand rule

Bd E dldt

Let us see an example . . .

7 4

0

6

4 10 500 100 4 10

25 10

2Wb turns A m

Am m s

V

Bd dB dIA n Adt dt dt

0

6 4

1 12 2

1 25 10 2 102 0.02

V V/m m

Bd dIE n Ar dt r dt

Homework sect. 29.5: Probl. 29.27 & 31 (11th Ed.),

or 28 & 30 (12th Ed.), or 36 & 40 (13th Ed.)

Summary of Section 29.5A changing flux induces an

EMF which induces an E-field

12

B Bd dE dl Edt r dt

For a loop:

PHYS 2421 - Fields and Waves