Two questions:(1) How to find the force, F on the electric charge, Q excreted by the field E and/or B?

(2) How fields E and/or B can be created?

BvqEqF

0/cos QAEE

2r

QkE

Gauss’s law for electric fieldElectric charges create electric field:

For one not moving (v<<c) charge:

Gauss’s law for magnetic field Magnetic charges do not exist:

0cos ABB

Amperes lawElectric current creates magnetic field:

IlB 0cos (As we will see later,this law should be extended)

Faraday’s law

Maxwell’s equations

tB

A changing magnetic flux induces an EMF

A changing magnetic field induces an electric field ! 1

8. Electromagnetic induction(Faraday’s law)

1) Flux (review)

cos

cos

AB

AB

B

B

2) EMF (review)

cos

cos

lE

q

lF

q

W

3) Faraday’s law

t

B

t

BA

cos t

BAlE

cos

cos

A changing magnetic flux induces an electric field !This electric field is not a potential field. The field lines form a closed loops.

A changing magnetic fluxinduces an EMF

Units: (weber) [ΦB] = 1 Wb = 1 T·m2

1 :loops For NN N

B

- EMF in the closed loop

- flux through a closed loop

2

4) Lenz’s law

The direction of any magnetic induction effect is such as to oppose the cause of the effect

For instance: a current produced by an induced emf moves in a direction so that its magnetic field opposes the original change in flux

N

S

v

B

N

S

v

B

S

N

v

B

S

N

v

B

I I I I

Example: If a North pole moves toward the loop in the plane of the page, in what direction is the induced current?

Since the magnet is moving parallel to the loop, there is no magnetic flux through the loop. Thus the induced current is zero.

3

1) drop the magnet2) move the magnet upwards3) move the magnet sideways4) all of the above5) only (1) and (2)

Example: In order to change the magnetic flux through the loop, what would you have to do?

Moving the magnet in any direction would change the

magnetic field through the loop and thus the magnetic flux.

1) tilt the loop2) change the loop area3) use thicker wires4) all of the above5) only (1) and (2)

Since, changing the area or tilting the loop (which varies the projected area) would change the magnetic flux through the loop.

cosBAB

4

Example 1: A wire loop is being pulled away from a current-carrying wire. What is the direction of the induced current in the loop?

I

1) Clockwise2) Counterclockwise 3) No induced current

On the right side of the wire the magnetic flux is into the page and decreasing due to the fact that the loop is being pulled away. By Lenz’s Law, the induced B field will oppose this decrease. Thus, the new B field points into the page, which requires an induced clockwise current to produce such a B field.

Example 2: What is the induced current if the wire loop moves down?

I

The magnetic flux through the loop is not changing as it moves parallel to the wire. Therefore, there is no induced current.

1) Clockwise2) Counterclockwise 3) No induced current

5

6

Example1: A wire loop is being pulled through a uniform magnetic field. What is the direction of the induced current? 1) Clockwise; 2) Counterclockwise; 3) No induced current 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

Since the magnetic field is uniform, the magnetic flux through the loop is not changing. Thus no current is induced.

Example 3: A wire loop is being pulled through a uniform magnetic field that suddenly ends.What is the direction of the induced current? 1) Clockwise 2) Counterclockwise 3) No induced current

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

The B field into the page is disappearing in the loop, so it must be compensated by an induced flux also into the page. This can be accomplished by an induced current in the clockwise direction in the wire loop.

Example2: What is the direction of the induced current if the B field suddenly increases while the loop is in the region? 1) Clockwise 2) Counterclockwise 3) No induced current

The increasing B field into the page must be countered by an induced flux out of the page. This can be accomplished by induced current in the counterclockwise direction in the wire loop.

Example: If a coil is shrinking in a magnetic field pointing into the page, in what direction is the induced current? 1) Clockwise 2) Counterclockwise 3) No induced current

The magnetic flux through the loop is decreasing, so the induced B field must try to reinforce it and therefore points in the same direction — into the page. According to the right-hand rule, an induced clockwise current will generate a magnetic field into the page.

Example: If a coil is rotated as shown, in a magnetic field pointing to the left, in what direction is the induced current? 1) Clockwise 2) Counterclockwise 3) No induced current

As the coil is rotated into the B field, the magnetic flux through it increases.

According to Lenz’s Law, the induced B field has to oppose this increase,

thus the new B field points to the right. An induced counterclockwise

current produces just such a B field.7

Example: Wire #1 (length L) forms a one-turn loop, & a bar magnet is dropped through. Wire #2 (length 2L) forms a two-turn loop, and the same magnet is dropped through. Compare the magnitude of the induced currents in these two cases.

NNSS

NNSS

1) I1 > I2 2) I1 < I2 3) I1 = I2 04) I1 = I2 = 0

Induced emf is twice as large in the wire with 2 loops. The current is given by Ohm’s law: I = V/R. Since wire #2 is twice as long as wire #1, it has twice the resistance, so the current in both wires is the same.

tN

t

1BB

Example: A bar magnet is held above the floor and dropped. In 1, there is nothing between the magnet and the floor. In 2, the magnet falls through a copper loop. How will the magnet in case 2 fall in comparison to case 1?

NS

NS

When the magnet is falling from above the loop in 2, the induced current will produce a North pole on top of the loop, which repels the magnet. When the magnet is below the loop, the induced current will produce a North pole on the bottom of the loop, which attracts the South pole of the magnet.

1) it will fall slower; 2) it will fall faster; 3) it will fall the same

8

Example 1: A coil of 600 turns with area 100 cm2 is placed in a uniform magnetic field. The angle between the direction of the field and the perpendicular to the loop is 60°. The field changes at the rate of 0.010 T/s. What is the magnitude of induced emf in the coil?

?

/010.0

100

60

600

2

sTt

B

cmA

N

VmsT

At

BN

tB

03.060cos10100/010.0600

cos

24

cos

coscos

At

BN

t

NBAAB

B

B

Example 2:

?

010.0

/100

60

600

2

TB

scmt

A

N

cos

coscos

t

ANB

t

NBAAB

B

B

VsmT

t

ANB

tB

03.060cos/10100010.0600

cos

24

9

Example 3: A 12.0-cm-diameter wire coil is initially oriented perpendicular to a 1.5 T magnetic field. The loop is rotated so that its plane is parallel to the field direction in 0.20 s. What is the average induced emf in the loop?

?

20.0

5.1

0.122

90

0

1

2

st

TB

rA

cmr

N

in

f

V

s

mT

t

BA

t

BABA

BA

B

inB

fBB

B

22

105.820.0

2/12.05.1

90cos0cos

cos

10

1) Rotating loop:

t

tBA

t

cosB

I

I

B

Tf 22

t 8a. Applications of Faraday’s law

tBABAB coscos

tBA sin

Example:

?

100

100

03.0

200

max

1

2

s

cmA

TB

N

VsmT 61001010003.0200 124max

max

NBAmax

11

Example: A generator rotates at 60 Hz in a magnetic field of 0.03 T. It has 1000 turns and produces voltage that is 120 V at a pick. What is the area of each turn of the coil?

?

120

1000

03.0

60

max

A

V

N

TB

Hzf

2210

03.01000602

120m

THz

VA

NBAmax

f 2 fNBA

2max

12

8b. EMF induced in a moving conductor

B

v

tvx

l

tlvxlA

A t

AB

tB

Another method:

B

v

FB

FE

qvBFB

qEFE

BlvElqvBqE

1) What is polarity of EMF?2) What would be the direction of the induced

current, if rod slides on a conducting track?

Blv

The B field points out of the page. The flux is increasing since the area is increasing. The induced B field opposes this change and therefore points into the page. Thus, the induced current runs clockwise according to the right-hand rule.

13

Example: A uniform magnetic field B is perpendicular to the area bounded by the U-shaped conductor and a movable metal rod of length l. The rod is moving along the conductor at a speed v. The total resistance of the loop is R. What is the induced emf, the current in the loop, the magnetic force on the moving rod, and power needed to move the rod?

R

vlBR

R

BlvRIP

R

vlBFvP

R

vlBlB

R

BlvIlBF

R

Blv

RI

Blv

dis

ext

22222

222

22

PFI

Find

RvlB

Given

,,,

:

,,,

:

14

8c. Transformers

tNV Bss

1

tNV Bpp

1

p

s

p

s

N

N

V

V

sss VIP ppp VIP

s

p

p

s

N

N

I

I

sp PP

15

120 V

Example: What is the voltage across the lightbulb?

The first transformer has a 2:1 ratio of turns, so the voltage doubles. But the second transformer has a 1:2 ratio, so the voltage is halved again. Therefore, the end result is the same as the original voltage.

240 V 120 V

Example: Given that the intermediate current is 1 A, what is the current through the lightbulb?

1 A

Power in = Power out240 V 1 A = 120 V ???

1 2

Example: A 6 V battery is connected to one side of a transformer. Compared to the voltage drop across coil 1, the voltage across coil 2 is:

Batteries provide DC current.

Only a changing magnetic flux induces an EMF.

Therefore, the voltage across coil 2 is zero.

The unknown current is 2 A.

16