Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at...

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Inductio n II

Transcript of Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at...

Page 1: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Induction II

Page 2: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.
Page 3: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Law of Induction• The magnitude of the induced emf in a

circuit is equal to the rate at which the magnetic flux through the circuit is changing with time.

dt

d B ||dt

dN B ||

If coil has N turns

Page 4: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Change in flux may be due to

• Change in magnetic field• Change in the area• Both.

AdBB

Page 5: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Lenz’s law

• The flux of the magnetic field due to the induced current opposes the change in the flux that causes the induced current.

dt

d B

Page 6: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Motional EMF

Page 7: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Induced current flows in the loop

External agent pulls the loop with constant speed

Page 8: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

BAB

BDxB

dt

d B ||

BDv||

R

BDv

RI ind

||

Page 9: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

F1 is the net magnetic force

• If external agent pulls with constant speed

• Fext = F1 = Iind DB

• Mechanical power

P = F1 v

Page 10: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

The power expended by the external agent

vFP 1DBvIP ind

R

vBDP

222

Page 11: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

• A conducting rod of length L is being pulled along horizontal, frictionless and conducting rails. A uniform magnetic field fills the region in which the rod moves. Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s, resistance of rod as 415 m.

Page 12: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

• Find Induced emf = BLv = 0.619 V

• Current in the conducting loop.

• I = /R = 1.49 A

•Assume B = 1.18 T, L = 10.8 cm, v = 4.86 m/s resistance of rod as 415 m

Page 13: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

•At what rate does the internal energy of rod increase?

•P = Iind = 0.922 W

•Force that must be applied by external agent to maintain its motion

•F = ILB = 0.190 N

•At what rate does this force do work on rod?

•P = F v = 0.922 W

Page 14: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Eddy Currents An emf and a current are induced in a

circuit by a changing magnetic flux.

When the magnetic flux through a large piece of conductor changes, induced current appear in the material in small loops.

These are called eddy currents as they induce in little swirls/eddies.

Page 15: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.
Page 17: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Eddy currents and energy loss

• They can increase internal energy and thus temperature of the material

• Big eddy currents larger energy loss

• Materials which are subjected to magnetic fields are often constructed in many small layers.

Page 18: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.
Page 19: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Eddy currents slow down the motion of the conductor

Page 20: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

A cylindrical bar magnet is dropped down a vertical aluminum pipe of slightly large diameter . It takes

several seconds to emerge at the bottom, whereas, identical piece of unmagnetized iron makes the trip in a fraction of a second. Explain why

magnet falls more slowly??

Ans: delay is due to forces exerted on the magnet by induced eddy currents in the pipe.

Page 21: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

•Advantage Heating effect can be used

in induction furnace.

Page 22: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Magnetic field cannot force a stationary charge to move. Then why the charges move?

Why there is an induced current?

Page 23: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Induced electric fields

A changing magnetic field induces an electric field.

Page 24: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

•Induced electric field exists, even when ring is removed.It is always tangential.

0EDiv

Page 25: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Some facts• The driving force for induced currents

is induced E-field

• It exists, even when ring is removed.

• It has no radial component.

• As real as that might be setup by a real stationary charge.

sdE

Page 26: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

dt

dsdE B

dt

BdECurl

adBdt

dsdE

Page 27: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

In the static case, Faraday’s law reduces to

dt

BdECurl

0ECurl

0 sdE

Page 28: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

You can not define a potential for an induced electric field.

Page 29: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

A uniform magnetic field B(t) pointing straight up fills the shaded circular

region. If B is changing with time what is the induced electric field ?

B(t)

adBdt

dsdE

Page 30: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

r

adBdt

dsdE

2)(2 rtBdt

drE

dt

dBrrE 22

If B is increasing with time, induced current will run clockwise as look from above.

Page 31: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

A line charge is glued onto the rim of a wheel of radius R, which is then suspended horizontally . It is free to rotate. The spokes are made of wood. In the central region out to radius a there is a uniform magnetic field pointing up. Now someone turns the field off. What happens?

dt

dBasdE 2

ds

B

Page 32: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Torque on the segment ds

RsdE

Rdt

dBa 2

Page 33: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Two parallel loops of wire are shown with common axis. Smaller loop is above the larger loop by a distance x>>R. Magnetic field due to current i in the larger loop is constant through the smaller loop and equal to the value on the axis. Suppose x is increasing with constant rate.

Page 34: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

(a) Determine the flux across the area bounded by smaller loop as a function of x.

2/322

20

2 xR

RIB

3

20

2 x

RIB

23

20

2r

x

RIBAB

Page 35: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Compute the emf generated in the smaller

loop

• Direction of current is anticlockwise as seen from above.

23

20

2r

x

RIBAB

vrx

RI

dt

d B 24

20

2

3

Page 36: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Two straight conducting rails form an angle where their ends are

joined. A conducting bar in contact with the rails and forming an isoscale triangle with them, starts at the vertex at time t =

0 and moves with constant velocity v to the right. A magnetic field points out of the

page.

Page 37: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Find emf induced as a function of

time.

2tan2 xA

2tan2 BxBAB

2tan2 2 tBv

Page 38: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

A square loop of wire lies on a table, a distance s from a very long

straight wire, which carries a current I. If someone pulls the loop away

from the wire at speed v, what emf is generated?

s

aa

a

Page 39: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

Flux through the loop

s

aa

a

adyy

Ias

s

B

2

0

s

asIaB ln

20

Page 40: Induction II. Law of Induction The magnitude of the induced emf in a circuit is equal to the rate at which the magnetic flux through the circuit is.

• Direction of induced current is anticlockwise.

s

asIaB ln

20

dt

ds

sdt

ds

as

Ia 11

20

vass

Ia

)(

1

2

20