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 changing with time.

dt

d B ||dt

dN B ||

If coil has N turns

Change in flux may be due to

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

AdBB

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

Motional EMF

Induced current flows in the loop

External agent pulls the loop with constant speed

BAB

BDxB

dt

d B ||

BDv||

R

BDv

RI ind

||

F1 is the net magnetic force

• If external agent pulls with constant speed

• Fext = F1 = Iind DB

• Mechanical power

P = F1 v

The power expended by the external agent

vFP 1DBvIP ind

R

vBDP

222

• 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.

• 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

•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

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.

• http://www.ndt-ed.org/EducationResources/HighSchool/Electricity/eddycurrents.htm

• http://www.ndt-ed.org/TeachingResources/NDT_Tips/LenzLaw.htm

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.

Eddy currents slow down the motion of the conductor

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.

•Advantage Heating effect can be used

in induction furnace.

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

Why there is an induced current?

Induced electric fields

A changing magnetic field induces an electric field.

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

0EDiv

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

dt

dsdE B

dt

BdECurl

adBdt

dsdE

In the static case, Faraday’s law reduces to

dt

BdECurl

0ECurl

0 sdE

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

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

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.

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

Torque on the segment ds

RsdE

Rdt

dBa 2

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.

(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

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

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.

Find emf induced as a function of

time.

2tan2 xA

2tan2 BxBAB

2tan2 2 tBv

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

Flux through the loop

s

aa

a

adyy

Ias

s

B

2

0

s

asIaB ln

20

• Direction of induced current is anticlockwise.

s

asIaB ln

20

dt

ds

sdt

ds

as

Ia 11

20

vass

Ia

)(

1

2

20