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Lecture 4

Physics 1502: Lecture 19Today’s Agenda

• Announcements:– Midterm 1 available

• Homework 06 next Friday

• Induction

Faraday's Law

dS

B Bv

BN S

vB

S N

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Lecture 4

Induction Effects

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

Faraday's Law• Define the flux of the magnetic field through a surface

(closed or open) from:

• Faraday's Law:The emf induced around a closed circuit is determined bythe time rate of change of the magnetic flux through thatcircuit.

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

dS

B B

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Lecture 4

Faraday’’s law for many loops

• Circuit consists of N loops:all same areaΦB magn. flux through one looploops in “series” emfs add!

Lenz's Law• Lenz's Law:

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

• Conservation of energy considerations:Claim: Direction of induced current must be so as tooppose the change; otherwise conservation of energywould be violated.» Why???

• If current reinforced the change, then the changewould get bigger and that would in turn induce alarger current which would increase the change,etc..

vB

S Nv

BN S

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Lecture 4

Lecture 19, ACT 1• A conducting rectangular loop moves with constant

velocity v in the +x direction through a region ofconstant magnetic field B in the -z direction as shown.

– What is the direction of the induced current in theloop?

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

4A

x

y

Lecture 19, ACT 1•A conducting rectangular loopmoves with constant velocity v in the-y direction away from a wire with aconstant current I as shown.

• What is the direction of theinduced current in the loop?4B

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

x

yi

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Lecture 4

Motional EMFx x x x x x x x x xx x x x x x x x x xx x x x x x x x x xx x x x x x x x x xx x x x x x x x x x

vl

-

B

FB

Charges in the conductor experience the force

The charges will be accumulated on both ends of the conductor producingthe electric field E.

The accumulation of charges will stop when the magnetic force qvB is balanced by electric force qE. Condition of equilibrium requires that

--

The electric field produced in the conductor is related to the potential difference across the ends of the conductor

++

Calculation

• Suppose we pull with velocity v acoil of resistance R through aregion of constant magnetic fieldB.– What will be the induced current?

» What direction?• Lenz’ Law ⇒ clockwise!!

x x x x x xx x x x x xx x x x x xx x x x x x

vw

x

⇒∴

I

– What is the magnitude?» Magnetic Flux:

» Faraday’s Law:

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Lecture 4

Power ProductionAn Application of Faraday’s Law

• You all know that we produce power from many sources. Forexample, hydroelectric power is somehow connected to therelease of water from a dam. How does that work?

• Let’s start by applying Faraday’s Law to the followingconfiguration:

N

S

N

S

Side View

N

S

End View

Power ProductionAn Application of Faraday’s Law

• Apply Faraday’s Law

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Lecture 4

Power ProductionAn Application of Faraday’s Law

• A design schematic

Water

ΔB → Ex x x x x x x x x xx x x x x x x x x xx x x x x x x x x xx x x x x x x x x xx 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 Efield is induced in the direction shown. To move a charge qaround the circle would require an amount of work =

• Faraday's law ⇒ a changing B inducesan emf which can produce a current ina loop.

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

· ∴ we can state Faraday's law moregenerally in terms of the E field whichis produced by a changing B field.

• This work can also be calculated from

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Lecture 4

ΔB → E• Putting these 2 eqns together:

⇒

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

x x x x x x x x x xx x x x x x x x x xx x x x x x x x x xx x x x x x x x x xx x x x x x x x x x

r

E

E

E

E

B

• Note that for E fields generated by charges atrest (electrostatics) since this would correspond to thepotential difference between a point and itself. Consequently,there can be no "potential function" corresponding to theseinduced E fields.

Lecture 19, ACT 2• The magnetic field in a region of space of

radius 2R is aligned with the z-direction andchanges in time as shown in the plot.

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

5A

t

Bz

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) ε < 0( E ccw)

(b) ε = 0 (c) ε > 0( E cw)

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Lecture 4

Lecture 19, ACT 25B

– What is the relation betweenthe magnitudes of the inducedelectric fields ER at radius R andE2R at radius 2R ?

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

t

Bz

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

ExampleAn instrument based on induced emf has been used to measure projectilespeeds up to 6 km/s. A small magnet is imbedded in the projectile, asshown in Figure below. The projectile passes through two coils separatedby a distance d. As the projectile passes through each coil a pulse of emfis induced in the coil. The time interval between pulses can be measuredaccurately with an oscilloscope, and thus the speed can be determined.

(a) Sketch a graph of ΔV versus t for the arrangement shown. Consider acurrent that flows counterclockwise as viewed from the starting point ofthe projectile as positive. On your graph, indicate which pulse is from coil1 and which is from coil 2.

(b) If the pulse separation is 2.40 ms and d = 1.50 m, what is the projectilespeed ?

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Lecture 4

A Loop Moving Through a Magnetic Field

Φ(t

) = ?

ε(t)

= ?

F(t)

= ?

Schematic Diagram of an AC Generator

ε = −d

dtBΦΝ = − ΝΑΒ

d (cos( ωt))dt

= − ΝΑΒ ω sin( ωt))

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Lecture 4

Schematic Diagram of an DC Generator

DemoE-M Cannon v

• Connect solenoid to a source ofalternating voltage.

~

side view

•

•

F

F

•B

B

B

top view

• The flux through the area ⊥ toaxis of solenoid thereforechanges in time.

• A conducting ring placed on topof the solenoid will have a currentinduced in it opposing thischange.

• There will then be a force on thering since it contains a currentwhich is circulating in thepresence of a magnetic field.

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Lecture 4

Lecture 19, ACT 3• Suppose two aluminum rings are used in

the demo; Ring 2 is identical to Ring 1except that it has a small slit as shown. LetF1 be the force on Ring 1; F2 be the force onRing 2.

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

Lecture 19, ACT 4• Suppose one copper and one aluminum

rings are used in the demo; theresistance of the two rings is similar butthe aluminum ring has less mass. Let a1be the acceleration of ring 1 and a2 bethe acceleration of Ring 2.

(a) a2 < a1 (b) a2 = a1 (c) a2 > a1

Ring 1

Ring 2

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Lecture 4

Lecture 19, ACT 5• Suppose you take the aluminum ring, shoot

it off the cannon, and try to nail yourannoying neighbor. Unfortunately, you justmiss. You think, maybe I can hit him (her) if Ichange the temperature of the ring. In orderto hit your neighbor, do you want to heat thering, cool the ring, or is it just hopeless?

(a) heat (b) cool (c) hopeless

Hot Ring

Cool Ring

Lecture 19, ACT 6• Suppose the alternating magnetic

field is kept at a level where the ringjust levitates, but doesn’t jump off. IfI keep the ring suspended for about5 minutes, is it safe to pick it up?

(a) No (b) Yeah, I’ll do it~

side view

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Lecture 4

InductionSelf-Inductance, RL Circuits

L/R

V

t0

L

ε

X X XX X X XX X

Recap from the last Chapter:

• Time dependent flux is generated bychange in magnetic field strength duemotion of the magnet

• Note: changing magnetic field canalso be produced by time varyingcurrent in a nearby loop

Faraday's Law of Induction

vB

N S

vB

S N

BdI/dtCan time varying current ina conductor induce EMF inin that same conductor ?

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Lecture 4

Self-Inductance

X X X X X X X

• ∴ emf induced in loop opposing initial emf

• Self-Induction: the act of a changing current through aloop inducing an opposing current in that same loop.

X X X X X X X

• Consider the loop at the right.

• ∴ magnetic field produced in the areaenclosed by the loop.

• ∴ flux through loop changes

• switch closed ⇒ current starts to flow inthe loop.

Self-Inductance

I• The magnetic field produced by the

current in the loop shown isproportional to that current.

• The flux, therefore, is also proportionalto the current.

• We define this constant of proportionalitybetween flux and current to be theinductance L.

• We can also define the inductance L,using Faraday's Law, in terms of theemf induced by a changing current.

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Lecture 4

Self-Inductance• The inductance of an inductor ( a set of coils in some

geometry, e.g., solenoid, toroid) then, like a capacitor, can becalculated from its geometry alone if the device is constructedfrom conductors and air.

• If extra material (e.g. iron core) is added, then we need to addsome knowledge of materials as we did for capacitors(dielectrics) and resistors (resistivity)