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Induction. Fall 2006. Magnetic Flux. For a CLOSED Surface we might expect this to be equal to some constant times the enclosed poles … but there ain’t no such thing!. CLOSED SURFACE. Examples. S N. A puzzlement . Let’s apply this to the gap of a capacitor.

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Induction

Fall 2006

Induction - Fall 2006


Magnetic Flux

For a CLOSED Surface we might expect this to be equal to some constant times the

enclosed poles … but there ain’t no such thing!

CLOSED SURFACE

Induction - Fall 2006


Examples

S N

Induction - Fall 2006


A puzzlement ..

Let’s apply this to the gap of a capacitor.

Induction - Fall 2006


Consider the poor little capacitor…

i

i

?

CHARGING OR DISCHARGING …. HOW CAN CURRENT

FLOW THROUGH THE GAP

In a FIELD description??

Induction - Fall 2006


Through Which Surface Do we measure the current for Ampere’s Law?

I=0

Huh??

Induction - Fall 2006


In the gap… DISPLACEMENT CURRENT

Fixes the Problem!

Induction - Fall 2006


Let's DO the Demo !

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OK

Let's take a look.

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From The Demo ..

A changing magnetic field INDUCES

a current in a circuit loop.

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Faraday’s Experiments

?

?

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Insert Magnet into Coil

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Remove Coil from Field Region

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That’s Strange …..

These two coils are perpendicular to each other

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Remember the Definition of TOTAL ELECTRIC FLUX through a CLOSED surface:

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Magnetic Flux:FB

  • Similar Definition with a special difference!

Faraday's Law

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Magnetic Flux

  • Applies to an OPEN SURFACE only.

  • “Quantity” of magnetism that goes through a surface.

surface

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Consider a Loop

  • Magnetic field passing through the loop is CHANGING.

  • FLUX is changing.

  • There must be an emf developed around the loop.

    • A current develops (as we saw in demo)

  • Work has to be done to move a charge completely around the loop.

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Faraday’s Law (Michael Faraday)

  • Again, for a current to flow around the circuit, there must be an emf.

  • (An emf is a voltage)

  • The voltage is found to increase as the rate of change of flux increases.

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Faraday’s Law (Michael Faraday)

We will get to the minus sign in a short time.

Induction - Fall 2006


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Faraday’s Law (The Minus Sign)

Using the right hand rule, we

would expect the direction

of the current to be in the

direction of the arrow shown.

Induction - Fall 2006


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Faraday’s Law (More on the Minus Sign)

The minus sign means that the current goes the other way.

This current will produce a magnetic field that would be coming OUT of the page.

The Induced Current therefore creates a magnetic field that OPPOSES the attempt to INCREASE the magnetic field! This is referred to as Lenz’s Law.

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How much work?

emf

Faraday's Law

A magnetic field and an electric field are

intimately connected.)

Induction - Fall 2006


The Strange World of Dr. Lentz

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MAGNETIC FLUX

  • This is an integral over an OPEN Surface.

  • Magnetic Flux is a Scalar

  • The UNIT of FLUX is the weber

    • 1 weber = 1 T-m2

Induction - Fall 2006


We finally stated

FARADAY’s LAW

Induction - Fall 2006


From the equation

Lentz

Lentz

Induction - Fall 2006


Flux Can Change

  • If B changes

  • If the AREA of the loop changes

  • Changes cause emf s and currents and consequently there are connections between E and B fields

  • These are expressed in Maxwells Equations

Induction - Fall 2006


Three of Maxwell’s Four Equations(Next Course .. Just a Preview!)

Ampere’s Law

Gauss

Faraday

Induction - Fall 2006


The Flux into the page begins to increase.

An emf is induced around a loop

A current will flow

That current will create a new magnetic field.

THAT new field will change the magnetic flux.

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Another View Of That damned minus sign again …..SUPPOSE that B begins to INCREASE its MAGNITUDE INTO THE PAGE

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Lenz’s Law

Induced Magnetic Fields always FIGHT to stop what you are trying to do!

i.e... Murphy’s Law for Magnets

Induction - Fall 2006


Example of Nasty Lenz

The induced magnetic field opposes the

field that does the inducing!

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Induction - Fall 2006


Don’t Hurt Yourself!

The current i induced in the loop has the direction

such that the current’s magnetic field Bi opposes the

change in the magnetic field B inducing the current.

Induction - Fall 2006


Let’s do the

Lentz Warp

again !

Induction - Fall 2006


OR

The toast will always fall buttered side down!

Lenz’s Law

An induced current has a direction

such that the magnetic field due to

the current opposes the change in

the magnetic flux that induces the

current. (The result of the negative sign!) …

Induction - Fall 2006


An Example

  • The field in the diagram

  • creates a flux given by

  • FB=6t2+7tin milliWebers

  • and t is in seconds.

  • What is the emf when

  • t=2 seconds?

  • (b) What is the direction

  • of the current in the

  • resistor R?

Induction - Fall 2006


This is an easy one …

Direction? B is out of the screen and increasing.

Current will produce a field INTO the paper

(LENZ). Therefore current goes clockwise and R

to left in the resistor.

Induction - Fall 2006


Figure 31-36 shows two parallel loops of wire having a common axis. The smaller loop (radius r) is above the larger loop (radius R) by a distance x >>R. Consequently, the magnetic field due to the currenti in the larger loop is nearly constant throughout the smaller loop. Suppose that x is increasing at the constant rate of dx/dt = v. (a) Determine the magnetic flux through the area bounded by the smaller loop as a function of x. (Hint: See Eq. 30-29.) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current.

v

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q

B is assumed to be constant through the center of the small loop and caused by the large one.

Induction - Fall 2006


q

The calculation of Bz

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dx/dt=v

More Work

In the small loop:

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q

Which Way is Current in small loop expected to flow??

B

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What Happens Here?

  • Begin to move handle as shown.

  • Flux through the loop decreases.

  • Current is induced which opposed this decrease – current tries to re-establish the B field.

Induction - Fall 2006


moving the bar

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Moving the Bar takes work

v

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What about a SOLID loop??

Energy is LOST

BRAKING SYSTEM

METAL

Pull

Eddy Currents

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Inductors

Back to Circuits for a bit ….

Induction - Fall 2006


Definition

Current in loop produces a magnetic field

in the coil and consequently a magnetic flux.

If we attempt to change the current, an emf

will be induced in the loops which will tend to

oppose the change in current.

This this acts like a “resistor” for changes in current!

Induction - Fall 2006


Remember Faraday’s Law

Lentz

Induction - Fall 2006


Look at the following circuit:

  • Switch is open

  • NO current flows in the circuit.

  • All is at peace!

Induction - Fall 2006


Close the circuit…

  • After the circuit has been close for a long time, the current settles down.

  • Since the current is constant, the flux through the coil is constant and there is no Emf.

  • Current is simply E/R (Ohm’s Law)

Induction - Fall 2006


Close the circuit…

  • When switch is first closed, current begins to flow rapidly.

  • The flux through the inductor changes rapidly.

  • An emf is created in the coil that opposes the increase in current.

  • The net potential difference across the resistor is the battery emf opposed by the emf of the coil.

Induction - Fall 2006


Close the circuit…

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Moving right along …

Induction - Fall 2006


Definition of Inductance L

UNIT of Inductance = 1 henry = 1 T- m2/A

FB is the flux near the center of one of the coils

making the inductor

Induction - Fall 2006


Consider a Solenoid

l

n turns per unit length

Induction - Fall 2006


So….

Depends only on geometry just like C and

is independent of current.

Induction - Fall 2006


Inductive Circuit

  • Switch to “a”.

  • Inductor seems like a short so current rises quickly.

  • Field increases in L and reverse emf is generated.

  • Eventually, i maxes out and back emf ceases.

  • Steady State Current after this.

i

Induction - Fall 2006


THE BIG INDUCTION

  • As we begin to increase the current in the coil

  • The current in the first coil produces a magnetic field in the second coil

  • Which tries to create a current which will reduce the field it is experiences

  • And so resists the increase in current.

Lenz with an ATTITUDE!

Induction - Fall 2006


i

Back to the real world…

Switch to “a”

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Solution

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Switch position “b”

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Max Current Rate of

increase = max emf

VR=iR

~current

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Solve the loop equation.

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IMPORTANT QUESTION

  • Switch closes.

  • No emf

  • Current flows for a while

  • It flows through R

  • Energy is conserved (i2R)

WHERE DOES THE ENERGY COME FROM??

Induction - Fall 2006


E=e0A/d

+dq

+q

-q

For an answerReturn to the Big C

  • We move a charge dq from the (-) plate to the (+) one.

  • The (-) plate becomes more (-)

  • The (+) plate becomes more (+).

  • dW=Fd=dq x E x d

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The calc

The energy is in

the FIELD !!!

Induction - Fall 2006


What about POWER??

power

to

circuit

power

dissipated

by resistor

Must be dWL/dt

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So

Energy

stored

in the

Capacitor

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WHERE is the energy??

l

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Remember the Inductor??

?????????????

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So …

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ENERGY IN THEFIELD TOO!

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IMPORTANT CONCLUSION

  • A region of space that contains either a magnetic or an electric field contains electromagnetic energy.

  • The energy density of either is proportional to the square of the field strength.

Induction - Fall 2006


END OF TOPIC

Induction - Fall 2006


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