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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

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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 !

Induction - Fall 2006

OK

Let's take a look.

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From The Demo ..

A changing magnetic field INDUCES

a current in a circuit loop.

Induction - Fall 2006

Faraday’s Experiments

?

?

Induction - Fall 2006

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

Induction - Fall 2006

Magnetic Flux:FB
• Similar Definition with a special difference!

Faraday's Law

Induction - Fall 2006

Magnetic Flux
• Applies to an OPEN SURFACE only.
• “Quantity” of magnetism that goes through a surface.

surface

Induction - Fall 2006

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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.

Induction - Fall 2006

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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.

Induction - Fall 2006

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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.

Induction - Fall 2006

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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

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

Induction - Fall 2006

Lenz’s Law

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

i.e... Murphy’s Law for Magnets

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Example of Nasty Lenz

The induced magnetic field opposes the

field that does the inducing!

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

Induction - Fall 2006

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

Induction - Fall 2006

dx/dt=v

More Work

In the small loop:

Induction - Fall 2006

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

Induction - Fall 2006

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…

Induction - Fall 2006

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”

Induction - Fall 2006

Solution

Induction - Fall 2006

Switch position “b”

Induction - Fall 2006

Max Current Rate of

increase = max emf

VR=iR

~current

Induction - Fall 2006

Solve the loop equation.

Induction - Fall 2006

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

Induction - Fall 2006

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

Induction - Fall 2006

So

Energy

stored

in the

Capacitor

Induction - Fall 2006

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