# Induction - PowerPoint PPT Presentation

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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

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### A puzzlement ..

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

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### Consider the poor little capacitor…

i

i

?

CHARGING OR DISCHARGING …. HOW CAN CURRENT

FLOW THROUGH THE GAP

In a FIELD description??

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

I=0

Huh??

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### In the gap… DISPLACEMENT CURRENT

Fixes the Problem!

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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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?

?

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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!

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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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• 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 (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.

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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

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

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### We finally stated

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### From the equation

Lentz

Lentz

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### 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

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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

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### Example of Nasty Lenz

The induced magnetic field opposes the

field that does the inducing!

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### 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.

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Let’s do the

Lentz Warp

again !

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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!) …

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### 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?

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### 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.

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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.

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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.

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### 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 ….

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### 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

Lentz

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### Look at the following circuit:

• Switch is open

• NO current flows in the circuit.

• All is at peace!

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### 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)

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### 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.

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### Close the circuit…

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### Moving right along …

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### 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

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### Consider a Solenoid

l

n turns per unit length

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### So….

Depends only on geometry just like C and

is independent of current.

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### 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

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### 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!

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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??

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E=e0A/d

+dq

+q

-q

• 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 !!!

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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.

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END OF TOPIC

Induction - Fall 2006