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## Induction

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### The Strange World of Dr. Lentz

### Lenz’s Law

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

Consider the poor little capacitor…

i

i

?

CHARGING OR DISCHARGING …. HOW CAN CURRENT

FLOW THROUGH THE GAP

In a FIELD description??

Induction - Fall 2006

Induction - Fall 2006

Insert Magnet into Coil

Induction - Fall 2006

Remove Coil from Field Region

Induction - Fall 2006

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

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

Induction - Fall 2006

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

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 PAGEInduction - Fall 2006

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!

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

The toast will always fall buttered side down!

Lenz’s LawAn 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

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

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

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

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

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

Solution

Induction - Fall 2006

Switch position “b”

Induction - Fall 2006

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

+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

So …

Induction - Fall 2006

Induction - Fall 2006

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

Induction - Fall 2006

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