Magnetic Induction April 1, 2005

Happenings Short Quiz Today New Topic: Magnetic Induction (Chapter 30) Quiz NEXT Friday Exam #3 – April 15 th. Should I do a class on Vectors and how we use them??? [vote]

From The Demo..

Faraday’s Experiments ????

Insert Magnet into Coil

Remove Coil from Field Region

That’s Strange ….. These two coils are perpendicular to each other

Remember Electric Flux? Did you really think you were through with this kind of concept???

We discussed the normal component of the Electric Field vector 

DEFINITION: Element of Flux through a surface E NORMAL AA E  =E NORMAL x  A (a scalar)

“Element” of Flux of a vector E leaving a surface n is a unit OUTWARD pointing vector.

This flux was LEAVING the closed surface 

Definition of TOTAL ELECTRIC FLUX through a surface:

There is ANOTHER Kind of FLUX  THINK OF MAGNETIC FLUX as the “AMOUNT of Magnetism” passing through a surface. Don’t quote me on this!!!

What is this thing called FLUX? Magnetic Flux is a Scalar The UNIT of FLUX is the weber  1 weber = 1 T-m 2

Consider a Loop Magnetic field passing through the loop is CHANGING. FLUX is changing. There is 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. xxxxxxxxxxxxxxx

Faraday’s Law (Michael Faraday) 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. xxxxxxxxxxxxxxx

Faraday’s Law (Michael Faraday) xxxxxxxxxxxxxxx We will get to the minus sign in a short time.

Faraday’s Law (The Minus Sign) xxxxxxxxxxxxxxx Using the right hand rule, we would expect the direction of the current to be in the direction of the arrow shown.

Faraday’s Law (More on the Minus Sign) xxxxxxxxxxxxxxx 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.

How much work? xxxxxxxxxxxxxxx A magnetic field and an electric field are intimately connected.) emf

So.. FARADAY’s LAW

Flux The MAGNITUDE of the emf induced in a loop is equal to the negative of the rate at which the magnetic flux through the loop changes with time.

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

Maxwell’s Equations (Next Course.. Just a Preview!) Gauss Faraday

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

Lenz’s Law Induced Magnetic Fields always FIGHT to stop what you are trying to do! i.e... Murphy’s Law for Magnets

Example of Nasty Lenz The induced magnetic field opposes the field that does the inducing!

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.

Take a deep breath!!

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!) … OR The toast will always fall buttered side down!

An Example The field in the diagram creates a flux given by  B =6t 2 +7t in milliWebers and t is in seconds. (a)What is the emf when t=2 seconds? (b) What is the direction of the current in the resistor R?

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.

Figure 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 current i 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 ) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current. v

This combines some stuff and is a good review for the final! Or Not.  B is assumed to be constant through the center of the small loop and caused by the large one.

The calculation of B z 

More Work In the small loop: dx/dt=v

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

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.

moving the bar

Moving the Bar takes work v

What about a SOLID loop?? METAL Pull Energy is LOST BRAKING SYSTEM

Back to Circuits for a bit ….

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!

Definition of Inductance L UNIT of Inductance = 1 henry = 1 T- m 2 /A   is the flux near the center of the coil

Consider a Solenoid n turns per unit length l

So…. Depends only on geometry just like C and is independent of current.

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

Consider a “uniform short” coil

From before

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.

Trying to make an equation out of this…..

Back to the real world… i Switch to “a”

Solution

Switch position “b”

Max Current Rate of increase = max emf

Solve the loop equation.

IMPORTANT QUESTION Switch closes. No emf Current flows for a while It flows through R Energy is conserved (i 2 R) WHERE DOES THE ENERGY COME FROM??

For an answer Return 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 +q -q E=  0 A/d +dq

The calc The energy is in the FIELD !!!

What about POWER?? power to circuit power dissipated by resistor Must be dW L /dt

So Energy stored in the Capacitor

WHERE is the energy?? l

Remember the Inductor?? ?????????????

So …

ENERGY IN THE FIELD TOO!

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.