Induction and Inductance Chapter 30

Magnetic Flux

Insert Magnet into Coil

Remove Coil from Field Region

A changing magnetic field INDUCES a current in a circuit loop. From The Demo .. First experiment Second experiment A changing magnetic field INDUCES a current in a circuit loop.

Faraday’s Experiments ?

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

Definition of TOTAL ELECTRIC FLUX through a surface:

passing through a surface. Magnetic Flux: F THINK OF MAGNETIC FLUX as the “AMOUNT of Magnetism” passing through a surface.

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) The minus sign means that the current goes the other way. xxxxxxxxxxxxxxx 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? Faraday's Law emf xxxxxxxxxxxxxxx A magnetic field and an electric field are intimately connected.)

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

We finally stated FARADAY’s LAW

From the equation Lentz Lentz

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 (chapter 32 .. Just a Preview!) Gauss Faraday

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

Lenz’s Law Induced Magnetic Fields always FIGHT to stop what you are trying to do!

Example of 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.

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!) …

#1 CHAPTER 30 The field in the diagram creates a flux given by FB=6t2+7t in 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?

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.

#21 Figure 30-50 shows two parallel loops of wire having a common axis #21 Figure 30-50 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. 29-27.) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current. v

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

The calculation of Bz q

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

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

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

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

Inductors 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!

Remember Faraday’s Law Lentz

Look at the following circuit: Switch is open NO current flows in the circuit. All is at peace!

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)

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.

Close the circuit…

Moving right along …

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

Consider a Solenoid l n turns per unit length

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

Inductive Circuit i 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.

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.

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

Solution

Switch position “b”

Max Current Rate of increase = max emf VR=iR ~current

Solve the loop equation.

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??

For an answer Return to the Big C +q -q E=e0A/d +dq 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

The calc The energy is in the FIELD !!!

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

So Energy stored in the Coil

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.

10. A uniform magnetic field B increases in magnitude with time t as given by Fig. 30-43b, where the vertical axis scale is set by Bs=9 mT and the horizontal scale is set by ts=3 s . A circular conducting loop of area A= 8x10-4 m2 lies in the field, in the plane of the page. The amount of charge q passing point A on the loop is given in Fig. 30-43c as a function of t, with the vertical axis scale set by qs=3 mC and the horizontal axis scale again set by ts=3 s. What is the loop's resistance?

29.  If 50.0 cm of copper wire (diameter=1mm ) is formed into a circular loop and placed perpendicular to a uniform magnetic field that is increasing at the constant rate of 10.0 mT/s, at what rate is thermal energy generated in the loop?

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