Induction Fall 2008 11/12/2018 Induction - Fall 2006

Dis here week We begin the study of magnetic induction There will be a quiz on Friday There is a new WebAssign. Exam #3 is next Monday 4/7 It will include the material on induction The end is in sight … Check the website for the Final Exam Schedule 11/12/2018 Induction - Fall 2006

Magnetic Flux   11/12/2018 Induction - Fall 2006

enclosed poles … but there ain’t no such thing! 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 11/12/2018

Examples S N Induction - Fall 2006 11/12/2018

A puzzlement .. Let’s apply this to the gap of a capacitor. Induction - Fall 2006 11/12/2018

Consider the poor little capacitor… ? CHARGING OR DISCHARGING …. HOW CAN CURRENT FLOW THROUGH THE GAP In a FIELD description?? Induction - Fall 2006 11/12/2018

Through Which Surface Do we measure the current for Ampere’s Law? Huh?? Induction - Fall 2006 11/12/2018

In the gap… DISPLACEMENT CURRENT Fixes the Problem! Induction - Fall 2006 11/12/2018

at some induction effects. Let's take a look at some induction effects. 11/12/2018 Induction - Fall 2006

A changing magnetic field INDUCES a current in a circuit loop. From The Demo .. A changing magnetic field INDUCES a current in a circuit loop. 11/12/2018 Induction - Fall 2006

Faraday’s Experiments ? 11/12/2018 Induction - Fall 2006

Insert Magnet into Coil 11/12/2018 Induction - Fall 2006

Remove Coil from Field Region 11/12/2018 Induction - Fall 2006

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That’s Strange ….. These two coils are perpendicular to each other 11/12/2018 Induction - Fall 2006

Remember the Definition of TOTAL ELECTRIC FLUX through a CLOSED surface: 11/12/2018 Induction - Fall 2006

Magnetic Flux: FB Faraday's Law Similar Definition with a special difference! Faraday's Law 11/12/2018 Induction - Fall 2006

Magnetic Flux Applies to an OPEN SURFACE only. “Quantity” of magnetism that goes through a surface. surface 11/12/2018 Induction - Fall 2006

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. xxxxxxxxxxxxxxx 11/12/2018 Induction - Fall 2006

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. xxxxxxxxxxxxxxx 11/12/2018 Induction - Fall 2006

Faraday’s Law (Michael Faraday) xxxxxxxxxxxxxxx We will get to the minus sign in a short time. 11/12/2018 Induction - Fall 2006

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. 11/12/2018 Induction - Fall 2006

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. 11/12/2018 Induction - Fall 2006

How much work? Faraday's Law emf xxxxxxxxxxxxxxx A magnetic field and an electric field are intimately connected.) 11/12/2018 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 11/12/2018 Induction - Fall 2006

We finally stated FARADAY’s LAW 11/12/2018 Induction - Fall 2006

From the equation Lentz Lentz 11/12/2018 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 11/12/2018 Induction - Fall 2006

Maxwell’s Four Equations Ampere’s Law Gauss Faraday 11/12/2018 Induction - Fall 2006

Another View Of That damned minus sign again … 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 11/12/2018 Induction - Fall 2006

The Strange World of Dr. Lentz 11/12/2018 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 11/12/2018 Induction - Fall 2006

Example of Nasty Lenz The induced magnetic field opposes the field that does the inducing! 11/12/2018 Induction - Fall 2006

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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. 11/12/2018 Induction - Fall 2006

Let’s do the Lentz Warp again ! 11/12/2018 Induction - Fall 2006

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! 11/12/2018 Induction - Fall 2006

An Example 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? 11/12/2018 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. 11/12/2018 Induction - Fall 2006

Figure 31-36 shows two parallel loops of wire having a common axis 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 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. 30-29.) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current. v 11/12/2018 Induction - Fall 2006

q B is assumed to be constant through the center of the small loop and caused by the large one. 11/12/2018 Induction - Fall 2006

The calculation of Bz q 11/12/2018 Induction - Fall 2006

More Work In the small loop: dx/dt=v 11/12/2018 Induction - Fall 2006

Which Way is Current in small loop expected to flow?? q B 11/12/2018 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. 11/12/2018 Induction - Fall 2006

moving the bar 11/12/2018 Induction - Fall 2006

Moving the Bar takes work 11/12/2018 Induction - Fall 2006

What about a SOLID loop?? Eddy Currents Energy is LOST BRAKING SYSTEM METAL Pull Eddy Currents 11/12/2018 Induction - Fall 2006

Inductors Back to Circuits for a bit …. 11/12/2018 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! 11/12/2018 Induction - Fall 2006

Remember Faraday’s Law Lentz 11/12/2018 Induction - Fall 2006

Look at the following circuit: Switch is open NO current flows in the circuit. All is at peace! 11/12/2018 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) 11/12/2018 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. 11/12/2018 Induction - Fall 2006

Close the circuit… 11/12/2018 Induction - Fall 2006

Moving right along … 11/12/2018 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 11/12/2018 Induction - Fall 2006

Consider a Solenoid l n turns per unit length 11/12/2018 Induction - Fall 2006

So…. Depends only on geometry just like C and is independent of current. 11/12/2018 Induction - Fall 2006

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. i 11/12/2018 Induction - Fall 2006

THE BIG INDUCTION Lenz with an ATTITUDE! 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! 11/12/2018 Induction - Fall 2006

Back to the real world… Switch to “a” i 11/12/2018 Induction - Fall 2006

Solution 11/12/2018 Induction - Fall 2006

Switch position “b” 11/12/2018 Induction - Fall 2006

Max Current Rate of increase = max emf VR=iR ~current 11/12/2018 Induction - Fall 2006

Solve the loop equation. 11/12/2018 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?? 11/12/2018 Induction - Fall 2006

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=e0A/d +dq 11/12/2018 Induction - Fall 2006

The calc The energy is in the FIELD !!! 11/12/2018 Induction - Fall 2006

What about POWER?? power to circuit power dissipated by resistor Must be dWL/dt 11/12/2018 Induction - Fall 2006

So Energy stored in the Capacitor 11/12/2018 Induction - Fall 2006

WHERE is the energy?? l 11/12/2018 Induction - Fall 2006

Remember the Inductor?? ????????????? 11/12/2018 Induction - Fall 2006

So … 11/12/2018 Induction - Fall 2006

ENERGY IN THE FIELD TOO! 11/12/2018 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. 11/12/2018 Induction - Fall 2006

END OF TOPIC 11/12/2018 Induction - Fall 2006