1. Induction
Fall 2008
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Induction - Fall 2006

2. 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
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Induction - Fall 2006

3. Magnetic Flux
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4. 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
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5. Examples
S N
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6. A puzzlement .. Let’s apply this to the gap of a capacitor.
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7. Consider the poor little capacitor…?
CHARGING OR DISCHARGING …. HOW CAN CURRENT
FLOW THROUGH THE GAP
In a FIELD description??
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8. Through Which Surface Do we measure the current for Ampere’s Law?
Huh??
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9. In the gap… DISPLACEMENT CURRENT
Fixes the Problem!
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10. at some induction effects.
Let's take a look
at some induction effects.
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Induction - Fall 2006

11. 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.
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Induction - Fall 2006

12. Faraday’s Experiments?
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13. Insert Magnet into Coil
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14. Remove Coil from Field Region
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15. 11/12/2018
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16. That’s Strange ….. These two coils are perpendicular to each other
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17. Remember the Definition of TOTAL ELECTRIC FLUX through a CLOSED surface:
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18. Magnetic Flux: FB Faraday's Law
Similar Definition with a special difference!
Faraday's Law
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19. Magnetic Flux Applies to an OPEN SURFACE only.
“Quantity” of magnetism that goes through a surface.
surface
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20. 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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Induction - Fall 2006

21. 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.
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Induction - Fall 2006

22. Faraday’s Law (Michael Faraday)
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We will get to the minus sign in a short time.
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23. Faraday’s Law (The Minus Sign)
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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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24. Faraday’s Law (More on the Minus Sign)
The minus sign means that the current goes the other way.
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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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25. How much work? Faraday's Law emf xxxxxxxxxxxxxxx
A magnetic field and an electric field are
intimately connected.)
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26. 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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27. We finally stated
FARADAY’s LAW
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28. From the equation
Lentz
Lentz
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29. 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
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30. Maxwell’s Four Equations
Ampere’s Law
Gauss
Faraday
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31. 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.
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32. The Strange World of Dr. Lentz
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33. 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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34. Example of Nasty Lenz The induced magnetic field opposes the
field that does the inducing!
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35. 11/12/2018
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36. 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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37. Let’s do the
Lentz Warp
again !
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38. 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!
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Induction - Fall 2006

39. 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?
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40. 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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41. Figure 31-36 shows two parallel loops of wire having a common axis
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
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42. q
B is assumed to be constant through the center of the small loop and caused by the large one.
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43. The calculation of Bz q
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44. More Work
In the small loop:
dx/dt=v
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45. Which Way is Current in small loop expected to flow??q B
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46. 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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47. moving the bar
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48. Moving the Bar takes work
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49. What about a SOLID loop?? Eddy Currents Energy is LOST BRAKING SYSTEM
METAL
Pull
Eddy Currents
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50. Inductors Back to Circuits for a bit …. 11/12/2018
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51. 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!
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52. Remember Faraday’s Law
Lentz
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53. Look at the following circuit:
Switch is open
NO current flows in the circuit.
All is at peace!
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54. 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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55. 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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56. Close the circuit…
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57. Moving right along …
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58. 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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59. Consider a Solenoid l n turns per unit length 11/12/2018
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60. So…. Depends only on geometry just like C and
is independent of current.
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61. 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
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62. 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!
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63. Back to the real world… Switch to “a” i 11/12/2018
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64. Solution
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65. Switch position “b”
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66. Max Current Rate of increase = max emf VR=iR ~current 11/12/2018
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67. Solve the loop equation.
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68. 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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69. 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
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70. The calc The energy is in the FIELD !!! 11/12/2018
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71. What about POWER?? power to circuit power dissipated by resistor
Must be dWL/dt
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72. So
Energy
stored
in the
Capacitor
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73. WHERE is the energy?? l
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74. Remember the Inductor??
?????????????
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75. So …
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76. ENERGY IN THE FIELD TOO!
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77. 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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78. END OF TOPIC
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