1. W10D1: Inductance and Magnetic Field Energy
Today’s Reading Assignment W10D1 Inductance & Magnetic Energy Course Notes: Sections
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2. Announcements Math Review Week 10 Tuesday from 9-11 pm in 32-082
PS 7 due Week 10 Tuesday at 9 pm in boxes outside or
Next Reading Assignment W10D2 DC Circuits & Kirchhoff’s Loop Rules Course Notes: Sections
Exam 3 Thursday April :30 pm –9:30 pm
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3. Outline Faraday Law Problem Solving Faraday Law Demonstrations
Mutual Inductance
Self Inductance
Energy in Inductors
Transformers
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4. Faraday’s Law of Induction
If C is a stationary closed curve and S is a surface spanning C then
The changing magnetic flux through S induces a non-electrostatic electric field whose line integral around C is non-zero
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5. Problem: Calculating Induced Electric Field
Consider a uniform magnetic field which points into the page and is confined to a circular region with radius R. Suppose the magnitude increases with time, i.e. dB/dt > 0. Find the magnitude and direction of the induced electric field in the regions (i) r < R, and (ii) r > R. (iii) Plot the magnitude of the electric field as a function r.
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6. Faraday’s Law Demonstrations
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7. Demonstration: Electric Guitar H32
Pickups
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8. Electric Guitar
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9. Demonstration: Aluminum Plate between Pole Faces of a Magnet H Copper Pendulum Between Poles of a Magnet H13
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10. What happened to kinetic energy of pendulum?
Eddy Current Braking
What happened to kinetic energy of pendulum?
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11. Eddy Current Braking
The magnet induces currents in the metal that dissipate the energy through Joule heating:
Current is induced counter-clockwise (out from center)
Force is opposing motion (creates slowing torque)
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12. Eddy Current Braking
The magnet induces currents in the metal that dissipate the energy through Joule heating:
Current is induced clockwise (out from center)
Force is opposing motion (creates slowing torque)
EMF proportional to angular frequency
Class 21

13. Demonstration: 26-152 Levitating Magnet H28 32-082 Levitating Coil on an Aluminum Plate H15
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14. Mutual Inductance
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15. Demonstration: Two Small Coils and Radio H31
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16. Mutual Inductance
Current I2 in coil 2, induces magnetic flux F12 in coil 1.
“Mutual inductance” M12:
Change current in coil 2?
Induce EMF in coil 1:
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17. Group Problem: Mutual Inductance
An infinite straight wire carrying current I is placed to the left of a rectangular loop of wire with width w and length l. What is the mutual inductance of the system?
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18. Self Inductance
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19. Self Inductance Faraday’s Law
What if is the effect of putting current into coil 1?
There is “self flux”:
Faraday’s Law
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20. Calculating Self Inductance
Unit: Henry
Assume a current I is flowing in your device
Calculate the B field due to that I
Calculate the flux due to that B field
Calculate the self inductance (divide out I)
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21. Worked Example: Solenoid
Calculate the self-inductance L of a solenoid (n turns per meter, length , radius R)
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22. Week 09, Day 2
Solenoid Inductance
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23. Concept Question: Solenoid
A very long solenoid consisting of N turns has radius R and length d, (d>>R).  Suppose the number of turns is halved keeping all the other parameters fixed. The self inductance
remains the same.
doubles.
is halved.
is four times as large.
is four times as small.
None of the above.
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24. Concept Q. Ans.: Solenoid
Solution 5. The self-induction of the solenoid is equal to the total flux through the object which is the product of the number of turns time the flux through each turn. The flux through each turn is proportional to the magnitude of magnetic field which is proportional to the number of turns per unit length or hence proportional to the number of turns. Hence the self-induction of the solenoid is proportional to the square of the number of turns. If the number of turns is halved keeping all the other parameters fixed then he self inductance is four times as small.
Class 23

25. Group Problem: Toroid
Calculate the self-inductance L of a toroid with a square cross section with inner radius a, outer radius b = a+h, (height h) and N square windings .
REMEMBER
Assume a current I is flowing in your device
Calculate the B field due to that I
Calculate the flux due to that B field
Calculate the self inductance (divide out I)
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26. Energy in Inductors
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27. Inductor Behavior L I Inductor with constant current does nothing
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28. Week 09, Day 2
Back EMF I I
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29. Demos: 26-152 Back “emf” in Large Inductor H17 32-082 Marconi Coil H12
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30. Marconi Coil: On the Titanic
Week 09, Day 2
Marconi Coil: On the Titanic
Another ship
Same era
Titanic
Marconi
Telegraph
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31. Marconi Coil: Titanic Replica
Week 09, Day 2
Marconi Coil: Titanic Replica
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32. Big L Big dI Small dt The Point: Big EMF Huge EMF Week 09, Day 2
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33. Energy To “Charge” Inductor
1. Start with “uncharged” inductor
Gradually increase current. Must do work:
3. Integrate up to find total work done:
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34. Energy Stored in Inductor
But where is energy stored?
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35. Example: Solenoid Ideal solenoid, length l, radius R,
n turns/length, current I:
Volume
Energy Density
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36. Energy is stored in the magnetic field
Energy Density
Energy is stored in the magnetic field
Magnetic Energy Density
Energy is stored in the electric field
Electric Energy Density
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37. Worked Example: Energy Stored in Toroid
Consider a toroid with a square cross section with inner radius a, outer radius b = a+h, (height h) and N square windings with current I. Calculate the energy stored in the magnetic field of the torus.
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38. Solution: Energy Stored in Toroid
The magnetic field in the torus is given by The stored energy is then  The self-inductance is
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39. Group Problem: Coaxial Cable
Inner wire: r = a
Outer wire: r = b
How much energy is stored per unit length?
What is inductance per unit length?
HINTS: This does require an integral
The EASIEST way to do (2) is to use (1)
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40. Transformer Step-up transformer Flux F through each turn same:
Ns > Np: step-up transformer
Ns < Np: step-down transformer
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41. Demonstrations: One Turn Secondary: Nail H Many Turn Secondary: Jacob’s Ladder H Variable Turns Around a Primary Coil H9
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42. Concept Question: Residential Transformer
Week 9, Day 2
Concept Question: Residential Transformer
If the transformer in the can looks like the picture, how is it connected?
House=Left, Line=Right
Line=Left, House=Right
I don’t know
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43. Answer: Residential Transformer
Week 9, Day 2
Answer: Residential Transformer
Answer: 1. House on left, line on right
The house needs a lower voltage, so we step down to the house (fewer turns on house side)
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44. Transmission of Electric Power
Power loss can be greatly reduced if transmitted at high voltage
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45. Electrical Power Power is change in energy per unit time
So power to move current through circuit elements:
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46. Power - Resistor
Moving across a resistor in the direction of current decreases your potential. Resistors always dissipate power
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47. Example: Transmission lines
An average of 120 kW of electric power is sent from a power plant. The transmission lines have a total resistance of 0.40 W. Calculate the power loss if the power is sent at (a) 240 V, and (b) 24,000 V.
(a)
83% loss!!
(b)
0.0083% loss
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48. Transmission lines We just calculated that I2R is smaller
for bigger voltages.
What about V2/R? Isn’t that bigger?
Why doesn’t that matter?
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