# Module 22: Inductance and Magnetic Field Energy

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1 Module 22: Inductance and Magnetic Field Energy 1

2 Module 22: Outline Self Inductance Energy in Inductors Circuits with Inductors: RL Circuit 2

3 Faraday s Law of Induction dφ = B dt Changing magnetic flux induces an EMF Lenz: Induction opposes change 3

4 Self Inductance 4

5 Self Inductance What if is the effect of putting current into coil 1? There is self flux : Φ LI B Faraday s Law ε = L di dt 5

6 Calculating Self Inductance Φ total L = Φ B,self I Unit: Henry 1 H = 1 V s A 1. Assume a current I is flowing in your device 2. Calculate the B field due to that I 3. Calculate the flux due to that B field 4. Calculate the self inductance (divide out I) 6

7 Problem: Solenoid Calculate the self-inductance L of a solenoid (n turns per meter, length l, radius R) L = Φ total B,self, REMEMBER I 1. Assume a current I is flowing in your device 2. Calculate the B field due to that I 3. Calculate the flux due to that B field 4. Calculate the self inductance (divide out I) 7

8 B w Solenoid Inductance B s = = μ = enc 0 enc 0 ( ) d B μ I μ n I l B = μ ni 0 Bturn, B da BA ni 0 R Φ = = = μ π 2 NΦ L = B,turn I = N μ 0 nπ R 2 = μ 0 n 2 π R 2 l 0 0 8

9 Energy in Inductors 9

10 Inductor Behavior ε = L di dt Inductor with constant current does nothing I L 10

11 Back EMF ε = L di dt I di di dt > 0, ε < 0 L dt < 0, ε L > 0 I 11

12 Energy To Charge Inductor 1. Start with uncharged inductor 2. Gradually increase current. Must work: dw = Pdt = ε Idt = L di dt Idt = LI di 3. Integrate up to find total work done: I W = dw = LI di = 1 2 L I 2 I =0 0 12

13 Energy Stored in Inductor U = 1 L I 2 L 2 But where eeis senergy egystoed stored? 13

14 Example: Solenoid Ideal solenoid, length l, radius R, n turns/length, current I: B = μ 0 ni 0 o L = μ o n 2 π R 2 l ( )I 2 U = 1 LI 2 = 1 μ n 2 π R 2 l B 2 2 o U B = B B 2 2μ o π R2 l Energy Density Volume 14

15 Energy Density Energy is stored in the magnetic field! B 2 u B = B2 B 2μ o : Magnetic Energy Density ε E 2 u = ε o E E 2 : Electric Energy Density 15

16 Problem: Coaxial Cable Inner wire: r = a Outer wire: r = b I X I 1. How much energy is stored per unit length? 2. What is inductance per unit length? HINTS: This does require an integral The EASIEST way to do (2) is to use (1) 16

17 Technology Many Applications of Faraday s Law 17

18 Demos: Breaking Circuits Big Inductor Marconi Coil The Question: What happens if big ΔI, small Δt 18

19 The Point: Big EMF L di dt ε = L dt Big L Big di Huge ε Small dt 19

20 First: Mutual Inductance 20

21 Demonstration: Remote Speaker 21

22 Mutual Inductance Current I 2 in coil 2, induces magnetic flux Φ 12 in coil 1. Mutual inductance M 12 : Φ M I M 12 = M 21 = M Change current in coil 2? Induce EMF in coil 1: di ε 12 M 12 di dt 22

23 Transformer Step-up transformer Flux Φ through each turn same: ε = N dφ ; ε = N dφ ε p N p dt ; ε s N s dt ε s = N s ε p N p N s > N p :step-up transformer N s < N p : step-down transformer 23

24 Demonstrations: One Turn Secondary: Nail Many Turn Secondary: Jacob s Ladder 24

25 Concept Question: Residential Transformer If the transformer in the can looks like the picture, how is it connected? 1. House=Left, Line=Right 2. Line=Left, House=Right 3. I don t know 25

26 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) 26

27 Transmission of Electric Power Power loss can be greatly reduced if transmitted at high voltage 27

28 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 Ω. Calculate the power loss if the power is sent at (a) 240 V, and (b) 24,000 V. (a) 5 I = P W = V V = 500A 83% loss!! P L = I 2 R = (500A) 2 (0.40Ω) = 100kW (b) I = P 5 V = W V = 5.0 A P L = I 2 R = (5.0 A) 2 (0.40Ω) = 10W % loss 28

29 Transmission lines We just calculated that I 2 R is smaller for bigger voltages. What about V 2 /R? Isn t that bigger? Why doesn t that matter? 29

30 Brakes 30

31 Magnet Falling Through a Ring Link to movie What happened to kinetic energy of magnet? 31

32 Demonstration: Eddy Current Braking 32

33 Eddy Current Braking What happened to kinetic energy of disk? 33

34 Eddy Current Braking The magnet induces currents in the metal that dissipate the energy through Joule heating: XX XX ω 1. Current is induced counter-clockwise (out from center) 2. Force is opposing motion (creates slowing torque) 34

35 Eddy Current Braking The magnet induces currents in the metal that dissipate the energy through Joule heating: XX XX ω 1. Current is induced clockwise (out from center) 2. Force is opposing motion (creates slowing torque) 3. EMF proportional to ω 4.. F ε 2 R 35

36 Demonstration: Levitating Magnet Link to Movie 36

37 MIT OpenCourseWare SC Physics II: Electricity and Magnetism Fall 2010 For information about citing these materials or our Terms of Use, visit:

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