lec30_032807_-_Inductance

lec30_032807_-_Inductance - Inductance Faraday's Law:...

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Unformatted text preview: Inductance Faraday's Law: Faraday' Big Picture B = E d = - d dt ( B dA) = - ddt Electrostatics Circuits Magnetics Chapter 27: Magnetic Force on Charges and currents Chapter 28: Sources of Magnetic Field Chapter 29: Electromagnetic Induction Chapter 30: Self Inductance and Inductors Lenz's Law: Lenz' Motional emf: emf: Induced magnetic field will oppose the change in the system. Lenz's Law Faraday's Law Faraday' Lenz' = Binduced Wednesday, 28 March 2007 d B =B v dt I= R = B v R 1 Optics Wednesday, 28 March 2007 2 B2 Mutual Inductance Coil 1: N1 turns, radius R1 Coil 2: N2 turns, radius R2 R1 << R2 What is the flux of 2 through 1? B2 Mutual Inductance Coil 1: N1 turns, radius R1 Coil 2: N2 turns, radius R2 R1 << R2 What is the flux of 2 through 1? Through 1 turn of Coil 1: A1 2 B1 = B2 dA1 = B2 dA1 = B2 dA1 = B2 A1 = B2 (R1 ) Through 1 turn of Coil 1: B1 = N 2 Through N1 turn of Coil 1: 2 1 o I2 2 B2 dA1 A1 B2 constant A1 = N1 N 2 o I 2 2 (d 2 R2 2 B2 = N 2 o I 2 2 (d 2 R2 2 2 + R2 ) 3/ 2 B1 = N 2 o I2 2 (d 2 R2 2 2 + R2 ) 3/ 2 (R ) 3 (d +R R 2 2 3/ 2 2 2 2 ) (R ) 2 1 +R Mutual Inductance: M = N1 Wednesday, 28 March 2007 Wednesday, 28 March 2007 B1 oR R = N1 N 2 I2 2 d 2 + R2 ( 2 3/ 2 2 2 2 1 2 2 3/ 2 ) (R ) 2 1 ) 4 1 B2 Mutual Inductance Coil 1: N1 turns, radius R1 Coil 2: N2 turns, radius R2 R1 << R2 What is the flux of 2 through 1? Mutual Inductance II Solenoid 1: N1 turns, radius R1 Solenoid 2: N2 turns, radius R2 21 = B2 dA1 = B2 dA1 = N1 B2 A2 N2 2 = N1 B2 R2 B2 dA1 12 = B1 dA2 = B1 dA2 = N 2 B1 A2 N 2 = N 2 1 o I1 R2 = MI1 M = o N1 N 2 Mutual Inductance: M = N1 B1 = N 2 B2 I2 I1 Units: H(Henries) = B2 0 outside of R2 Tm 2 Vs J = = A A A2 ( ) o I 2 (R22 ) ( ) d B = -N dt Wednesday, 28 March 2007 dI Mutual = -M 2 Induced emf: 1 emf: dt dI 2 = -M 1 dt 5 = N1 = MI 2 (R ) 2 2 6 Wednesday, 28 March 2007 Question The mutual inductance M = N1 A. Current in Coil 1 B1 = N 2 B2 depends on: depends I2 I1 Tesla Coil Consider the following circuit: Vac -Vsolenoid 1 = 0 Vsolenoid 1 = Vac = = d turn 1 Vac = dt N1 Vac A1 d turn 2 = N1 A2 dt d total 1 d turn 1 = N1 dt dt A1 turn 2 A2 d turn 1 A1 d turn 2 = dt A2 dt B. Magnetic flux in Coil 2 C. N1 D. N2 E. C & D turn 1 = d turn 2 Vac A2 = dt N1 A1 N1 = 10 N 2 = 10,000 Vac 15,000 V 8 total 2 Wednesday, 28 March 2007 7 AN d turn 2 = Vac 2 2 = N2 A1 N1 dt Wednesday, 28 March 2007 2 Tesla Coil in Reverse - Transformer When electricity moves from a power plant it is put into a very high voltage to be able to travel long distances (~155,000 V to ~765,000 V). ~765,000 For us to use it, the electricity is "stepped-down" to a lower level using a stepped- down" transformer (maintaining the same frequency). This lower voltage electricity is put into the local electric wires at a substation. Once smaller substation. transformers take that voltage down to usually 7,200, the power leaves this substation. A transformer on top of a utility pole, or one connected to underground underground wires, transforms the 7,200 volts into 220-240 volts. This is then sent into 220your home over three wires. The three wires go through the electric electric meter, which measures how much electricity you use. One of the three three wires is the ground, and the other two are the positives. Some of the electrical appliances in your home use the 220-240 volts. 220These are things like a water heater, stove and oven, or air conditioner. conditioner. They have very special connections and plugs. Other devices, like your like TV or computer only use one-half of the electricity -- 110-120 volts. one110Wednesday, 28 March 2007 9 B Can Crusher C C = 147 F = 3500 V 1 U = CV 2 = 900 J 2 Must be a net force! B d B Lenz's Law: Lenz' dt Oppose the change in flux! Change the area! I induced F is inward everywhere! Wednesday, 28 March 2007 10 Self Inductance A current loop can create a flux through itself! Consider a solenoid: Energy in Inductors N I dI dt dI = 0 Multiply through by I dt dI Power Equation! I - I 2 R - LI =0 dt 1 dU L dU = Pdt = LIdI U L = LI 2 PL = 2 dt Apply Kirchhoff's Rule: Kirchhoff' B = o nI = o N I (R N ) = 2 - IR - L = B dA = BA = o L= o N 2R 2 I = LI PL = LI o N R 2 2 = o n2 A is the self inductance of the solenoid. What if the current changes? =- dI d B d = - (LI ) = - L dt dt dt 11 L = on2 A B2 B2 A 1 1 2 = U = LI 2 = o n A 2 o n2 2o 2 2 Solenoids: B = o nI Compare to the energy density of E-field, found Efrom capacitors: Inductors oppose change in the current through them! Wednesday, 28 March 2007 uB = B2 U = vol 2 o Wednesday, 28 March 2007 Energy is stored in the magnetic field through the inductor! 1 uB = o E 2 2 12 3 Energizing an RL Circuit At t = 0: 0: I=0 At t : I = /R For 0 < t < , apply Kirchhoff's Rule: Kirchhoff' De-Energizing an RL Circuit DedI dt - IR - L dI =0 dt - IR = L I= ( - IR )dt = LdI R - I dt = LdI R Rdt dI = L ( R - I ) eL = R t ( R R t L R-I) R 1- e R - t L For 0 < t < , apply Kirchhoff's Rule: Kirchhoff' dI = 0 VR2 = 0 dt I R1 = IL = R1 R1 At t = 0 (when switch is opened): Change is bad! I L = opened): R1 I R1 Switch has been closed "a long time" time" IL VL = 0, b c ( ( R - I )e = R R - t L R - I ) = ( R )e R - t L I = I f 1- e L = R - t L L dI - IR2 = 0 dt dI = IR2 dt dI is negative dt dI R = - 2 dt I L ln R R t = ln ( R - I ) L Wednesday, 28 March 2007 I R =- 2t Io L I = Ioe - t = - IR = e -1 I = I oe - R2 t L L R2 14 13 Wednesday, 28 March 2007 Good for surge protectors! R4 I1 R1 RLC Circuits I2 I3 R3 Material for Exam 3 IC = 0 IR2 = 0 L1 C I1 L2 R2 Immediately after closing the switch ... At t = 0: 0: IL1 = 0 IL2 = 0 Chapter 27: All Chapter 28: Sections 2 through 7 28.8: 3 slides from lecture Not 28.1 = (R1+ R3+ R4) At t : IL1 = 0 because the capacitor is charged! IC = 0 IL2 = 0 IR2 = 0 Chapter 29: Sections 1 through 6 Not 29.7, 29.8 I1 = I 2 + I 3 I 2 + I 3 = I1 Wednesday, 28 March 2007 - I1R4 + - I1R1 - I 3 R3 = 0 - I1R4 + - I1R1 - I 2 R2 = 0 15 Chapter 30: Sections 1 through 4 Not 30.5, 30.6 Wednesday, 28 March 2007 16 4 ...
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