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lec31_030207_-_Inductance_II

lec31_030207_-_Inductance_II - Inductance II B2 Mutual...

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Unformatted text preview: Inductance II B2 Mutual Inductance: I R22 (R12 ) B1 = N1 N 2 o 2 2 2 2 (d + R2 )3 / 2 Self Inductance A current loop can create a flux through itself! Consider a solenoid: B = o nI = o N I oR12 R22 M = N1 B1 = N 2 B2 = N1 N 2 2 3/ 2 I2 I1 2 d 2 + R2 ( ) = B dA = BA = o L= N I (R 2 N ) = o N 2R 2 I = LI Mutual Induced emf: emf: dI dI 2 2 = -M 1 1 = - M dt dt 21 = B2 dA1 = N1 M = o Monday, 2 April 2007 o N 2R 2 = o n2 A is the self inductance of the solenoid. What if the current changes? N2 2 2 o I 2 (R22 ) = MI 2 =- dI d B d = - (LI ) = - L dt dt dt 2 N1 N 2 (R ) 1 Inductors oppose change in the current through them! Monday, 2 April 2007 Energy in Inductors dI - IR - L = 0 Multiply through by I dt dI 2 Power Equation! I - I R - LI =0 dt 1 dU L dU = Pdt = LIdI U L = LI 2 PL = 2 dt Compare to the energy density of E-field, found Efrom capacitors: Apply Kirchhoff's Rule: Kirchhoff' Energizing an RL Circuit At t = 0: 0: I=0 At t : I = /R For 0 < t < , apply Kirchhoff's Rule: Kirchhoff' - IR - L PL = LI dI dt dI =0 dt - IR = L I= dI dt ( - IR )dt = LdI R 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 uB = B2 U = vol 2 o Monday, 2 April 2007 Energy is stored in the magnetic field through the inductor! 1 uE = o E 2 2 3 - I dt = LdI R Rdt dI = L ( R - I ) R R t = ln ( R - I ) L eL = R t ( R R t L R-I) = R R - t L R 1- e R - t L ( ( R - I )e R - I ) = ( R )e R - t L I = I f 1- e - t - IR = e -1 = L R 4 Monday, 2 April 2007 1 De-Energizing an RL Circuit DedI IR VL = 0, b c L = 0 VR2 = 0 IL dt I R1 = IL = R1 R1 At t = 0 (when switch is opened): Change is bad! I L = opened): R1 1 Switch has been closed "a long time" time" R4 I1 R1 RLC Circuits I2 I3 R3 L1 C I1 L2 R2 Immediately after closing the switch ... At t = 0: 0: IL1 = 0 IL2 = 0 IC = 0 IR2 = 0 For 0 < t < , apply Kirchhoff's Rule: Kirchhoff' = (R1+ R3+ R4) At t : IL1 = 0 because the capacitor is charged! IC = 0 IL2 = 0 IR2 = 0 L L dI - IR2 = 0 dt dI = IR2 dt dI is negative dt dI R = - 2 dt I L ln I R =- 2t Io L I = I oe - I = Ioe - t = R2 t L L R2 5 I1 = I 2 + I 3 I 2 + I 3 = I1 Monday, 2 April 2007 - I1R4 + - I1R1 - I 3 R3 = 0 - I1R4 + - I1R1 - I 2 R2 = 0 6 Monday, 2 April 2007 Good for surge protectors! Material for Exam 3 Chapter 27: All Chapter 28: Sections 2 through 7 28.8: 3 slides from lecture Not 28.1 Material for Exam 3 Chapter 27: All Chapter 29: Sections 1 through 6 Not 29.7, 29.8 Chapter 30: Sections 1 through 4 Not 30.5, 30.6 Monday, 2 April 2007 7 Monday, 2 April 2007 8 2 Chapter 27 Magnetic Force All sections! Force on individual moving charges: FB = qv B Magnetic Flux: Gauss's Law: Gauss' Material for Exam 3 Chapter 27: All Chapter 28: Sections 2 through 7 28.8: 3 slides from lecture Not 28.1 B = B dA B dA = 0 Motion of charges in B-field circles, helixes, traps B Velocity Selector, E/M Experiments, Mass Spectrometers Force on current-carrying wire: dFB = Id B current Force & Torque on a current loop: = B Monday, 2 April 2007 9 Monday, 2 April 2007 10 Chapter 28 Sources of B Sections 2 through 7 (NOT 1) (NOT 1) Biot-Savart Law Biot- Material for Exam 3 Chapter 27: All Chapter 28: Sections 2 through 7 28.8: 3 slides from lecture Not 28.1 ^ I d r Magnetic Field of current element: dB = o 2 4 r Forces between parallel current-carrying wires current Ampere's Law: Ampere' B d = o I thru Chapter 29: Sections 1 through 6 o NI 2r Not 29.7, 29.8 Solenoids B = o nI ; Toroids B = Magnetic Materials: Ferromagnetic, Paramagnetic, & Diamagnetic Monday, 2 April 2007 11 Monday, 2 April 2007 12 3 Chapter 29 Electromagnetic Induction Sections 1 through 6 (NOT 7 or 8) (NOT 8) Faraday's Law: Faraday' Lenz's Law: Lenz' Change is Bad! Material for Exam 3 Chapter 27: All Chapter 28: Sections 2 through 7 28.8: 3 slides from lecture Not 28.1 d d B B dA = - = Ed = - dt dt Chapter 29: Sections 1 through 6 Not 29.7, 29.8 Chapter 30: Sections 1 through 4 Not 30.5, 30.6 Eddy Currents Monday, 2 April 2007 13 Monday, 2 April 2007 14 Chapter 30 Inductance Sections 1 through 4 (NOT 5 or 6) (NOT 6) Mutual Inductance: M = N1 Easy & Hard B1 = N 2 B2 I2 I1 Right Hand Rule Cross Product: FINGERS point in 1st vector, PALM points in 2nd vector THUMB points in the resulting vector Self Inductance: L = B I Inductors oppose change in current through them Current Carrying Wire: THUMB point in current, FINGERS curl in B Current Loop: FINGERS curl in current loop, THUMB (on axis) in Energy in Inductors: RL Circuits: = L R Energizing I = I f 1 - e Monday, 2 April 2007 - t U= 1 2 LI 2 uB = B2 2 o B, , A & magnetic north pole ; De-Energizing I = I o e De- - t Induced Current: THUMB point in 15 Monday, 2 April 2007 Binduced , FINGERS curl in I induced 16 4 ...
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