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 Efield, 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 RI) = 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 DeEnergizing 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 Bfield circles, helixes, traps B Velocity Selector, E/M Experiments, Mass Spectrometers Force on currentcarrying 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) BiotSavart 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 currentcarrying 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 ; DeEnergizing 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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 Fall '07
 JapGuy
 Magnetic Field, DI, RL circuit, DT DT DT

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