Lesson 10 Calculation of Inductance LR circuits Lesson 10: Inductance Lesson 10 Calculation of Inductance LR circuits Oscillations in LC circuits LRC circuits EMF due to mutual inductance 1

I E Field in Capacitor - - q By passing an electric current through a capacitor we store energy as an electric E Field in Capacitor field between the plates which has potential energy 1 U = CV 2 2 and energy density 1 u = e E 2 2 I - + q - + 2

By passing an electric current through a solenoid we form a magnetic Field B which also stores energy B Field in Inductor I - + L INDUCTOR 3

Charging Inductor I Source of Back EMF 4

Charging an Inductor Charging Inductor II Current flows through coil producing a magnetic field B As current builds up to its equilibrium value it is changing thus B is changing and hence  is changing Changing  produces an EMF 5

Charging Inductor III This EMF ind will produce a changing current Iind in the opposite direction to the charging current Thus producing an induced magnetic field Bind that opposes B After some time this results in equilibrium magnetic field Beq and current Ieq 6

Ieq is actually equal to the steady state current R --why? and Beq is the magnetic field corresponding to this current The effect of the inductor is to cause a gradual increase of current from zero to the steady state value Charging Inductor IV 7

e Inductance = - N d F dt L dI µ changing flux in coil B dt L dI induced emf µ changing flux in coil changing current For soleniod BA m nIA thus nA n 2 Al ; l length hence solenoid which depends on geometry of coil Inductance 8

SI units I e e [ ] [ ] S . I . units of Inductance L = - dI dt é ù ê ú V Vs [ ] L = = = = H ( Henry ) ê dI ú A A ë û dt s Also d F N B dt L = - dI dt é ù d F Wb ê N B ú Wb [ ] dt s L = = = = H ( Henry ) ê ú dI A A ê ú ë dt û s 9

SI units II 10

RL Circuits: Charging I - L + Kirchoffs Law e e ( ) ( ) - I + = t R t L 11

e e e e e Charging II [ ] ( ) ( ) ( ) ( ) Increasing current - I + = t L e dI - I ( ) - = t R L dt e e æ ö æ ö t Rt - - Þ ( ) = I t ç - ÷ = ç - 1 e L 1 e t ÷ ç L ÷ R è ø R è ø L t = Time constant for RL circuit : L R é ù L H [ ] t = = = ê ú s L ë û R W 12

Discharging 13

Energy Stored in the Magnetic Field Energy in Inductor e dI - I ( ) t R - L = dt e dI Þ ( ) ( ) 2 ( ) I t - I t R - L I t = dt Power supplied Joule heat Rate at which by source emf dissipated energy is stored by load by Inductor dU dI ( ) = B = LI t dt dt ò U ( t ) ò I ( t ) 1 Þ ( ) = = ( ) 2 U t dU L I ¢ d I ¢ = LI t B 2 14

Comparing Inductor and capacitor 15

Energy densities ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) energy Energy Density = volume Energy densities For solenoid ( ) B t L = m n 2 = m 2 ( ) Al n Vol and I t = m n æ ( ) ö 2 æ ö 1 1 B t ( ) ( ) 2 U t = LI t = ç m n 2 Vol ÷ ç ÷ B è ø 2 2 è m n ø ( ) ( ) 2 U t B t 1 ( ) = ( ) u t B = = 2 B t B Vol 2 m 2 m compare to energy density in capacitor e ( ) ( ) 2 u t = E t E 2 16

Oscillations in LC circuits V(t) I(t) t) L 17

e 1 Start with fully charged capacitor Q Q U = ; I ( ) = ; V ( ) = 2 C m ; I ( ) = ; V ( ) = m E 2 C C 2 . Close switch , charge flows from + plate to plate - e - = Kirchoffs Rule : V ( t ) ( t ) Q ( t ) dI Q ( t ) d 2 Q Þ + L = Û + L = C dt C dt 2 d 2 Q 1 Þ = - Q dt 2 LC 18

to equation for Harmonic Oscillator of mass on spring d 2 Q 1 compare = - Q dt 2 LC to equation for Harmonic Oscillator of mass on spring for displacement X ( t ) from equilibrium position d 2 X k k = - X = - w 2 X ; w = = angular frequency m dt 2 m This has solution X ( t ) = A cos ( w t + d ) ; A = amplitude d = phase shift 19

Solution to Equations I 20

Solution to Equations II + + - + - Solution to Equations II +Qm Q(t) t -Qm +Im t I(t) -Im 21

Analogy to Harmonic Oscillator I 22

Analogy to Harmonic Oscillator II + t = 0 L Analogy to Harmonic Oscillator II - t = T/4 L - L t = T/2 + L t = 3T/4 + L t = T - + 23 -

Analogy to Harmonic Oscillator III Total Energy of Harmonic Oscillator Analogy to Harmonic Oscillator III 1 1 = U + K = kX 2 + mv 2 2 2 æ ö 2 1 1 dX 1 = kX 2 + m ç ÷ = kA 2 ; A is amplitude è ø 2 2 dt 2 Total Energy of LC Circuit 1 1 = U + U = Q 2 + LI 2 E B 2 C 2 æ ö 1 1 dQ 2 1 = = 2 Q 2 + L ç ÷ Qmax è ø 2 C 2 dt 2 C 24

PE / KE K.E. P.E. UB UE 25

In real circuits always have some resistance RLC Circuit (dc) I V(t) C I(t) R t) L 26

RLC Circuit (dc) II In RLC circuit get energy lost as joule heat and oscillations decay in amplitude and one has a damped harmonic oscillator . 27

Picture 28

Flux Current and Inductance

Mutual Inductance Mutual Inductance e e e e Flux through coil 2 Nearby circuits effect each other define MUTUAL INDUCTION M of coil 2 with 21 F respect to coil 1 as M = N 21 2 21 I 1 F Flux through coil 2 = 21 N = number of turns in coil 2 2 I = current in coil 1 1 Induced emf in coil 2 by coil 1 is e d F dI = - N 21 = - M 1 2 2 dt 21 dt if rate at which currents change are equal e e e dI = = = - M 2 1 dt 29