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Lectures 18-19 (Ch. 30) Inductance and Self-inductunce 1. 2. 3. 4. 5. 6. Mutual inductunce Tesla coil Inductors and self-inductance Toroid and long solenoid Inductors in series and parallel Energy stored in the inductor, energy density 7. LR circuit 8. LC circuit 9. LCR circuit Mutual inductance d 2 dt N 2 2 M 21i1 2 N2 N 2 2 M 21 i1 2 M 21 di1 dt Virce verse: if current in coil 2 is changing, the changing flux through coil 1 induces emf in coil 1. d1 1 N1 dt N11 M 12i2 N11 M 12 i2 di2 1 M 12 dt M 12 M 12 M N11 N 2 2 M i2 i1 Units of M 1Wb M ,[M ] 1H (henry ) i 1A 1T 1m 2 Vs 1H s 1A A 1Ns 1Vs F qv B 1T 2 Cm m F V N V E ,[E] q m C m 1H s Typical magnitudes: 1μH-1mH Joseph Henry (1797-1878) Examples where mutual inductance is useful Tesla coil 1.r2 r1 , M B1 0 n1i1 M 3 0 N1i1 l1 0 A1 N1 N 2 why M Estimate. N 2 2 , 2 B1 A1 i1 l1 0 A1 N1 N 2 l2 ? l1 0.5m, A1 10cm 10 m , N1 1000, N 2 10, i2 2 106 2 2 4 10 7 Tm10 3 m 210310 M 25H 0.5mA di A 1 M 2 25 10 6 H 2 106 50V dt s With ferromagnetic core A1 N1 N 2 M , K m 0 l1 Nikola Tesla (1856 –1943) [B]=1T to his honor take K 1000 M 25mH , 50kV A t s 2.if r2 r1 2 B1 A2 M 0 A2 N1 N 2 l1 3.if r2 r and is an angle between axises M 0 A2 N1 N 2 l1 cos Example: M=? Mutual inductance may induce unwanted emf in nearby circuits. Coaxial cables are used to avoid it. Self-inductance d N , N Li dt di N L , L dt i Thin Toroid Thin solenoid with approximately equal inner and outer radius. N L , BA i Ni B 2r N 2 A L 2r Long solenoid N L , max Bmax A i Ni Bmax l N 2 A Lmax l Example. Toroidal solenoid with a rectangular area. N L , d Bhdr i Ni Nih b dr Nih b B , ln 2r 2 a r 2 a N 2 h b L ln 2 a Inductors in circuits di L dt di Vab L dt Energy stored in inductor di Pin IVab iL dt dW Pin dt Lidi I LI 2 W Lidi 2 0 LI 2 UL 2 Compare to Q 2 CV 2 QV UC 2C 2 2 Magnetic energy density UL UL uB volume 2rA LI 2 N 2 A UL ,L 2 2r N 2 AI 2 uB 2r 2 2rA Let’s consider a thin toroidal solenoid, but the result turns out to be correct for a general case NI B 2r B I 2r N N 2 AB 2 (2r ) 2 uB 2 2 2r 2 2rA N B2 uB 2 Energy is stored in B inside the inductor Compare to: uE Energy is stored in E inside the capacitor E 2 2 Example. Find U of a toroidal solenoid with rectangular area LI 2 N 2 h b 1.U L ,L ln 2 2 a b I 2 N 2 h ln a UL 4 2.U L u B dV volune B2 NI uB ,B 2 2r dV h 2rdr b 2 2 I N h ln a UL 4 LR circuit, storing energy in the inductor di 0 dt di R (i ) dt R L di dt L , R (i ) R iR L (i ) t R ln( ) ε R L t i I (1 e ), I R t di L L e dt Energy conservation law di di 2 iR L 0, i i R Li , dt dt di 1 dLi 2 dU L Li 2 Li ,U L dt 2 dt dt 2 Power output of the battery =power dissipated in the resistor dU L 2 i i R the rate at which the energy is stored in inductor dt General solution t i I (1 e ), I R t di L L e dt Initial conditions (t=0) i0 L Steady state (t→∞) iI L 0 R + LR circuit, delivering energy from inductor ε ε L di di R iR L 0, i dt dt L i i di dt L I i I , R t di idi 2 iR L 0, L i R 0 dt dt d ( Li 2 / 2) 2 dU L i R, i2R dt dt i t ln , i Ie , I t t t t di IL L L e RIe e dt The rate of energy decrease in inductor is equal to the power input to the resistor. Oscillations in LC circuit Oscillations in LC circuit q di dq d 2 q q L 0, i , 2 0, C dt dt dt LC 1 q q 0, LC 2 2 q Q cos(t ) i Q sin( t ) Qand are defined by the initial conditions Compare to mechanical oscillator mx kx, F k x x 0, m 2 0 x x X cos(t ) v X sin( t ) Xand are defined by initial conditions x q, v i kx2 q 2 mv2 Li 2 , 2 2C 2 2 1 k m L, k , C m 1 LC General solution 1 q q 0, LC 2 2 Qand are defined by initial conditions 1.q(t 0) q0 , i (t 0) 0 q i (t 0) 0 0 T q Q cos t q(t 0) Q q0 q q0 cos t i q0 sin t 2 t i T 2 t 2.q (t 0) 0, i (t 0) i0 i general solution q Q cos(t ) i Q sin( t ) q (t 0) Q cos 0 / 2 q Q cos(t / 2) Q sin t i Q sin( t / 2) Q cos t i (t 0) Q i0 Q i i0 cos t q i0 sin t i0 T 2 q T 2 t t 3. Arbitrary initial conditions : q (t 0) q0 , i (t 0) i0 q Q cos(t ) i Q sin( t ) q0 Q cos i0 Q sin i0 i0 tan ar tan q0 q 0 (q0 ) i0 (Q) Q 2 2 2 2 (q0 ) i0 2 2 2 q0 2 i0 2 2 Energy conservation law q di dq L 0, i C dt dt q dq di Li 0 C dt dt d q2 d Li 2 ( ) ( )0 dt 2C dt 2 2 2 q0 Li0 q 2 Li 2 Q 2 LI 2 const 2C 2 2C 2 2C 2 UC+UL=const UL UC UL UC T/2 T 2 t -Q Q q Example. In LC circuit C=0.4 mF, L=0.09H. The initial charge on the capacitor is 0.005mC and the initial current is zero. Find: (a) Maximum charge in the capacitor (b) Maximum energy stored in the inductor; (c) the charge at the moment t=T/4, where T is a period of oscillations. 1.q(t 0) q0 , i (t 0) 0 i (t 0) 0 0 q Q cos t q(t 0) Q q0 0.005mC 2 q0 Q2 2.U max L U max C 3.12 10 6 J 2C 2C 3)q q0 cos(T / 4) q0 cos( / 2) 0 Example. In LC circuit C=250 ϻF, L=60mH. The initial current is 1.55 mA and the initial charge is zero. 1) Find the maximum voltage across the capacitor . At which moment of time (closest to an initial moment) it is reached? 2) What is a voltage across an inductor when a charge on the capacitor is 1 ϻ C? q Q I 1)V , Q i (t 0) LC C i (t 0) LC L V i (t 0) 2.4mV , t T / 4 C C di q 2)V L 4mV dt C T 2 Example. In LC circuit C=18 ϻF, two inductors are placed in parallel: L1=L2=1.5H and mutual inductance is negligible. The initial charge on the capacitor is 0.4mC and the initial current through the capacitor is 0.2A. Find: (a) the current in each inductor at the instant t=3π/ω, where ω is an eigen frequency of oscillations; (b) what is the charge at the same instant? (c) the maximum energy stored in the capacitor;(d) the charge on the capacitor when the current in each inductor is changing at a rate of 3.4 A/s. a ) i1 ( 3 ) i1 ( 2 3 b) q ( ) 0.4mC 2 c) q0 2C Lef i0 2 T 3 ) i1 (T ) 0.1A, i2 ( ) 0.1A 2 2 2.25 10 2 J , 1 1 1 Lef L1 L2 q di1 di1 d ) L1 q CL1 108C C dt dt LCR circuit q di dq L iR 0, i C dt dt q Lq Rq 0 C q 2q 02 q 0, q ~ e t R , 0 2L 2 2 02 0 1 LC Characteristic equation 1, 2 (02 2 ) , 2 02 2 2 1 R 4L 2 2 2 2 0 0 2 R LC 4 L C Critical damping 4L a) Underdamped oscillations: 02 2 R2 C 1, 2 i q(t ) Q(t ) cos(t ) i(t ) Q(t )[ cos(t ) sin( t )], Q(t ) Qe t 4L R2 b) Critically damped oscillations: 02 2 C 4 L 02 2 R2 C 1, 2 q (t ) (C1 C2t )e 1t i (t ) [ (C1 C2t ) C2 ]e 2t 2 2 c) Overdamped oscillations: 0 1, 2 are real numbers q (t ) C1e 1t C2 e 2t i (t ) 1C1e 1t 2C2 e 2t if 0 1 0, 2 2 4L R2 C Example. The capacitor is initially uncharged. The switch starts in the open position and is then flipped to position 1 for 0.5s. It is then flipped to position 2 and left there. 1) What is a current through the coil at the moment t=0.5s (i.e. just before the switch was flipped to position 2)? 2) If the resistance is very small, how much electrical energy will be dissipated in it? 3) Sketch a graph showing the reading of the ammeter as a function of time after the switch is in position 2, assuming that r is small. t 25Ω 1 2 50V 10µF r 10mH A 2)U dis 50V 1)i I (1 e ), I 2A R 25 L 10 2 H 0.4ms 0.5s R 25 i (t 0.5s ) I 2 A LI 2 UL 20mJ 2 3) Induced oscillations in LRC circuit, resonance q di L Ri cos t 0 C dt dq 1 R i , 0 , ,f , dt 2L L LC ~ 2 q 2q 0 q f cos t q Q cos(t ) q Q sin( t ) q 2Q cos(t ) Q (0 2 )Q cos(t ) 2Q sin( t ) f cos(t ) 2 f [cos(t ) cos sin( t ) sin ] (0 2 )Q f cos 2 2Q f sin f2 2 Q , tan 2 2 [(0 2 ) 2 (2 ) 2 ] ( 2 0 ) 2 0 At the resonance condition: 0 an amplitude greatly insreases 0