Lectures 18-19 (Ch. 30) Inductance and Self-inductunce

1. Lectures 18-19 (Ch. 30)Inductance and Self-inductunce • 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

2. Mutual inductance Virce verse: if current in coil 2 is changing, the changing flux through coil 1 induces emf in coil 1.

3. Units of M Joseph Henry (1797-1878) Typical magnitudes: 1μH-1mH

4. Examples where mutual inductance is useful

5. Tesla coil Estimate. Nikola Tesla (1856 –1943) [B]=1T to his honor

6. Example: M=? Mutual inductance may induce unwanted emf in nearby circuits. Coaxial cables are used to avoid it.

7. Self-inductance

8. Thin Toroid Thin solenoid with approximately equal inner and outer radius. Long solenoid

9. Example. Toroidal solenoid with a rectangular area.

10. Inductors in circuits

11. Energy stored in inductor Compare to

12. Magnetic energy density Let’s consider a thin toroidal solenoid, but the result turns out to be correct for a general case Compare to: Energy is stored in E inside the capacitor Energy is stored in B inside the inductor

13. Example. Find U of a toroidal solenoid with rectangular area

14. LR circuit, storing energy in the inductor εL

15. Energy conservation law Power output of the battery =power dissipated in the resistor + the rate at which the energy is stored in inductor General solution Initial conditions (t=0) Steady state (t→∞)

16. LR circuit, delivering energy from inductor εL ε t The rate of energy decrease in inductor is equal to the power input to the resistor.

17. Oscillations in LC circuit

18. Oscillations in LC circuit

19. Compare to mechanical oscillator F 0 x

20. General solution q t i t

21. i t q t

23. Energy conservation law UC+UL=const UC UL UL UC t Q -Q q T/2

24. 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.

25. 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

26. 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.

27. LCR circuit Characteristic equation Critical damping

28. a) Underdamped oscillations: b) Critically damped oscillations: c) Overdamped oscillations:

29. 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. 10µF 25Ω 2 1 50V r 10mH A 3)

30. Induced oscillations in LRC circuit, resonance ~ Q At the resonance condition: an amplitude greatly insreases