Mutual Inductance and Circulation of Currents

1. Addendum – Chapter 21

2. Mutual inductance – • Circulation of currents in one coil can generate a field in the coil that will extend to a second, close by device. Flux Changes Suppose i1 CHANGES Current (emf) is induced in 2nd coil.

3. Mutual Inductance • i1 creates a field that (partially) passes through the second coil. • As i1 changes, the flux through coil 2 changes and an emf (and current i2) are created. • The two coils are mutually linked by what we call an “inductance” i2 Induction

4. Watch Out! • Exam #2 one week from today. • Chapters 20 & 21 • Same format but possibly one set of multiple choice questions that you hate. • You should already be studying. • QUIZ on Friday – Chapter #21 • Today we continue with the chapter. We should finish it on Friday. Maybe. • No study session on Monday next week • We will have a study session on Tuesday morning like last time. Details to follow. Induction

5. This schedule is now in effect: If I am not there … find me! Induction

6. Mutual Inductance i2 mutual Inductance Induction

7. Note the form: UNIT: henry Think of this when we define INDUCTANCE (L) of a small coil in the next section. Induction

8. The two coils Remember – the magnetic field outside of the solenoid is pretty much zero. Two fluxes (fluxi?) are the same! Induction

9. One solenoid is centered inside another. The outer one has a length of 50.0 cm and contains 6750 coils, while the coaxial inner solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 A/s. (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the inner solenoid Length = 0.5 meters N=6750 coils n=6750/.5=1.35E04 turn/meter Magnetic field INSIDE the smaller coil is the same as in the larger coil and is given by: Check My Arithmetic Please! Induction

10. One solenoid is centered inside another. The outer one has a length of 50.0 cm and contains 6750 coils, while the coaxial inner solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 A/s. (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the inner solenoid Induction

11. One solenoid is centered inside another. The outer one has a length of 50.0 cm and contains 6750 coils, while the coaxial inner solenoid is 3.0 cm long and 0.120 cm in diameter and contains 15 coils. The current in the outer solenoid is changing at 37.5 A/s. (a) What is the mutual inductance of these solenoids? (b) Find the emf induced in the inner solenoid Check My Arithmetic Please! Induction

12. Self-inductance – • Any circuit which carries a varying current self-induced from it’s own magnetic field is said to have INDUCTANCE (L).

13. An inductor resists CHANGESin the current going through it. Induction

14. An inductor resists CHANGESin the current going through it. Induction

15. An inductor resists CHANGESin the current going through it. Induction

16. Inductance Defined If the FLUX changes a bit during a short time Dt, then the current will change by a small amount Di. Faraday says this is the emf! This is actually a calculus equation Induction

17. So … There should be a (-) sign but we use Lenz’s Law instead! E= The UNIT of “Inductance – L” of a coil is the henry. SYMBOL: Induction

18. Induction

19. Consider “AC” voltage Minimum Change@Dt V1 Maximum Change@Dt Induction

20. The transformer FLUX is the same through both coils (windings). Induction

21. Induction

22. Input/Output Impedance (Resistance) Induction

23. Remember that a Capacitor stored ENERGY? U=(1/2)CV2 U=Area=(1/2)LI2 i Li LI Li DU i Di I Induction Induction

24. SO … Energy Stored in a capacitor The energy stored in a capacitor with capacitance C and a voltage V is U=(1/2)LI2 Induction

25. The Energy stored is in the Magnetic Field Consider a solenoid with N turns that is very long. We assume that the field is uniform throughout its length, ignoring any “end effects”. For a long enough solenoid, we can get away with it for the following argument. Maybe. Induction

26. Energy Storage in Inductor Induction

27. Back to Circuits Induction

28. Series LR Circuit Induction

29. RL or LR Series Circuit • Switch is open .. no current flows for obvious reasons. • Switch closed for a long time: • Steady current, voltage across the inductor is zero. All voltage (E) is across the resistor. • i=E/R Induction

30. RL or LR Series Circuit When the switch opens, current change is high and back emf from L is maximum. i E/R t As the current increases, more voltage is across R, the rate of change of I decreases and as the current increases, it increases more slowly. Induction

31. RL Circuit • When L=0, the current rises very rapidly (almost instantly) • As L increases, it takes longer for the current to get to its maximum. Induction

32. RL Circuit - Kirchoff Stuff Induction

33. The Graphic Result – Current Growth } 63% of maximum e= 2.71828… Induction

34. Decay – Short out the battery • Magnetic field begins to collapse, sending its energy into driving the current. • The energy is dissipated in the resistor. • i begins at maximum (E/R) and decays. Induction

35. Solution Induction

36. Up and Down and Up and Down and ….. Induction

37. NEXT: AC Circuits Induction