CHAPTER 30 INDUCTANCE BASIC CONCEPTS Mutual Inductance Self Inductance Magnetic Field Energy Circuits with Inductors 1 Mutual Inductance I1 #1 If I1 changes with time t 2 Put second coil at P I1 #1 #2 3 If I1 changes with time For example Then And B
4 Or And I1 #1 #2 I2 5 be the flux through coil #2 due to B of coil #1. Let be the number of turns of coil #2. depends on the Then magnitude of B. Magnitude of B depends on 6 Therefore depends on In fact Call constant of proportionality Then is the Mutual Inductance 7 Solve for Faraday’s Law 8 is a constant depending only on geometry‐ position, size and shape. By Symmetry And is Mutual Inductance. Units are Henry (H). 9 Example 10 From Chapter 28 11 Only depends on geometry, number of turns, area and length. 12 Self Inductance If the current is changing then there will be a changing magnetic flux through the coil. 13 A changing magnetic flux will induce an emf. Therefore we define in analogy to the mutual inductance This is Self‐Inductance Solve for 14 Faraday’s Law Self‐Induced emf 15 What is the inductance, , of the black coil? Area A B
16 Magnetic Field Energy Consider the coil Current starts at zero and increases to . Or 17 And Power is work per unit time. 18 How much work to bring current through inductor from zero to ? Integrate. 19 Apply to solenoid 20 So Put into equation for 21 And So Energy stored in magnetic field . 22 R‐L Circuit Consider this circuit 23 First consider that switch is closed and calculate how the current will increase with time. Second, then while there is a current in the circuit consider opening and closing and calculate how the current will decay. Close 24 At time close switch. Kirchkoff’s loop starting at the switch and going ccw. 25 Integrate 26 Raise to the power of each side Solve for i 27 Let to get 28 29 The L‐C Circuit The capacitor has been charged to an initial charge and then placed across the inductor. 30 Kirchhoff’s Law A solution of this differential equation is Where is the initial charge on C. 31 Differentiate to get current Remember energy in Capacitor And Inductor Then study 32 33