1. Capacitance in a quasi-steady current circuit Section 62

2. Linear circuit connected to a capacitor Charges and discharges Source and sink of current Volume between plates is small The magnetic energy of the circuit is (L/2c 2 ) J 2 L = self inductance of the circuit of the capacitor was shorted by a wire

3. EMF of circuit Equation for circuit without capacitor J = rate of change of charge on plate Voltage drop across capacitor

4. Periodic emf Impedance Ohm’s law

6.  = 0 (No driving forcefree oscillations) Since and Solving for  gives complex frequency

7. Since Damping term with decay rate Rc 2 /2L Periodic oscillations if Otherwise it’s an aditional damping term (with “-” sign) and no oscillations

8. For free oscillations of an LCR circuit, the characteristic decay time depends on 1.L, C, and R 2.L and R 3.R only

9. As R is increased in an LCR circuit, the free oscillation frequency 1.Increases 2.Decreases 3.Stays the same

10. If R0, there is no damping at all (no dissipation) W. Thomson (1853)

11. Consider several inductively-coupled circuits containing capacitors B-field goes everywhere E-field is contained within the capacitors a th circuit where e a is the charge on capacitor a Sum is over all circuits, including the a th

12. Periodic monochromatic circuits

13. Complex impedance matrix Free oscillations at “eigen” frequencies of the system When A system of homogeneous linear equations For non-trivial solution, This equation gives the eigen- frequencies of the system

14. All oscillations are damped if any R is nonzero By analogy with coupled damped oscillators, the Lagrangian is “kinetic” energy = magnetic energy “potential” energy = electric energy Work done by external “forces”  a moving a “distance” e a

15. “Dissipative function” Then is the analog of Lagrange equations