Section 62 1 Linear circuit connected to a capacitor Charges and discharges Source and sink of current Volume between plates is small 2 Differential Equation for charge on capacitor The magnetic energy of the circuit is ID: 1031580
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1. Capacitance in a quasi-steady current circuitSection 62
2. 1. Linear circuit connected to a capacitorCharges and dischargesSource and sink of currentVolume between plates is small
3. 2. Differential Equation for charge on capacitorThe magnetic energy of the circuit isself inductance of the circuit if the capacitor was shorted by a wireEquation for circuit without capacitorJ = rate of change of charge on plate Voltage drop across capacitorEMF of circuit
4. 3. Periodic emfImpedance Ohm’s law
5.
6. If e = 0 (No driving force free oscillations)Since and Solving for w gives a complex frequency 4. Free electric oscillations in LCR circuits
7. Since Damping term with decay rate Rc2/2LPeriodic oscillations if Otherwise it’s an aditional damping term (with “-” sign) and no oscillations
8. If R 0, there is no damping at all (no dissipation)W. Thomson (1853)
9. 5. Consider several inductively-coupled circuits containing capacitorsB-field goes everywhereE-field is contained within the capacitorsath circuitwhere ea is the charge on capacitor aSum is over all circuits, including the ath
10. Periodic monochromatic circuits
11. Complex impedance matrixFree oscillations at “eigen” frequencies of the systemWhen A system of homogeneous linear equationsFor non-trivial solution, This equation gives the eigen-frequencies of the systemAll oscillations are damped if any R is nonzero
12. By analogy with forced coupled damped mechanical oscillators, the Lagrangian is “kinetic” energy = magnetic energy“potential” energy = electric energyWork done by external “forces” ea moving a “mass” by “distance” ea6. Analogy with mechanical systemsLL1 Classical Mechanics, sections 25 and 26.
13. “Dissipative function” Thenis the analog of Lagrange equations See LL1 (25.11)