Capacitance in a quasi-steady current circuit

Section 62

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/2c2) J2

L = self inductance of the circuit of the capacitor was shorted by a wire

EMF of circuit

Equation for circuit without capacitor

J = rate of change of charge on plate

Voltage drop across capacitor

Periodic emf

Impedance

Ohm’s law

e = 0 (No driving force free oscillations)

Since and

Solving for w gives complex frequency

Since

Damping term with decay rate Rc2/2L

Periodic oscillations if

Otherwise it’s an aditional damping term (with “-” sign) and no oscillations

For free oscillations of an LCR circuit, the characteristic decay time depends on

1. L, C, and R2. L and R3. R only

As R is increased in an LCR circuit, the free oscillation frequency

1. Increases2. Decreases3. Stays the same

If R 0, there is no damping at all (no dissipation)

W. Thomson (1853)

Consider several inductively-coupled circuits containing capacitors

B-field goes everywhere

E-field is contained within the capacitors

ath circuit

where ea is the charge on capacitor a

Sum is over all circuits, including the ath

Periodic monochromatic circuits

Complex impedance matrix

Free oscillations at “eigen” frequencies of the systemWhen

A system of homogeneous linear equations

For non-trivial solution, This equation gives

the eigen-frequencies of the system

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” ea moving a “distance” ea

“Dissipative function”

Then

is the analog of Lagrange equations