Damped Oscillations in an RLC Circuit — Question 8

Question 8

Distributed series resistance shares all the eventual oscillator heat

A source-free series RLC loop has initial charge Q0, initial current I0, and two positive resistances R1 and R2 in series. All components are constant and ideal except these resistances.

1. Find the eventual heat in each resistor.

2. Find how the partition changes if R1 is doubled with R2 fixed, without solving q(t).

Question 8 - Solution

Distributed series resistance shares all the eventual oscillator heat

(a) Method and calculation

$$U_0=Q_0^2/(2C)+LI_0^2/2,\quad\int_0^\infty i^2(R_1+R_2)dt=U_0.$$$$\boxed{H_1=\frac{R_1}{R_1+R_2}U_0,\quad H_2=\frac{R_2}{R_1+R_2}U_0}.$$

(b) Interpretation and further calculation

$$R_1\to2R_1:\quad H_1^{\mathrm{new}}=\frac{2R_1}{2R_1+R_2}U_0.$$$$\boxed{H_1^{\mathrm{new}}/H_1=\frac{2(R_1+R_2)}{2R_1+R_2}}.$$

Physical check

The time history changes with resistance, but the common-current heat ratio remains fixed.