Resonant RLC Circuit Homework - Purcell 8.4

AI Thread Summary
The discussion focuses on solving the differential equation for a resonant RLC circuit with a resistor connected in parallel to the LC combination, as presented in Purcell 8.4. The goal is to derive an equation analogous to the series circuit equation, taking into account the different current behaviors through the resistor and inductor. The user is grappling with the relationships between the currents in the circuit and how to express them in terms of each other. They have made some progress by setting up loop equations but seek further guidance on how to proceed with the derivation. The conversation emphasizes the need to understand the changing current dynamics in the parallel configuration.
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Homework Statement


Purcell 8.4: In the resonant circuit of the figure(I'll reproduce the image below) the dissipative element is a resistor R' connected in parallel, rather than in series, with the LC combination. Work out the equation analogous to Eq 2. (d^2V/dt^2 + (R/L)(dV/dt) + (1/LC)V = 0, this was for a series circuit), which applies to this circuit. Find also the conditions on the solution analogous to those that hold in the series RLC circuit. If a series RLC and a parallel R'LC circuit have the same L, C, and Q, how must R' be related to R

Crude drawing of the circuit:
_____C___
|________|
|____R'__ |
|________|
|____L___|
(ignore the white lines)

Homework Equations


Well, in deriving Eq 2 the book uses the following equations:
I = -dQ/dt
Q = CV
V(inductor) = L(dI/dt)
V(resistor) = IR

The Attempt at a Solution


I really only think I need help on the first part of the problem (finding the differential equation).
The problem that arises from the fact that the I over R' is different from the I over L and that the proportion changes (I(R') + I(L) = I(total) but all of these change don't they? The capacitor runs out of charge and the Inductor depends on the changing current.
Any push in the right direction would be appreciated, Thanks.
 
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I think I may have made a little progress.

Loop 1:
When V is Q/C
V = I_{1}R
0 = I_{1}R - V

Loop 2:
V = LdI_{2}/dt
0 = LdI_{2}/dt - V

Combining:
I_{1}R = LdI_{2}/dt
(I'm having a problem with latex it seems, those superscripts are supposed to be subscripts)

Now, this looks like it may end up giving me I_{1} in terms of I_{2}, but that's not really what I'm looking for. . . I suppose I may be able to re-plug into one of these equations when I'm done. Does this seem right at all?
 
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