# Resonant RLC Circuit

1. 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)

2. 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

3. 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 thats 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?