(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given a circuit with two resistors, R and r, and a capacitor of C, and EMF of V_{0}as shown in the diagram, find the voltage across the capacitor during charging. Prove that this voltage, V is given by V = V_{0}(r/(R+r)) (1-e^{-((R+r)t)/(RrC)})

2. Relevant equations

N.A.

3. The attempt at a solution

This is what I have

Loop A (with C): V_{0}+ RI + q/c = 0 ==> I = (V_{0}- V_{C})/R

Loop B (with r): V_{0}+ RI + rI_{r}= 0

I = I_{r}+ I_{c}==> I = V/r + C(dv/dt)

From first and third,

(V_{0}- V_{C})/R = V/r + C(dv/dt)

Simplify to get,

V_{0}- RC(dv/dt) = V(1+R/r)

V = (r/[R+r])(V_{0}- RC(dV/dt))

The shape of the equation is getting there (I hope), but what do I do next? To get the given equation, RC(dV/dt) must be V_{0}e^{-((R+r)t)/(RrC)}.

RC (dV/dt) = V, solving this differential equation to get ln (V) = -t/(RC) + k, hence, RC (dV/dt) = e^{-t/(RC) + k}. And I am totally stuck.

Did I do something wrong somewhere? I can't think of anyway to get the RrC term in e^{-((R+r)t)/(RrC)}, not to mention the V_{0}and the (R+r) terms, unless k is like Rt/r. But that still does not give me a V_{0}?

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# RC Circuit with two resistors

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