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## Homework Statement

An RC circuit is given with a 1μF capacitor, a 25kΩ resistor and a 4μF capacitor, all in series. Initially, the 1μF capacitor has a voltage of 40V, while the 4μF capacitor has a voltage of 0V. The circuit, starting from the left and going clockwise is, 1μF (reverse polarity), the resistor and then the 4μF capacitor (regular polarity).

The goal is to find the functions governing current and the voltage of each capacitor (total of three functions).

i(t)=?, v1(t)=?, v4(t)=?

## Homework Equations

[itex]\sum[/itex]V=0

V=IR

Q=CV

I=dQ/dt

## The Attempt at a Solution

Applying KVL I get:

V1=Vr+V4

Q1/C1=IR+Q2/C2

Q1/C1=Q'R+Q2/C2

This equation becomes very confusing to me. Firstly there are two charge functions of time. I can eliminate this problem by noting that since the circuit is a series, the current is constant throughout all elements meaning that dQ1/dt=dQ2/dt, so I can differentiate this DE and get:

Q1'/C1=Q1''R+Q1'/C2

Q1''+[(C1-C2)/(C1C2R)]Q1'=0

Which leads me to conclude that:

Q1(t)= k1+k2exp([(C2-C1)/(C1C2R)]t

however I can go further, because since I=dQ1/dt=dQ2/dt, then Q2=∫(dQ1/dt)dt, so I get the following:

Q1(t)= k1+k2exp([(C2-C1)/(C1C2R)]t

Q2(t)= k3+k2exp([(C2-C1)/(C1C2R)]t

I(t)= ([(C2-C1)/(C1C2R)]k2exp([(C2-C1)/(C1C2R)]t

However, this leads to a problem. The initial conditions are:

Q1(0)=C1V1i=(1μF)(40V)=40μC

Q2(0)=C2V2i=(4μF)(0V)=0μC

I(0)=0 A

Which gives me:

k1=40 μC

k2=0 μC

k3=0 μC

and:

Q1(t)= 40 μC

Q2(t)= 0 μC

I(t)= 0 A

Can someone point out the error in my reasoning? I've been over it countless times and it all makes sense to me, but there is obviously a problem.