Finding q(t) on a capacitor in an RC circuit

[Robert S]

The following circuit is given:
[PLAIN]http://img822.imageshack.us/img822/6369/image2lz.jpg

Switch S is closed until the charge q₂ on C₂ reaches its maximum, then at t=0 the switch is opened. Find q₁(t).

Using Kirchkoff's rules I found:
EMF= (q₁)/(C₁+I₂R
I₂R= q₂/C₂
I₁= I₂+I₃

In order to solve for q₁ I need to substitute $I_1=\frac{dq_1}{dt}$ and solve the differential equation. But I need one more equation to eliminate I₃ and q₂.
I was thinking of relating the charge q₂ to the total charge in the circuit, but after some calculation I found that the total charge isn't constant.

At t=0 the total charge q₂ on C₂ is EMF*C₂ and the charge on C₁ is zero.
At t=$\inf$ the total charge on both capacitors is $EMFC_1C_2 / (C_1 + C_2)$ (since the current is zero you can combine the two capacitors using $C_3^{-1}=C_1^{-1} + C_2^{-1}$)

any help is appreciated :-)

edit: I tried to put the kirchkoff equations in Latex, but for some reason the rest of the post becomes unreadable.

 

[berkeman]

The following circuit is given:
[PLAIN]http://img822.imageshack.us/img822/6369/image2lz.jpg

Switch S is closed until the charge q₂ on C₂ reaches its maximum, then at t=0 the switch is opened. Find q₁(t).

Using Kirchkoff's rules I found:
EMF= (q₁)/(C₁+I₂R
I₂R= q₂/C₂
I₁= I₂+I₃

In order to solve for q₁ I need to substitute $I_1=\frac{dq_1}{dt}$ and solve the differential equation. But I need one more equation to eliminate I₃ and q₂.
I was thinking of relating the charge q₂ to the total charge in the circuit, but after some calculation I found that the total charge isn't constant.

At t=0 the total charge q₂ on C₂ is EMF*C₂ and the charge on C₁ is zero.
At t=$\inf$ the total charge on both capacitors is $EMFC_1C_2 / (C_1 + C_2)$ (since the current is zero you can combine the two capacitors using $C_3^{-1}=C_1^{-1} + C_2^{-1}$)

any help is appreciated :-)

edit: I tried to put the kirchkoff equations in Latex, but for some reason the rest of the post becomes unreadable.

I would label the output of the voltage source as V1, and the voltage between the caps as V2. Write the KCL equation at that V2 node, and solve for V2(t). Then use Q=CV to calculate the charge across C1 as a function of time.

 

[ehild]

Using Kirchkoff's rules I found:
EMF= (q₁)/(C₁+I₂R

The correct form is: EMF= q₁/C₁+I₂R

I₂R= q₂/C₂
I₁= I₂+I₃

In order to solve for q₁ I need to substitute $I_1=\frac{dq_1}{dt}$ and solve the differential equation. But I need one more equation to eliminate I₃ and q₂.

You can write the first equation also as EMF=q₁/C₁+q₂/C₂, and differentiating it, you get

I₁/C₁+I₃/C₂=0

I₂ is obtained from the second equation: I₂=q₂/(C₂R)

I₃=dq₂/dt.

Now you can apply the Nodal Law:

I₁=I₂+I₃

I suggest to solve for q_2(t), it is easy to get q1(t) from it.

ehild

 

[Robert S]

I didn't realize I could differentiate one equation to get a new one. Thanks! :-)