Find the functions of V1, V2, V3 in an RC circuit

[Zoja]

Homework Statement

Consider the following electrical network with three capacitors $C (1 nF)$ and two resistors $R=1M\Omega$.
It represents a model of three cells of capacitance $C$ connected by a gap junction with resistance$R$.
At time$t=0$ (initial condition), the potential at point $1 (V_1)$ is $100 mV$ (with respect to ground) and the potential at point $2 (V_2)$ and at point $3 (V_3)$ is $0 mV$.
How do the potentials $V_1$ , $V_2$ and $V_3$ evolve with time? Find the functions $V_1=f_1(t)$ , $V_2=f_2(t)$ and $V_3=f_3(t)$
Prove that$V1 + V2 + V3$ does not change with time. To what physical principle does this correspond?

(I apologize if I posted in the wrong section, but it is homework given to me..and I am also new to the forum)

Relevant Equations

$I=\frac{dQ}{dt}$
$V_r=IR$
$V_c=\frac{Q}{C}$

$Q_1+Q_2+Q_3=Q_1(0)$

I tried using Kirchhof's current law, and to pose the problem in matrix form as $\frac{dv}{dt}=Mv$ with$v$ the vector of the $3$ potentials at nodes $1, 2$ and $3$, and $M$ is a $3x3$ matrix.
it would be enough to show me which will be the differential equations, I would proceed by solving them by myself.

 

[berkeman]

Problem Statement: Consider the following electrical network with three capacitors $C (1 nF)$ and two resistors $R=1M\Omega$.
It represents a model of three cells of capacitance $C$ connected by a gap junction with resistance$R$.
At time$t=0$ (initial condition), the potential at point $1 (V_1)$ is $100 mV$ (with respect to ground) and the potential at point $2 (V_2)$ and at point $3 (V_3)$ is $0 mV$.
How do the potentials $V_1$ , $V_2$ and $V_3$ evolve with time? Find the functions $V_1=f_1(t)$ , $V_2=f_2(t)$ and $V_3=f_3(t)$
Prove that$V1 + V2 + V3$ does not change with time. To what physical principle does this correspond?

(I apologize if I posted in the wrong section, but it is homework given to me..and I am also new to the forum)
Relevant Equations: $I=\frac{dQ}{dt}$
$V_r=IR$
$V_c=\frac{Q}{C}$

$Q_1+Q_2+Q_3=Q_1(0)$

I tried using Kirchhof's current law

Welcome to the PF.

Can you show us the equations you got for the KCL nodes?

 

[Zoja]

$i_1=-I_1+I_2$

$i_2=I_1-2I_2+I_3$

$i_3=I_2-I_3$

 

[berkeman]

Um, no. That's obviously of no help.

KCL equations involve the sum of all currents out of each node, sure, but you need to express those currents in terms of the voltage difference across impedances leading out of each node. If it's all resistors, that's just linear equations. When there are inductors and capacitors, you write the differential equations and solve them.

Have at it!