Given this diagram:
With S1 switch closed and S2 switch left open, I am trying to find the time constant
I know τ = RC for a basic circuit, but how would you calculate it for a complex circuit? Is R the equivalent resistance to the battery?
i = dq/dt dq/dt + Q/τ - emf / R = 0
The attempt at a solution
I start with junction rule and loop rule
I1 = I2 + I3 -emf + I1R1 + Q/C + I2R2 = 0 =emf + I1R1 + I3R3 + I3R4 = 0
At this point the teacher says I2 = dq/dt and we need to get rid of I1 so we can put something next to the C in Q/C.
I use I1 = I2+I3 in junction rule and put it into loop rule #2, getting:
-emf + (I2 + I3)R1 + I3R3 + I3R4 = 0 I2R1 + I3(R1+R3+R4) = emf I3 = (emf - I2R1) / (R1+R3+R3)
Then I put I3 in for the I1 eq.
I1 = I2 + (emf - I2R1) / (R1+R3+R3)
At this point it's so messy and confusing I think I am doing it all wrong.
The baseline she is giving us is that
dq/dt + Q/[foo] - emf/[bar] = 0
where [foo] would be the time constant. In a simple circuit she gives [foo] = RC and [bar] = R