Assume a circuit like that in the picture is used to charge a 470-uF capacitor to 50 volts, which is then discharged through a 1.8-k ohhm resistor. How do I find the time constant. I know it has something to do with T=RC...
Let us suppose the capacitor C =470 microF has already been charged to a voltage U0= 50 Volts and is now connected in series with the resistor R =1.8 kOhms.
The voltage on the capacitor VC (equal to the voltage on the resistor VR) will be a function of time and will decrease with time as the capacitor is discharging.
We write now these two voltages (on the capacitor and on the resistor):
`V_C(t) = -(Q(t))/C = -(int(I*dt))/C` and `V_R = I*R`
where the sign minus in the expression of VC comes from the fact that the capacitor is discharging with time.
and impose the condition that `V_C = V_R`
`I*R + (int(I*dt))/C =0`
by differentiating and rearranging we obtain
`(dI)/dt +I/(R*C) =0` where `I =I(t)` is a function of time.
to solve we separate variables I and t
`(dI)/I = -dt/(RC)`
and by integration from t=0 to t (from I0 to I(t))
`ln((I(t))/I_0) = -t/(RC)` or equivalent `I(t) =I_0*e^(-t/(RC)) =I_0*e^(-t/tau)`
We can now define the time constant `tau = R*C =1800*470*10^-6 =0.846 seconds`
as the time that the current needs to decrease to a value of (1/e) from its initial value I0.