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Differential equation problem?
You have an idea that maybe the time rate of discharge on a capacitor should be proportional to the actual charge on a capacitor. To test this idea, you grab your multi-meter and RC-circuit. So, you power up the capacitor and then switch your circuit off. At that instant (at t=0), you find that the capacitor has 1 coulomb of charge. Ten minutes later, you come back to the circuit and measure 0.89 coulombs (or 89% of the initial charge). Wanting to see how accurate your mathematical model is, you decide that you will predict just how long it will take for the capacitor to reach 1% of its initial charge. How many minutes does your model predict this will take?
If the rate of discharge is proportional to the charge, then:
dq(t)/dt = -k*q(t)
where k is a positive constant of proportionality. (The minus sign appears because the capacitor is *discharging*, i.e, q is decreasing with time).
This is a simply first-order, separable equation:
dq/q = - k dt
ln(q/Qo) = -k*t
where Qo is the charge at time t = 0.
Rearrange this to be:
k = -(1/t)*ln(q(t)/Qo)
We know that at t = 10 min, q(t)/Qo = 0.89, so we can use this to solve for k:
k = -1/(10min) * ln(0.89) = 1/(85.812 min) = 0.012/min
So:
ln(q/Qo) = -t/(85.812 min)
t = (-85.812 min)*ln(q/Qo)
For q/Qo = 0.01 (i.e. 1%), it will take:
t = (-85.812 min)*ln(0.01) = 395.2 min
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