If you’ve studied capacitors, you’ll have (very probably) come across the decay equation:
where is resistance, is capacitance, is the charge on the capacitor at time and is the initial charge.
Where does this come from?
Firstly from the basics of circuits, we know that (yes, I’m using the british convention with regards the symbol for voltage, I understand elsewhere is used instead).
Now when we discharge a capacitor through a circuit, the e.m.f.s are supplied by the capacitor.
Now substitutiting for V in the above equation gives
So we now have an equation for the current in the circuit, but wait, where does this current come from? It’s the capacitor discharging – this current is equal to the rate of discharge of the capacitor, so:
So substituting the above expression for I
Ooh look, a differential equation.
Seperate the variables giving:
where k is some constant.
If we define the initial charge to be – the charge when t = 0
so we now can replace giving:
MY WORK HERE IS DONE