Charging and Discharging a Capacitor in a RC Circuit (1.1M)
The exponential nature of charging or discharging a capacitor:
Mathematically, an exponential change occurs when the derivative of a quantity with time is proportional to the quantity itself. For example if,
Let us apply this to the discharge of a capacitor through a resistor when the switch is closed and the capacitor is initially charged. After the switch is closed Kirchoff's voltage rule applies at all times and gives the equation.
Using Ohm's Law V = IR, the definition Capacitance C= Q/V, and the definition of current I = dQ/dt, we can rewrite this last equation so that the derivative of the charge is proportional to the negative of the amount of charge on the capacitor.
Thus the charge on the capacitor decays exponentially with a time constant τ.
Instead of Q_{o} we have use Q_{max} since the capacitor starts off with its maximum charge and decays away exponentially.
Mathematically, discharging a capacitor takes an infinite amount of time.
The time constant τ represent the time for the system to make significant change in charge, voltage, or current whenever a capacitor is charging or discharging. After a time equal to one time constant, t = τ, the charge on the capacitor has dropped to e^{1} = 36.8% of its maximum value. After 5τ the charge has dropped to 0.7% of its maximum value. 
t

e^{–}^{t/}^{τ}

τ  .368  
2τ  .135  
3τ  .050  
4τ  .018  
5τ  .007 
Other quantities such as the current and the voltage drop across the resistor or the capacitor can be found using the definition of current I = dQ/dt, Ohms Law V = IR, and the definition of capacitance C= Q/V.
Thus the current also decays away from its initial value when the switch is first closed. Ohms law shows that this also true for the voltage across the resistor.
The mathematical results for charging a capacitor are similar but more complicated. The main difference is that the charge and voltage across the capacitor approach a maximum value exponentially. For example, the charge on the capacitor start at zero an goes to a maximum value,
Transient RC Conditions:
Charging a Single Capacitor in a Series with a Resistor
t = 0 
Later at t 
Later t >> t 
























Transient RC Conditions:
Discharging a Single Capacitor Across a Resistor
t = 0 
at some time t 
t >> τ 























