Exponential decay When discharging the capacitor, the current time graph has this particular form. It is exponential in form. (The “mathematical” form of a curve like this never actually falls to zero though in practice it does). Current μA Time s
Exponential decay The equation of the curve can be shown to be Io Current μA Where C is the capacitance of the capacitor and R is the resistance of theFIXED series resistor Time s
Exponential decay The form of the graph is exactly the same for the charge stored on the capacitor. We can multiply both sides of the equation by t Charge stored μC As Q=It we have This is the form you find the equation in in your specification Time s
Exponential decay Note that the only variable on the right is t. When t=CR Charge stored μC e = 2.718 so 1/e = 0.368 Time s So C x R is an important value and is known as the time constant.
Exponential decay Current μA Io Q = 0.368Qo 0.368Io (0.368)2Io (0.368)3Io RC 2RC 3RC Time s The time it takes the current to fall by a factor of 1/e is a constant. That time interval is RC the time constant. What are the units of the time constant?
Calculate the time constant in each case: In each case calculate the length of time it would take A B C and D to fall to a)0.638 b) 0.135 of its initial value?