Whenever the rate of change of a quantity is proportional to the quantity its se

Question

Whenever the rate of change of a quantity is proportional to the quantity its self, exponential behavior results. For example, if each algae cell on the surface of a pond bifurcates into two algae cells every hour, the number of algae cells increases exponentially in time. Another example, due originally to Newton, is a cooling cup of tea. Hotter objects give away their energy to their surroundings more readily than cooler objects do, so as the tea cup’s temperature decreases its rate of cooling decreases as well. The temperature of the tea cup therefore decreases exponentially in time (“exponential decrease” is also called “exponential decay”). With these examples and your answers to the questions above in mind, how does the charge on a capacitor change as a function of time while it’s charging or discharging? Be sure to explain your answer thoroughly and in your own words.

Explanation / Answer

When a capacitor is charging, the charging current decreases since the potential across the resistance decreases as the potential across the capacitor increases. As more and more charge accumulates on the capacitor, charge at any time 't' is given by

Capacitor charging : Q = CVo[1-e-(t/RC)] where Vo is the final potential across plates or the applied potential and RC is the time constant

When a capacitor is discharging, the current through capacitor keeps on increasing and the charge keeps on decreasing.Charge at any time 't' is given by:-

Capacitor discharge (voltage decay): Q = Qoe-(t/RC) ,where Qo is the initial charge on the capacitor

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