16.9 Exponential Decay of Discharging Capacitor Through Resistor

16.9 Exponential Decay of Discharging Capacitor Through Resistor

An RC circuit for the discharging capacitor through a resistor was shown in Fig. 16.13(a) in the previous section. A graph charge vs time for that is shown in Fig. 16.14. The charge ($Q$) decreases exponentially overtime ($t$). In general, exponential means increasing or decreasing very rapidly. It can also refer to a mathematical functions involving exponents where a variable appears as an exponent. We can write this function in the form, $f(x) = a^{x}$ where $a$ is a constant and $x$ is the exponent (variable). The general exponential decay formula for a discharging capacitor in an RC circuit is of the form;

or $x = x_{o} e^{\frac{-t}{RC}}$

(Fig. 16.14)

where $x$ = quantity (such as charge, voltage or current) at time $t$.

$x_{o}$ = initial value of the quantity at $t = 0$.

$R$ = resistance in ohms ($\Omega$).

$C$ = capacitance in farads (F).

$t$ = time in seconds.

$RC$ = time constant ($\tau$) which determines how quickly the capacitor discharges.

$e$ = a mathematical constant called Euler's constant which is also used as of the natural logarithm ($\log_{e}$). Its value is approximately $2.718$.

The negative exponent ($-t/RC$) in the formula indicates that the quantity decreases exponentially over time.

Using Eq. 16.43, the decrease in charge, voltage and current at any time $t$, can be described by the following equations.

$Q = Q_{o} e^{-t/RC}$, $V = V_{o} e^{-t/RC}$ and $I = I_{o} e^{-t/RC}$.

In the graph for the decrease in all the three quantities, they start with a steep slope (fast discharge) and gradually flattens out as the time progresses, indicating a slower discharge rate. This is shown in Fig. 16.15 by the charge decay graph.

(Fig. 16.15)

The initial charge is $Q_{o}$ at $t = 0$ and it drops to $Q$ after a time interval $t$. Now, if at another stage during the discharging of capacitor, let the charge be $Q'$. After the same interval of time $t$, it drops to $Q''$. We can see that the rate of decrease of charge in the beginning was $(Q_{o} - Q)$, but as the process goes on, the rate of decrease of charge becomes $(Q' - Q'')$ which is less than $(Q_{o} - Q)$.

The decay of charge is very much similar to flow of water out of a tank through a narrow pipe. When the tank is full of water, its level and pressure is analogous to charge and voltage, and flow rate of water analogous to the current. In the beginning, the water level is high and it decreases rapidly and the flow of water through the pipe is fast. As the water level comes down, the rate of flow slows down. This flowing down is due to the decreasing pressure.

For your information

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Exponential functions frequently and quantitatively describe a number of phenomena in physics, such as radioactive decay, in which the rate of change of a process or substance depends directly on its current value.