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When you use a flash camera, it takes a few seconds to charge the capacitor that powers the flash. The light flash discharges the capacitor in a tiny fraction of a second. Why does charging take longer than discharging? This question and several other phenomena that involve charging and discharging capacitors are discussed in this module.
An RC circuit is a circuit containing resistance and capacitance. As presented in Capacitance , the capacitor is an electrical component that stores electric charge, storing energy in an electric field.
[link] (a) shows a simple RC circuit that employs a dc (direct current) voltage source $\epsilon $ , a resistor R , a capacitor C , and a two-position switch. The circuit allows the capacitor to be charged or discharged, depending on the position of the switch. When the switch is moved to position A , the capacitor charges, resulting in the circuit in part (b). When the switch is moved to position B , the capacitor discharges through the resistor.
We can use Kirchhoff’s loop rule to understand the charging of the capacitor. This results in the equation $\epsilon -{V}_{R}-{V}_{c}=0.$ This equation can be used to model the charge as a function of time as the capacitor charges. Capacitance is defined as $C=q\text{/}V,$ so the voltage across the capacitor is ${V}_{C}=\frac{q}{C}$ . Using Ohm’s law, the potential drop across the resistor is ${V}_{R}=IR$ , and the current is defined as $I=dq\text{/}dt.$
This differential equation can be integrated to find an equation for the charge on the capacitor as a function of time.
Let $u=\epsilon C-q$ , then $du=\text{\u2212}dq.$ The result is
Simplifying results in an equation for the charge on the charging capacitor as a function of time:
A graph of the charge on the capacitor versus time is shown in [link] (a). First note that as time approaches infinity, the exponential goes to zero, so the charge approaches the maximum charge $Q=C\epsilon $ and has units of coulombs. The units of RC are seconds, units of time. This quantity is known as the time constant :
At time $t=\tau =RC$ , the charge is equal to $1-{e}^{\mathrm{-1}}=1-0.368=0.632$ of the maximum charge $Q=C\epsilon $ . Notice that the time rate change of the charge is the slope at a point of the charge versus time plot. The slope of the graph is large at time $t=0.0\phantom{\rule{0.2em}{0ex}}\text{s}$ and approaches zero as time increases.
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