An RCRC size 12{ ital "RC"} {} circuit is one containing a resistorRR size 12{R} {} and a capacitor CC size 12{C} {}. The capacitor is an electrical component that stores electric charge.

Figure 1 shows a simple RCRC size 12{ ital "RC"} {} circuit that employs a DC (direct current) voltage source. The capacitor is initially uncharged. As soon as the switch is closed, current flows to and from the initially uncharged capacitor. As charge increases on the capacitor plates, there is increasing opposition to the flow of charge by the repulsion of like charges on each plate.

In terms of voltage, this is because voltage across the capacitor is given by Vc=Q/CVc=Q/C size 12{V rSub { size 8{c} } =Q/C} {}, where QQ size 12{Q} {} is the amount of charge stored on each plate and CC size 12{C} {} is the capacitance. This voltage opposes the battery, growing from zero to the maximum emf when fully charged. The current thus decreases from its initial value of I0=emfRI0=emfR size 12{I rSub { size 8{0} } = { {"emf"} over {R} } } {} to zero as the voltage on the capacitor reaches the same value as the emf. When there is no current, there is no IRIR size 12{ ital "IR"} {} drop, and so the voltage on the capacitor must then equal the emf of the voltage source. This can also be explained with Kirchhoff’s second rule (the loop rule), discussed in Kirchhoff’s Rules, which says that the algebraic sum of changes in potential around any closed loop must be zero.

The initial current is I0=emfRI0=emfR size 12{I rSub { size 8{0} } = { {"emf"} over {R} } } {}, because all of the IRIR size 12{ ital "IR"} {} drop is in the resistance. Therefore, the smaller the resistance, the faster a given capacitor will be charged. Note that the internal resistance of the voltage source is included in RR size 12{R} {}, as are the resistances of the capacitor and the connecting wires. In the flash camera scenario above, when the batteries powering the camera begin to wear out, their internal resistance rises, reducing the current and lengthening the time it takes to get ready for the next flash.

Voltage on the capacitor is initially zero and rises rapidly at first, since the initial current is a maximum. Figure 1(b) shows a graph of capacitor voltage versus time (tt size 12{t} {}) starting when the switch is closed at t=0t=0 size 12{t=0} {}. The voltage approaches emf asymptotically, since the closer it gets to emf the less current flows. The equation for voltage versus time when charging a capacitor CC size 12{C} {} through a resistor RR size 12{R} {}, derived using calculus, is

V=emf(1−e−t/RC) (charging),V=emf(1−e−t/RC) (charging), size 12{V="emf" \( 1 - e rSup { size 8{ - t/ ital "RC"} } \) } {}

(1)where VV size 12{V} {} is the voltage across the capacitor, emf is equal to the emf of the DC voltage source, and the exponential e = 2.718 … is the base of the natural logarithm. Note that the units of RCRC size 12{ ital "RC"} {} are seconds. We define

τ=RC,τ=RC, size 12{τ= ital "RC"} {}

(2)where ττ size 12{τ} {} (the Greek letter tau) is called the time constant for an RCRC size 12{ ital "RC"} {} circuit. As noted before, a small resistance RR size 12{R} {} allows the capacitor to charge faster. This is reasonable, since a larger current flows through a smaller resistance. It is also reasonable that the smaller the capacitor CC size 12{C} {}, the less time needed to charge it. Both factors are contained in τ=RCτ=RC size 12{τ= ital "RC"} {}.

More quantitatively, consider what happens when t=τ=RCt=τ=RC size 12{t=τ= ital "RC"} {}. Then the voltage on the capacitor is

V=emf1−e−1=emf1−0.368=0.632⋅emf.V=emf1−e−1=emf1−0.368=0.632⋅emf. size 12{V="emf" left (1 - e rSup { size 8{ - 1} } right )="emf" left (1 - 0 "." "368" right )=0 "." "632" cdot "emf"} {}

(3)This means that in the time τ=RCτ=RC size 12{τ= ital "RC"} {}, the voltage rises to 0.632 of its final value. The voltage will rise 0.632 of the remainder in the next time ττ size 12{τ} {}. It is a characteristic of the exponential function that the final value is never reached, but 0.632 of the remainder to that value is achieved in every time, ττ size 12{τ} {}. In just a few multiples of the time constant ττ size 12{τ} {}, then, the final value is very nearly achieved, as the graph in Figure 1(b) illustrates.

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