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Part a shows a circuit with a cell of e m f script E connected in series with a resistor R, a capacitor C, and a switch to close the circuit. The current is shown flowing in a clockwise direction. The capacitor plates are shown to have a charge positive q and negative q respectively. Part b shows a graph of the variation of voltage of the capacitor with time. The voltage is plotted along the vertical axis and the time is along the horizontal axis. The graph shows a smooth upward rising curve which approaches a maximum and flattens out at maximum voltage equal to e m f script E over time.
(a) An RC size 12{ ital "RC"} {} circuit with an initially uncharged capacitor. Current flows in the direction shown (opposite of electron flow) as soon as the switch is closed. Mutual repulsion of like charges in the capacitor progressively slows the flow as the capacitor is charged, stopping the current when the capacitor is fully charged and Q = C emf size 12{Q=C cdot "emf"} {} . (b) A graph of voltage across the capacitor versus time, with the switch closing at time t = 0 size 12{t=0} {} . (Note that in the two parts of the figure, the capital script E stands for emf, q stands for the charge stored on the capacitor, and τ is the RC time constant.)

Voltage on the capacitor is initially zero and rises rapidly at first, since the initial current is a maximum. [link] (b) shows a graph of capacitor voltage versus time ( t size 12{t} {} ) starting when the switch is closed at t = 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 C size 12{C} {} through a resistor R size 12{R} {} , derived using calculus, is

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

where V 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 RC size 12{ ital "RC"} {} are seconds. We define

τ = RC , size 12{τ= ital "RC"} {}

where τ size 12{τ} {} (the Greek letter tau) is called the time constant for an RC size 12{ ital "RC"} {} circuit. As noted before, a small resistance R 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 C size 12{C} {} , the less time needed to charge it. Both factors are contained in τ = RC size 12{τ= ital "RC"} {} .

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

V = emf 1 e 1 = emf 1 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"} {}

This means that in the time τ = 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 τ 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, τ size 12{τ} {} . In just a few multiples of the time constant τ size 12{τ} {} , then, the final value is very nearly achieved, as the graph in [link] (b) illustrates.

Discharging a capacitor

Discharging a capacitor through a resistor proceeds in a similar fashion, as [link] illustrates. Initially, the current is I 0 = V 0 R size 12{I rSub { size 8{0} } = { {V rSub { size 8{0} } } over {R} } } {} , driven by the initial voltage V 0 size 12{V rSub { size 8{0} } } {} on the capacitor. As the voltage decreases, the current and hence the rate of discharge decreases, implying another exponential formula for V size 12{V} {} . Using calculus, the voltage V size 12{V} {} on a capacitor C size 12{C} {} being discharged through a resistor R size 12{R} {} is found to be

V = V 0 e t / RC (discharging). size 12{V=`V"" lSub { size 8{0} } `e rSup { size 8{ - t/ ital "RC"} } } {}
Part a shows a circuit with a capacitor C connected in series with a resistor R and a switch to close the circuit. The current is shown flowing in a counterclockwise direction. The capacitor plates are shown to have a charge positive q and negative q respectively. Part b shows a graph of the variation of voltage across the capacitor with time. The voltage is plotted along the vertical axis and the time is along the horizontal axis. The graph shows a smooth downward falling curve which approaches a minimum and flattens out close to zero over time.
(a) Closing the switch discharges the capacitor C size 12{C} {} through the resistor R size 12{R} {} . Mutual repulsion of like charges on each plate drives the current. (b) A graph of voltage across the capacitor versus time, with V = V 0 size 12{V=V rSub { size 8{0} } } {} at t = 0 . The voltage decreases exponentially, falling a fixed fraction of the way to zero in each subsequent time constant τ size 12{τ} {} .

The graph in [link] (b) is an example of this exponential decay. Again, the time constant is τ = RC size 12{τ= ital "RC"} {} . A small resistance R size 12{R} {} allows the capacitor to discharge in a small time, since the current is larger. Similarly, a small capacitance requires less time to discharge, since less charge is stored. In the first time interval τ = RC size 12{τ= ital "RC"} {} after the switch is closed, the voltage falls to 0.368 of its initial value, since V = V 0 e 1 = 0 . 368 V 0 size 12{V=V rSub { size 8{0} } cdot e rSup { size 8{ - 1} } =0 "." "368"V rSub { size 8{0} } } {} .

Practice Key Terms 3

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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