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  • Explain the importance of the time constant, τ , and calculate the time constant for a given resistance and capacitance.
  • Explain why batteries in a flashlight gradually lose power and the light dims over time.
  • Describe what happens to a graph of the voltage across a capacitor over time as it charges.
  • Explain how a timing circuit works and list some applications.
  • Calculate the necessary speed of a strobe flash needed to “stop” the movement of an object over a particular length.

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 a number of other phenomena that involve charging and discharging capacitors are discussed in this module.

RC Circuits

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

[link] shows a simple RC 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 V c = Q / C size 12{V rSub { size 8{c} } =Q/C} {} , where Q size 12{Q} {} is the amount of charge stored on each plate and C 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 I 0 = emf R 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 IR 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 I 0 = emf R size 12{I rSub { size 8{0} } = { {"emf"} over {R} } } {} , because all of the IR 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 R 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.

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.)

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Source:  OpenStax, General physics ii phy2202ca. OpenStax CNX. Jul 05, 2013 Download for free at http://legacy.cnx.org/content/col11538/1.2
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