# 4.6 Dc circuits containing resistors and capacitors  (Page 2/9)

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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 ( $t$ ) starting when the switch is closed at $t=0$ . 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 $C$ through a resistor $R$ , derived using calculus, is

$V=\text{emf}\left(1-{e}^{-t/\text{RC}}\right) \left(charging\right),$

where $V$ 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 $\text{RC}$ are seconds. We define

$\tau =\text{RC},$

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

More quantitatively, consider what happens when $t=\tau =\text{RC}$ . Then the voltage on the capacitor is

$V=\text{emf}\left(1-{e}^{-1}\right)=\text{emf}\left(1-0\text{.}\text{368}\right)=0\text{.}\text{632}\cdot \text{emf}.$

This means that in the time $\tau =\text{RC}$ , the voltage rises to 0.632 of its final value. The voltage will rise 0.632 of the remainder in the next time $\tau$ . 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, $\tau$ . In just a few multiples of the time constant $\tau$ , 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 ${I}_{0}=\frac{{V}_{0}}{R}$ , driven by the initial voltage ${V}_{0}$ on the capacitor. As the voltage decreases, the current and hence the rate of discharge decreases, implying another exponential formula for $V$ . Using calculus, the voltage $V$ on a capacitor $C$ being discharged through a resistor $R$ is found to be

$V=V{}_{0}\text{}\phantom{\rule{0.25em}{0ex}}{e}^{-t/\text{RC}}\text{(discharging).}$

The graph in [link] (b) is an example of this exponential decay. Again, the time constant is $\tau =\text{RC}$ . A small resistance $R$ 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 $\tau =\text{RC}$ after the switch is closed, the voltage falls to 0.368 of its initial value, since $V={V}_{0}\cdot {e}^{-1}=0\text{.}\text{368}{V}_{0}$ .

During each successive time $\tau$ , the voltage falls to 0.368 of its preceding value. In a few multiples of $\tau$ , the voltage becomes very close to zero, as indicated by the graph in [link] (b).

Now we can explain why the flash camera in our scenario takes so much longer to charge than discharge; the resistance while charging is significantly greater than while discharging. The internal resistance of the battery accounts for most of the resistance while charging. As the battery ages, the increasing internal resistance makes the charging process even slower. (You may have noticed this.)

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
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absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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is Bucky paper clear?
CYNTHIA
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Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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China
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types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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what is the function of carbon nanotubes?
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I'm interested in nanotube
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what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
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