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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 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} } } {} .

During each successive time τ size 12{τ} {} , the voltage falls to 0.368 of its preceding value. In a few multiples of τ size 12{τ} {} , 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.)

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
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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what's the easiest and fastest way to the synthesize AgNP?
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China
Cied
types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
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what is the function of carbon nanotubes?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, College physics ii. OpenStax CNX. Nov 29, 2012 Download for free at http://legacy.cnx.org/content/col11458/1.2
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