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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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