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

 Page 2 / 9

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

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

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!   By By    By By By