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By the end of the section, you will be able to:
  • Describe the charging process of a capacitor
  • Describe the discharging process of a capacitor
  • List some applications of RC circuits

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 several other phenomena that involve charging and discharging capacitors are discussed in this module.

Circuits with resistance and capacitance

An RC circuit    is a circuit containing resistance and capacitance. As presented in Capacitance , the capacitor is an electrical component that stores electric charge, storing energy in an electric field.

[link] (a) shows a simple RC circuit that employs a dc (direct current) voltage source ε , a resistor R , a capacitor C , and a two-position switch. The circuit allows the capacitor to be charged or discharged, depending on the position of the switch. When the switch is moved to position A , the capacitor charges, resulting in the circuit in part (b). When the switch is moved to position B , the capacitor discharges through the resistor.

Part a shows an open circuit with three branches, left branch is a voltage source with upward positive terminal connected to point A, the middle branch is a short circuit with point B and right branch is a resistor with a capacitor. Part b shows circuit of part a with first branch connected to third branch. Part c shows circuit of part a with second branch connected to third branch.
(a) An RC circuit with a two-pole switch that can be used to charge and discharge a capacitor. (b) When the switch is moved to position A , the circuit reduces to a simple series connection of the voltage source, the resistor, the capacitor, and the switch. (c) When the switch is moved to position B , the circuit reduces to a simple series connection of the resistor, the capacitor, and the switch. The voltage source is removed from the circuit.

Charging a capacitor

We can use Kirchhoff’s loop rule to understand the charging of the capacitor. This results in the equation ε V R V c = 0 . This equation can be used to model the charge as a function of time as the capacitor charges. Capacitance is defined as C = q / V , so the voltage across the capacitor is V C = q C . Using Ohm’s law, the potential drop across the resistor is V R = I R , and the current is defined as I = d q / d t .

ε V R V c = 0 , ε I R q C = 0 , ε R d q d t q C = 0.

This differential equation can be integrated to find an equation for the charge on the capacitor as a function of time.

ε R d q d t q C = 0 , d q d t = ε C q R C , 0 q d q ε C q = 1 R C 0 t d t .

Let u = ε C q , then d u = d q . The result is

0 q d u u = 1 R C 0 t d t , ln ( ε C q ε C ) = 1 R C t , ε C q ε C = e t R C .

Simplifying results in an equation for the charge on the charging capacitor as a function of time:

q ( t ) = C ε ( 1 e t R C ) = Q ( 1 e t τ ) .

A graph of the charge on the capacitor versus time is shown in [link] (a). First note that as time approaches infinity, the exponential goes to zero, so the charge approaches the maximum charge Q = C ε and has units of coulombs. The units of RC are seconds, units of time. This quantity is known as the time constant :

τ = R C .

At time t = τ = R C , the charge is equal to 1 e −1 = 1 0.368 = 0.632 of the maximum charge Q = C ε . Notice that the time rate change of the charge is the slope at a point of the charge versus time plot. The slope of the graph is large at time t = 0.0 s and approaches zero as time increases.

Questions & Answers

Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
Michael Reply
how was the check your understand 12.7 solved?
Bysteria Reply
Who is ISSAAC NEWTON
LOAK Reply
he's the father of 3 newton law
Hawi
he is Chris Issaac's father :)
Ethem
how to name covalent bond
Bryan Reply
Who is ALEXANDER BELL
LOAK
what do you understand by the drift voltage
Brunelle Reply
what do you understand by drift velocity
Brunelle
nothing
Gamal
well when you apply a small electric field to a conductor that causes to add a little velocity to charged particle than usual, which become their average speed, that is what we call a drift.
graviton
drift velocity
graviton
what is an electromotive force?
Danilo Reply
It is the amount of other forms of energy converted into electrical energy per unit charge that flow through it.
Brunelle
How electromotive force is differentiated from the terminal voltage?
Danilo
in the emf power is generated while in the terminal pd power is lost.
Brunelle
what is then chemical name of NaCl
Sagar Reply
sodium chloride
Azam
sodium chloride
Brunelle
Sodium Chloride.
Ezeanyim
How can we differentiate between static point and test charge?
Comfort Reply
Wat is coplanar in physics
Humble Reply
two point charges +30c and +10c are separated by a distance of 80cm,compute the electric intensity and force on a +5×10^-6c charge place midway between the charges
Tijani Reply
0.0844kg
Humble
what is the difference between temperature and heat
Ishom Reply
Heat is the condition or quality of being hot While Temperature is ameasure of cold or heat, often measurable with a thermometer
Abdul
Temperature is the one of heat indicators of materials that can be measured with thermometers, and Heat is the quantity of calor content in material that can be measured with calorimetry.
Gamma
the average kinetic energy of molecules is called temperature. heat is the method or mode to transfer energy to molecules of an object but randomly, while work is the method to transfer energy to molecules in such manner that every molecules get moved in one direction.
Adnan
2. A brass rod of length 50cm and diameter 3mm is joined to a steel rod of the same length and diameter. What is the change in length of the combined rod at 250°c( degree Celsius) if the original length are 40°c(degree Celsius) is there at thermal stress developed at the junction? The end of the rod are free to expand (coefficient of linear expansion of brass = 2.0×10^-5, steel=1.2×10^-5k^1)
TONEJIT Reply
A charge insulator can be discharged by passing it just above a flame. Explain.
Mudassar Reply
of the three vectors in the equation F=qv×b which pairs are always at right angles?
DRASHTI Reply
what is an ideal gas?
Justine Reply
What is meant by zero Kelvin ?
Justine
Why does water cool when put in the pot ?
Justine
when we pour the water in a vessel(pot) the hot body(water) loses its heat to the surrounding in order to maintain thermal equilibrium.Thus,water cools.
rupendra
when we drop water in the pot, the pot body loses heat to surrounded in order to maintain thermal equilibrium thus,water cool.
Srabon
my personal opinion ideal gas means doesn't exist any gas that obey all rules that is made for gases, like when get the temp of a gas lower, it's volume decreases.since the gas will convert to liquid when the temp get lowest.. so you can imagine it, but you can't get a gas at the lowest T
Adnan
Edit An ideal gas is a theoretically gascomposed of many randomly moving point particles whose only interactions are perfectly elastic collisions.
Gamma
ideal gases are real gases at low temperature
Brunelle
Practice Key Terms 1

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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