Charging and Discharging a Capacitor - Revision history

Ghood6 at 01:24, 27 April 2026

2026-04-27T01:24:25Z

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	'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).		'''Diagrams (external):''' For a clean RC circuit schematic and the exponential charging/discharging curves, see [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit] (figures licensed CC BY-SA 4.0).

	===A Computational Model===		==A Computational Model==

	The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.		The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.

	~~<!-- TRINKET IFRAME: paste the embed snippet from trinket.io here once the GlowScript model is built. Example~~:		'''Live simulation:''' [https://www.glowscript.org/#/user/gabrielhoodjr/folder/MyPrograms/program/Wiki Run the GlowScript RC-circuit model on glowscript.org] (opens in a new tab; click "Run this program" to start the simulation, then use the ''Charge / Discharge'' button and the sliders for ε, ''R'', and ''C'').
	~~<iframe src="~~https://~~trinket~~.io/~~embed~~/glowscript~~/XXXXXXXXXX~~" ~~width=~~"~~100%" height="600" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen><~~/~~iframe>~~
	~~-->~~

	''~~Live simulation pending; sample output of charge buildup~~ is shown below.''		[[File:RC_glowscript_screenshot_GHood.png\|center\|frame\|Screenshot of the GlowScript model running with default values (ε = 9 V, R = 1 kΩ, C = 1 mF, τ = 1 s). The top scene shows the circuit and capacitor; the live readouts beneath give t, Q, I, V<sub>C</sub>, and τ; the bottom plot tracks Q(t)/Q<sub>max</sub> and I(t)/(ε/R) over time. Click the link above to interact with it directly.]]

			''For comparison, the textbook analytic Q(t) charging curve is shown below — the GlowScript model's blue trace should reproduce this same exponential approach to Q<sub>max</sub> = Cε.''

	[[File:Capacitor Charging.svg\|center\|frame\|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]		[[File:Capacitor Charging.svg\|center\|frame\|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

Ghood6 at 01:05, 27 April 2026

2026-04-27T01:05:47Z

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	The capacitor was discovered independently in 1745–46 by Ewald Georg von Kleist in Germany and Pieter van Musschenbroek at the University of Leiden, the latter giving the device its early name: the Leyden jar. Both used a glass jar partially filled with water, with a wire passing through the stopper, and charged it with an electrostatic generator; touching the wire produced a painful shock as the jar discharged.		The capacitor was discovered independently in 1745–46 by Ewald Georg von Kleist in Germany and Pieter van Musschenbroek at the University of Leiden, the latter giving the device its early name: the Leyden jar. Both used a glass jar partially filled with water, with a wire passing through the stopper, and charged it with an electrostatic generator; touching the wire produced a painful shock as the jar discharged.

	Benjamin Franklin investigated the Leyden jar in the 1740s and showed that the stored charge resided on the glass dielectric itself rather than on the water, and he coined the term "battery" for an array of such jars wired together. The modern parallel-plate geometry and the formal capacitance relation <~~math~~>~~C = \varepsilon_0~~ A / s~~</math>~~ followed in the 19th century alongside the development of electromagnetism by Faraday, Maxwell, and others. The unit of capacitance, the farad, is named for Michael Faraday.		Benjamin Franklin investigated the Leyden jar in the 1740s and showed that the stored charge resided on the glass dielectric itself rather than on the water, and he coined the term "battery" for an array of such jars wired together. The modern parallel-plate geometry and the formal capacitance relation ''C'' = ''ε''<sub>0</sub>''A''/''s'' followed in the 19th century alongside the development of electromagnetism by Faraday, Maxwell, and others. The unit of capacitance, the farad, is named for Michael Faraday.

	== See also ==		== See also ==

Ghood6 at 01:04, 27 April 2026

2026-04-27T01:04:54Z

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	==Connectedness==		==Connectedness==

	~~Capacitor can be temporary batteries~~. ~~Capacitors~~ in ~~parallel can continue to supply~~ current to the ~~circuit if the battery runs out~~. ~~This~~ is ~~interesting because~~ the ~~capacitor gets its charge from being connected to~~ a ~~chemical battery, but the capacitor itself supplies voltage without chemicals~~.		RC circuits show up everywhere in real electronics. The same exponential charge/discharge behavior derived above sets the timing in camera flashes (a capacitor stores energy and dumps it into the bulb), defibrillators (a large capacitor delivers a controlled current pulse), and the smoothing capacitors in any DC power supply. The time constant <math>\tau = RC</math> is also the basic building block for low-pass and high-pass filters in audio gear and signal processing — picking <math>R</math> and <math>C</math> is literally how you choose a cutoff frequency.

	Capacitors are ~~being researched for applications~~ in ~~electromagnetic armour~~ and ~~electromagnetic weapons~~. ~~Currently capacitors~~ are ~~used as detonators~~ in ~~nuclear weapons. Capacitors also are largely involved in separations of AC~~ and ~~DC components~~.		Capacitors are not chemical batteries: they store energy in an electric field rather than in chemical bonds, so they can charge and discharge much faster but hold far less total energy per unit mass. That trade-off is why supercapacitors are showing up alongside (not replacing) lithium batteries in electric vehicles and regenerative braking systems, and why pulsed-power applications — from particle accelerators to high-energy physics experiments — rely on capacitor banks rather than batteries.

	==History==		==History==

	~~Put this idea~~ in ~~historical context. Give the reader the Who, What, When, Where, and Why.~~		The capacitor was discovered independently in 1745–46 by Ewald Georg von Kleist in Germany and Pieter van Musschenbroek at the University of Leiden, the latter giving the device its early name: the Leyden jar. Both used a glass jar partially filled with water, with a wire passing through the stopper, and charged it with an electrostatic generator; touching the wire produced a painful shock as the jar discharged.
	~~In 1745~~ Ewald Georg von Kleist ~~was the first to "discover" capacitors~~ in Germany. ~~He connected~~ a ~~generator to~~ glass ~~jars of~~ water and charged ~~them. When he touched~~ the wire ~~they were connected to he shocked himself (discharged the capacitor). At the same time Pieter van Musschenbroek made~~ a ~~similar capacitor and named it~~ the ~~Leyden Jar~~. ~~When~~ Benjamin Franklin ~~studied~~ the Leyden ~~Jar he determined, among other things,~~ that the charge ~~was stored~~ on the glass~~. During his studies Franklin was~~ the ~~first to give~~ the ~~capacitor the name~~ battery. ~~Since then batteries have most often been electro~~-~~chemical cells~~ of ~~capacitors made~~ of ~~sheets of conducting and dielectric material~~.
			Benjamin Franklin investigated the Leyden jar in the 1740s and showed that the stored charge resided on the glass dielectric itself rather than on the water, and he coined the term "battery" for an array of such jars wired together. The modern parallel-plate geometry and the formal capacitance relation <math>C = \varepsilon_0 A / s</math> followed in the 19th century alongside the development of electromagnetism by Faraday, Maxwell, and others. The unit of capacitance, the farad, is named for Michael Faraday.

	== See also ==		== See also ==

			* [[RC Circuit]]
			* [[Capacitor]]
			* [[Resistors and Conductivity]]
			* [[Kirchhoff's Loop Rule]]

	===Further reading===		===Further reading===
	#Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered"
	#Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s		* Williams, Henry Smith. ''A History of Science, Volume II, Part VI: The Leyden Jar Discovered''.
			* Keithley, Joseph F. (1999). ''The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s''. IEEE Press.

	===External links===		===External links===

	~~#Wikipedia Page "Capacitor"~~[https://en.wikipedia.org/wiki/Capacitor]		* [https://en.wikipedia.org/wiki/Capacitor Wikipedia: Capacitor]
	~~#Khan Academy~~[https://~~www~~.~~khanacademy~~.org/~~science~~/~~physics/circuits-topic~~]		* [https://en.wikipedia.org/wiki/RC_circuit Wikipedia: RC circuit]
	~~#WebAssign "Lab 4 - Charge and Discharge of a Capacitor"~~[~~http~~://www.~~webassign~~.~~net~~/~~labsgraceperiod~~/~~ncsulcpem2~~/~~lab_4/manual.html~~]		* [https://www.khanacademy.org/science/physics/circuits-topic Khan Academy: Circuits]
	~~#"Charge of Capacitor vs Current"~~ [~~http~~://~~www.physicsbook~~.~~gatech~~.edu/~~Charging_and_Discharging_a_Capacitor~~]		* [https://phet.colorado.edu/en/simulations/capacitor-lab-basics PhET: Capacitor Lab Basics simulation]

	==References==		==References==

	#~~Chabay, Ruth W~~., ~~and Bruce A. Sherwood. Matter and Interactions. 3rd ed. Vol. 2. N.p~~.: ~~John Wiley and Sons, 2002~~. ~~Print~~.		# "Capacitor." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Capacitor
	#"~~Capacitor~~." Wikipedia. Wikimedia Foundation, n.d. ~~Web~~. ~~05 Dec~~. ~~2015.~~		# "RC circuit." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/RC_circuit
			# Griffiths, David J. ''Introduction to Electrodynamics''. 4th ed. Cambridge University Press, 2017.

	[[Category:~~Which Category did you place this in?~~]]		[[Category:Electric Circuits]]

Ghood6 at 01:02, 27 April 2026

2026-04-27T01:02:01Z

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	==Examples==		==Examples==

	~~'''Question 1'''~~		===Simple===

			'''Question 1.''' For a parallel-plate capacitor with capacitance <math>C = \varepsilon_0 A / s</math>, predict how each change affects <math>C</math> and the field <math>E</math> between the plates (the plates carry a fixed charge <math>Q</math>).

	Doubling the radius ~~of the capacitor~~		(a) Doubling the plate radius:
			:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

	A) quarters ~~the capacitance~~		(b) Doubling the plate radius (effect on <math>E</math> with fixed <math>Q</math>):
			:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

	B) halves ~~the capacitance~~		(c) Doubling the plate separation <math>s</math>:
			:A) quarters <math>C</math> B) halves <math>C</math> C) doubles <math>C</math> D) quadruples <math>C</math>

	C) doubles ~~the capacitance~~		(d) Doubling the capacitance (with <math>Q</math> fixed):
			:A) quarters <math>E</math> B) halves <math>E</math> C) doubles <math>E</math> D) quadruples <math>E</math>

	D) ~~quadruples the capacitance~~		'''Answers:''' (a) D, (b) A, (c) B, (d) B.

			'''Why:'''
			* (a) Plate area is <math>A = \pi r^2</math>. Doubling <math>r</math> gives <math>A \to 4A</math>, and since <math>C = \varepsilon_0 A/s</math>, we get <math>C \to 4C</math>.
			* (b) The field is <math>E = \sigma/\varepsilon_0 = Q/(\varepsilon_0 A)</math>. With <math>Q</math> fixed and <math>A \to 4A</math>, <math>E \to E/4</math>.
			* (c) <math>C \propto 1/s</math>, so doubling <math>s</math> halves <math>C</math>.
			* (d) With <math>Q</math> fixed, doubling <math>C</math> halves <math>V = Q/C</math>, and since <math>E = V/s</math>, the field also halves.

	~~Doubling the radius of the capacitor~~		===Middling===

	A) ~~quarters the electric field between the plates~~		'''Question 2.''' A battery is connected through a switch to a resistor (light bulb) in series with an initially uncharged capacitor.

	B) ~~halves the electric field between the plates~~		[[File:Circuits problem 1 wiki.PNG\|center\|frame\|Series circuit: battery, switch, bulb (resistor), capacitor.]]

	C) ~~doubles~~ the ~~electric field between~~ the ~~plates~~		What is the current at points A, B, and C (a) the instant the switch closes, and (b) after a long time? What is the final charge on the capacitor?

	~~D) quadruples~~ the ~~electric field between~~ the ~~plates~~		'''Solution.''' At <math>t=0</math> the capacitor is uncharged, so <math>V_C = 0</math> and the loop equation gives <math>I_0 = \varepsilon/R</math>. Because the circuit is a single series loop, the current is the same at A, B, and C: all equal to <math>\varepsilon/R</math>.

			As <math>t \to \infty</math>, the capacitor charges until <math>V_C = \varepsilon</math>. With no voltage across <math>R</math>, the current at every point is zero. The final charge is <math>Q_\infty = C\varepsilon</math>.

	~~Doubling the distance between the plates~~		[[File:Circuits problem 1 wiki answers.PNG\|center\|frame\|Worked answers for Question 2.]]

	~~A) quarters the capacitance~~		===Difficult===

	~~B) halves~~ the ~~capacitance~~		'''Question 3.''' The switch in the circuit below has been closed for a long time, then is opened.

	~~C) doubles~~ the ~~capacitance~~		[[File:Circuits problem 2 wiki.PNG\|center\|frame\|Circuit with a battery, capacitor, and bulb; analyze before and after opening the switch.]]

	D) ~~quadruples~~ the ~~capacitance~~		(a) Before the switch opens (steady state): What is the current at each point? What is the charge on the capacitor? Is the bulb lit?

			(b) Immediately after the switch opens: Is the bulb lit? After a long time? What is the initial current through the bulb, and in what direction?

	~~Doubling the capacitance~~		'''Solution.'''

	A) ~~quarters~~ the ~~electric field between~~ the ~~plates~~		(a) In steady state the capacitor is fully charged and acts as an open circuit, so no current flows in the branch containing the capacitor. Current still flows through any branch that bypasses the capacitor (for example the bulb in parallel with the source), set by Ohm's law on that loop. The capacitor charge is <math>Q = C V_C</math>, where <math>V_C</math> equals the steady-state voltage across its terminals.

	B) ~~halves~~ the ~~electric field between~~ the ~~plates~~		(b) When the switch opens, the battery is disconnected. The charged capacitor now acts as the only EMF source and discharges through the bulb. Immediately after, the bulb is lit with current <math>I_0 = V_C/R</math>, where <math>V_C</math> is the capacitor voltage just before opening. The current direction reverses relative to the charging direction because the capacitor is now driving the loop. The current decays as <math>I(t) = I_0 e^{-t/RC}</math>, so after several time constants the bulb goes dark.

	~~C) doubles the electric field between the plates~~		[[File:Circuits problem 2 wiki answers.PNG\|center\|frame\|Worked answers for Question 3.]]

	~~D) quadruples the electric field between the plates~~


	~~ANS: D, B, A, C~~


	~~'''Question 2'''~~

	[[File:Circuits problem 1 wiki.PNG]]


	~~What is the current at points A,B, and C when the capacitor is not yet charged and when the capacitor is fully charged?~~

	~~When the capacitor is fully charged what is the charge on the plates?~~

	~~'''Answer:'''~~

	~~[[File:Circuits problem 1 wiki~~ answers~~.PNG]]~~


	~~'''~~Question 3~~'''~~

	~~[[File:Circuits problem 2 wiki.PNG]]~~


	~~The switch has been closed for a long time.~~

	~~What is the current at each point?~~

	~~What is the charge on the capacitor?~~

	~~Is the light bulb lit?~~


	~~The switch is opened.~~

	~~Immediately after the switch is opened is the bulb lit? After a while?~~

	~~What current is initially running through the bulb?~~

	~~Which direction is the current moving?~~

	~~'''Answer:'''~~

	~~[[File:Circuits problem 2 wiki answers~~.~~PNG~~]]

	==Connectedness==		==Connectedness==

Ghood6 at 00:44, 27 April 2026

2026-04-27T00:44:51Z

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	===A Computational Model===		===A Computational Model===

	[[File:Capacitor Charging.svg\|~~Capacitor Charging~~]]		The model below numerically integrates Kirchhoff's loop rule for the same series RC circuit derived above. Pick values for the battery emf <math>\mathcal{E}</math>, resistance <math>R</math>, and capacitance <math>C</math> at the top of the script and the simulation will plot <math>Q(t)</math> on the capacitor and <math>I(t)</math> through the resistor as the capacitor charges, then discharges. The numerical curves should match the analytic <math>Q(t) = C\mathcal{E}(1 - e^{-t/RC})</math> and <math>Q(t) = Q_0\,e^{-t/RC}</math> from the section above.

			<!-- TRINKET IFRAME: paste the embed snippet from trinket.io here once the GlowScript model is built. Example:
			<iframe src="https://trinket.io/embed/glowscript/XXXXXXXXXX" width="100%" height="600" frameborder="0" marginwidth="0" marginheight="0" allowfullscreen></iframe>
			-->

			''Live simulation pending; sample output of charge buildup is shown below.''

			[[File:Capacitor Charging.svg\|center\|frame\|Sample output: Q(t) for a charging capacitor showing the characteristic exponential approach to Q_max = Cε.]]

	===Current and Charge within the Capacitors===		===Current and Charge within the Capacitors===

Ghood6 at 00:41, 27 April 2026

2026-04-27T00:41:32Z

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	==The Main Idea==		==The Main Idea==

	~~'''Charging~~ a ~~Capacitor''~~'		A capacitor in a circuit acts like a charge reservoir. While charging, current flows from the battery and piles charge onto the plates; while discharging, the plates push current back through the resistor until they're empty. In both cases, the current at any instant is set by how much voltage the capacitor still has left to gain or lose, which is why both processes are exponential.

	~~[[File:~~Charging ~~a capacitor wiki.PNG]]~~		===Charging===

			[[File:Charging a capacitor wiki.PNG\|center\|frame\|Charging a capacitor through a resistor (here, a light bulb). The field across the gap drops as charge accumulates, and current decays toward zero.]]

	~~Charging a~~ capacitor ~~isn’t much more difficult than discharging and the same principles still apply. The circuit consists of two batteries,~~ a ~~light bulb,~~ and a ~~capacitor~~. ~~Essentially~~, the ~~electron current from~~ the ~~batteries will continue to run until~~ the ~~circuit reaches equilibrium (the capacitor~~ is ~~“full”)~~. ~~Just like when discharging~~, the ~~bulb starts out bright while~~ the ~~electron~~ current ~~is running, but it slowly dims~~ and ~~goes out as the capacitor charges~~.		Connect an uncharged capacitor in series with a battery (emf <math>\mathcal{E}</math>) and a resistor. Initially there is no charge on the plates, so the full battery voltage drops across the resistor and the current is at its maximum, <math>I_0 = \mathcal{E}/R</math>. As charge builds up on the plates, it creates a back-voltage <math>V_C = Q/C</math> that opposes the battery. The net driving voltage <math>\mathcal{E} - Q/C</math> shrinks, and so does the current. Charging stops when <math>V_C = \mathcal{E}</math> and <math>Q = C\mathcal{E}</math>.

	~~The electron current will flow out~~ the ~~negative end of the battery as usual (conventional current will exit the positive end). Positive charges begin to build up on the right plate~~ and ~~negative charges on~~ the ~~left. The electric field slowly decreases until the net electric field~~ is 0~~. The fringe field is equal~~ and ~~opposite~~ to ~~the electric field caused by everything else.~~		In the bulb-and-capacitor demo, the bulb is brightest at <math>t = 0</math> and dims to dark as <math>Q \to C\mathcal{E}</math>.

	~~If you were~~ to draw a box around the capacitor and label it with positive and negative ends it would look like a battery. It also behaves like a battery. The electron current will continue to flow and the electric field will continue to exist until the potential difference across the capacitor is equal to that of the batteries (sum of emf of all batteries in the circuit).

	~~The following link shows~~ the ~~relationship~~ of ~~capacitor~~ plate charge ~~to current:~~		'''Plate polarity.''' Conventional current flows out the + terminal of the battery. The plate it reaches first becomes positive (charge accumulates there); the other plate becomes negative. This determines the sign of <math>V_C</math> when applying Kirchhoff's loop rule.
	~~[https:~~/~~/trinket~~.~~io/glowscript/cd909d0091?outputOnly=true Capacitor Charge Vs Current]~~

			===Discharging===

	~~'''~~Discharging a ~~Capacitor'''~~		[[File:Discharging a capacitor wiki.PNG\|center\|frame\|Discharging a charged capacitor through a resistor. The capacitor itself drives the current, which decays as Q drops.]]

	~~[[File~~:~~Discharging~~ a capacitor ~~wiki~~.~~PNG]]~~		Disconnect the battery and connect the charged capacitor across a resistor. Now the capacitor's voltage <math>V_C = Q/C</math> drives the current: <math>I = V_C/R = Q/(RC)</math>. As <math>Q</math> falls, so does the current — exponentially. The bulb starts bright and dims to dark, just like in charging, but for a different reason: now it's the capacitor running out of charge, not the capacitor opposing the battery.

	A circuit with a charged capacitor has an electric fringe field inside the wire. This field creates an electron current. The electron current will move opposite the direction of the electric field. However, so long as the electron current is running, the capacitor is being discharged. The electron current is moving negative charges away from the negatively charged plate and towards the positively charged plate. Once the charges even out or are neutralized the electric field will cease to exist. Therefore the current stops running.		'''Role of <math>R</math>.''' The resistance sets the timescale through <math>\tau = RC</math>. Larger <math>R</math> ⇒ slower charge/discharge. (A bulb with a thinner filament has higher <math>R</math>, so it lights more dimly but for longer.) Smaller <math>R</math> means faster, brighter, and shorter.

	In the example where the charged capacitor is connected to a light bulb you can see the electric field is large in the beginning but decreases over time. The electron current is also greater in the beginning and decreases over time. Because of this the light bulb starts out shining brightly but slowly dims and goes out.

	~~RIVASH DEEPNARAIN (SPRING 2023):~~

	~~Determining Which side of the Capacitor becomes Positive and Negative~~

	A common thing that confused me was which side of the capacitor acquires a positive charge and which side is negative. You need to know this because when calculating the voltage across a capacitor, you need to know whether your path goes against the electric field or in the same direction as the electric field that is in between the two plates. Since conventional current, denoted with a capital I, is actually the flow of positive charges, the side that the current meets first will therefore become positive. The other side would then become negative. Examples of this can be seen in the images below. This was confusing to me at first but after I realized this, calculating voltage across capacitors became much simpler.



	'''~~Resistors~~'''

	The ~~amount of~~ resistance in the ~~circuit will determine how long it takes a capacitor to~~ charge or discharge. ~~The less resistance~~ (~~a light~~ bulb with a ~~thicker~~ filament~~) the faster the capacitor will charge or discharge. The~~ more ~~resistance (a light bulb with a thin filament) the~~ longer ~~it will take the capacitor to charge or discharge~~. ~~The thicker filament bulb will be~~ brighter, ~~but won't last as long as a thin filament bulb. '''V = IR''', The larger the resistance the smaller the current~~.

	===A Mathematical Model===		===A Mathematical Model===

Ghood6 at 00:36, 27 April 2026

2026-04-27T00:36:07Z

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	<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>		<math>\mathcal{E} - IR - \frac{Q}{C} = 0</math>

			[[File:RC_circuit_schematic_GHood.png\|center\|frame\|Series RC circuit: battery emf ε, switch S, resistor R, capacitor C with charges +Q and −Q. Conventional current I = dQ/dt flows clockwise during charging.]]

	Substituting <math>I = dQ/dt</math>:		Substituting <math>I = dQ/dt</math>:
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	<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>		<math>Q(t) = Q_0\,e^{-t/RC}, \qquad I(t) = -\frac{dQ}{dt} = \frac{Q_0}{RC}\,e^{-t/RC}</math>

			[[File:RC_charging_discharging_curves_GHood.png\|center\|frame\|Charge on the capacitor vs. time. Charging (blue) reaches 63.2% of Q_max at t = τ; discharging (red) falls to 36.8% of Q₀ at t = τ. After 5τ both processes are essentially complete.]]

	'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.		'''Time constant.''' The product <math>\tau = RC</math> has units of seconds and sets the timescale of every RC process. After one <math>\tau</math>, a charging capacitor reaches <math>1 - e^{-1} \approx 63.2\%</math> of its final charge; a discharging one falls to <math>e^{-1} \approx 36.8\%</math>. After <math>5\tau</math> both are within 1% of their asymptote.

Ghood6 at 00:16, 27 April 2026

2026-04-27T00:16:22Z

@@ Line 1: / Line 1: @@
-'''Edited by Gabriel Hood Spring 2025 Physics 2212)'''
+'''Edited by Gabriel Hood Spring 2026 Physics 2212)'''
 A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.
@@ Line 44: / Line 44: @@
 ===A Mathematical Model===
-<math>V = IR</math>
+'''Parallel-plate capacitance.''' Treat the plates as two oppositely charged infinite sheets with surface charge density <math>\sigma = Q/A</math>. Each sheet contributes a field of magnitude <math>\sigma/(2\varepsilon_0)</math>, and between the plates these fields add:
-<math>E = (Q/A)/\varepsilon_0</math>
+<math>E = \frac{\sigma}{\varepsilon_0} = \frac{Q}{\varepsilon_0 A}</math>
-<math>C = Q/V = \varepsilon_0A/s</math>
+The voltage across a gap of size <math>s</math> is <math>V = E\,s = Qs/(\varepsilon_0 A)</math>, so
-<math>V = (Q/A)s/ε0</math>
+<math>C \equiv \frac{Q}{V} = \frac{\varepsilon_0 A}{s}</math>
 ===A Computational Model===

Ghood6 at 00:07, 27 April 2026

2026-04-27T00:07:02Z

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	~~[https://trinket.io/glowscript/31d0f9ad9e]~~		'''Edited by Gabriel Hood Spring 2025 Physics 2212)'''
	~~The main purpose of having a capacitor in a circuit is to store electric charge. For intro physics you can almost think of them as a battery.~~

			A capacitor is a two-conductor device that stores energy in the electric field between its plates. When you connect an uncharged capacitor to a battery through a resistor, charge piles up on the plates over time — that's '''charging'''. Disconnect the battery and let the capacitor drive current through a resistor, and the stored charge bleeds back off — that's '''discharging'''. Both processes are governed by the same RC time constant <math>\tau = RC</math>, and both follow exponential curves in time.

	''~~'Edited by ROHAN NANDAKUMAR (SPRING 2021)~~'''		A capacitor is ''not'' a battery. A battery maintains a roughly constant emf from chemistry; a capacitor's voltage rises or falls as charge moves on or off its plates.

	==The Main Idea==		==The Main Idea==

Rivash Deepnarain: /* The Main Idea */

2023-04-24T01:41:50Z

The Main Idea

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	A common thing that confused me was which side of the capacitor acquires a positive charge and which side is negative. You need to know this because when calculating the voltage across a capacitor, you need to know whether your path goes against the electric field or in the same direction as the electric field that is in between the two plates. Since conventional current, denoted with a capital I, is actually the flow of positive charges, the side that the current meets first will therefore become positive. The other side would then become negative. Examples of this can be seen in the images below. This was confusing to me at first but after I realized this, calculating voltage across capacitors became much simpler.		A common thing that confused me was which side of the capacitor acquires a positive charge and which side is negative. You need to know this because when calculating the voltage across a capacitor, you need to know whether your path goes against the electric field or in the same direction as the electric field that is in between the two plates. Since conventional current, denoted with a capital I, is actually the flow of positive charges, the side that the current meets first will therefore become positive. The other side would then become negative. Examples of this can be seen in the images below. This was confusing to me at first but after I realized this, calculating voltage across capacitors became much simpler.



	'''Resistors'''		'''Resistors'''