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Capacitance

Energy stored in a capacitor

Energy stored in a capacitor

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Summary

Energy stored in a capacitor

​​In a nutshell

The work done depositing the charge onto the plates of the capacitor is the same as the energy stored by the capacitor. The area under a VVV against QQQ graph gives the energy stored by a capacitor.


Equations

​​Description

​​Equation

Capacitance equation
​​​​​C=QVC=\frac QVC=VQ​​​
Energy stored by a capacitor
​​​W=12QVW=12V2CW=12Q2CW=\frac 12 QV \newline \\[0.1in]W=\frac 12 V^2C \newline \\[0.1in]W=\frac 12 \frac {Q^2}{C} \newline \\[0.1in]W=21​QVW=21​V2CW=21​CQ2​​​


Variable definitions

​​Quantity Name

Symbol

​​Derived Unit

SI BASE Units

​​capacitancecapacitancecapacitance​
​​​​​CCC​​
​​​​​FFF​​
​​​​​kg−1 m−2 s4 A2kg^{-1}\,m^{-2}\,s^4\,A^2kg−1m−2s4A2​​
​chargechargecharge​​
​QQQ​​
​CCC​​
​A sA\,sAs​​
​potential differencepotential \ differencepotential difference​​
​VVV​​
​VVV​​
​kg m2 s−3 A−1kg\,m^2\,s^{-3}\,A^{-1}kgm2s−3A−1​​
​enery stored by capacitorenery \ stored \ by \ capacitorenery stored by capacitor​​
​WWW​​
​JJJ​​
​kg m2 s−2kg\,m^2\,s^{-2}kgm2s−2​​


Energy stored by capacitors

When a capacitor is charged, one of the plates becomes negatively charged and the opposite plate becomes positively charged.


The work done depositing electrons on the negative plate and removing the electrons from the positive plate, is supplied by the power supply. This work is then stored as the electric potential between the plates. 


The equation for charge stored on a capacitor is given as:


​Q=CVQ=CVQ=CV​​


The charge stored by a capacitor is directly proportional to the potential difference across it. This means that the graph will be a straight line which passes through the origin: 


Physics; Capacitance; KS5 Year 12; Energy stored in a capacitor
A
Potential difference
B
Charge


The work done on the capacitor plate, or the energy stored, is the product of the charge and the average potential difference. 

 

The average potential difference can be calculated as:


Vav=Vi+Vf2Vav=0+V2Vav=V2V_{av}=\frac {V_i+V_f }2\newline \\[0.1in]V_{av}=\frac {0+V }2\newline \\[0.1in]V_{av}=\frac {V }2Vav​=2Vi​+Vf​​Vav​=20+V​Vav​=2V​​​


Therefore the energy stored by the capacitor is:


​W=Q×V2W=12QVW=Q\times \frac V2 \newline \\[0.1in]W=\frac 12 QVW=Q×2V​W=21​QV​​


It is also the area under a VVV​ against QQQ​ graph.


Physics; Capacitance; KS5 Year 12; Energy stored in a capacitor
A
Potential difference
B
Charge
1
Energy stored


Equation variations

Using variations of the capacitance equation, two other derivations for the energy stored by a capacitor can be derived: 


Using Q=CVQ=CVQ=CV to eliminate QQQ​:


W=12(CV)VW=12V2CW=\frac 12 (CV)V \newline \\[0.1in]W=\frac 12 V^2CW=21​(CV)VW=21​V2C​​


Using V=QCV=\frac QCV=CQ​ to eliminate VVV​:​


​W=12Q(QC)W=12Q2CW=\frac 12 Q(\frac QC) \newline \\[0.1in]W=\frac 12 \frac {Q^2}C W=21​Q(CQ​)W=21​CQ2​​​


​​Example

A 48 μF48 \, \mu F48μF capacitor is charged using a 12 V12 \, V 12V battery. Calculate the amount of energy stored in the capacitor.


Firstly, write down the known values: 


C=48 μF=48×10−6 FV=12 VC=48 \, \mu F = 48 \times 10^{-6} \, F \newline \\[0.1in]V=12 \, VC=48μF=48×10−6FV=12V


​Next, write down the equations needed and rearrange if necessary:


W=12V2CW=\frac 12 V^2CW=21​V2C


Then, substitute the values into the equation:


W=12×122×48×10−6W=3.456×10−3W=\frac 12 \times 12^2\times 48\times 10^{-6} \newline \\[0.1in]W=3.456\times 10^{-3}W=21​×122×48×10−6W=3.456×10−3


Make sure to include units and round to the lowest number of significant figures of the values given by the question:


energy stored by the capacitor, W=3.5×10−3 Jenergy \ stored \ by \ the \ capacitor, \ W=3.5 \times 10^{-3} \, Jenergy stored by the capacitor, W=3.5×10−3J​​​​​


The energy stored by the capacitor is 3.5×10−3 J‾.\underline{3.5 \times 10^{-3} \, J}.3.5×10−3J​.​

​


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FAQs - Frequently Asked Questions

How can the energy stored by a capacitor be obtained from a graph?

The area under a V against Q graph gives the energy stored by a capacitor. ​

Where does the work come from for depositing electrons onto the negative plate in a capacitor?

The work done depositing electrons on the negative plate and removing the electrons from the positive plate is supplied by the power supply. The work is then stored as the electric potential between the two plates.

What is the relationship between charge and potential difference for a capacitor?

The charge stored by a capacitor is directly proportional to the potential difference across it.

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