AcademyCapacitors and Dielectrics
Academy
Stored Energy and Field Energy
Level 1 - Physics topic page in Capacitors and Dielectrics.
Principle
A capacitor stores work done to move charge across an increasing potential difference.
Notation
\(U\)
energy stored in the capacitor
\(\mathrm{J}\)
\(q\)
charge already moved during charging
\(\mathrm{C}\)
\(Q\)
final charge magnitude
\(\mathrm{C}\)
\(V\)
final voltage
\(\mathrm{V}\)
\(C\)
capacitance
\(\mathrm{F}\)
\(u_E\)
electric field energy density
\(\mathrm{J\,m^{-3}}\)
Method
Derivation 1: Integrate the charging work
During charging, the voltage is not constant. When charge \(q\) is already on the capacitor, the voltage is \(q/C\).
Instantaneous voltage
\[v(q)=\frac{q}{C}\]
Small work
\[dW=v(q)\,dq=\frac{q}{C}\,dq\]
Stored energy
\[U=\int_0^Q\frac{q}{C}\,dq=\frac{Q^2}{2C}\]
Derivation 2: Rewrite the same energy
Use \(Q=CV\) to express the energy with whichever pair of variables is known.
Charge form
\[U=\frac{Q^2}{2C}\]
Voltage form
\[U=\frac{1}{2}CV^2\]
Mixed form
\[U=\frac{1}{2}QV\]
Derivation 3: Field energy density
For a parallel-plate capacitor, write the stored energy as energy in the space between the plates.
Capacitor energy
\[U=\frac{1}{2}CV^2\]
Plate substitutions
\[C=\epsilon_0\frac{A}{d},\quad V=Ed\]
Field energy
\[U=\frac{1}{2}\epsilon_0E^2(Ad)\]
Energy density
\[u_E=\frac{U}{Ad}=\frac{1}{2}\epsilon_0E^2\]
Rules
These are equivalent forms of the stored energy.
Charge form
\[U=\frac{Q^2}{2C}\]
Voltage form
\[U=\frac{1}{2}CV^2\]
Mixed form
\[U=\frac{1}{2}QV\]
Field density
\[u_E=\frac{1}{2}\epsilon_0E^2\]
Examples
Question
Find the energy stored in a
\[10\,\mu\mathrm{F}\]
capacitor charged to \[12\,\mathrm{V}\]
Answer
\[U=\frac{1}{2}CV^2=\frac{1}{2}(10\times10^{-6})(12^2)=7.2\times10^{-4}\,\mathrm{J}\]
Checks
- Use the energy form that matches the fixed quantity: fixed \(Q\) or fixed \(V\).
- Stored energy is never negative.
- The factor \(1/2\) appears because the voltage rises from zero during charging.
- Field energy density scales as \(E^2\), so reversing field direction does not change it.