Energy density in charged parallel plate capacitor

In this post, we will derive the equation of the Energy density in charged parallel plate capacitor.

Energy stored in parallel plate capacitor U = (1/2) C V² where, C is the capacitance of the capacitor and V is the potential difference between the two parallel plates of the above said capacitor.

We know, that the capacitance C of the parallel plate capacitor can be expressed as: C = Aε₀ /d. Here, ε₀ = permittivity of air, and d = the separation between the plates, A = area of each plate of the capacitor

The Potential difference V can be expressed as: V = Ed, where E = electric field between the plates

So, Energy stored in parallel plate capacitor U = (1/2) C V²

=> U = (1/2) ( Aε₀ /d) (Ed)² = (1/2) ( Aε₀) (E²d) ………….. (1)

Now, the volume of the space between the plates of the parallel plate capacitor: volume = A d ……………….. (2)

Hence, Energy density u = Energy stored in parallel plate capacitor / volume = (1/2) ( Aε₀) (E²d) / A d

=> u = (1/2) ε₀ E²

Energy density of parallel plate capacitor = (1/2) ε₀ E²