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