Physics Capacitors

Energy Density

Energy Density ie energy stored per unit volume.
For a parallel plate capacitors `U = 1/2 CV^2` where `C = epsilon_0 A/d` and `V = Ed`

`U = 1/2 (epsilon_0A/d)E^2d^2 => (1/2epsilon_0E^2)Ad`

`=>U = 1/2 epsilon_0 E^2 tau` where `tau` is volume of the capacitor

`U/tau = U_e = 1/2 epsilon_0 E^2 = sigma^2/ (2epsilon_0)`

This energy is stored in the capacitor in the form of electrostatic field
electrostatic energy/volume = Energy Density

Energy Density ` = 1/2 epsilon_0E^2 = sigma^2 /(2epsilon_0)` , `{E = sigma/epsilon_0}`

Energy for series combination

For series combination charge Q will be same in all capcitor.

`1/C_s = 1/C_1 + 1/C_2 + 1/C_3`

`Q^2/2C_s = Q^2/(2C_1) + Q^2/(2C_2) + Q^2/(2C_3)`

`U_s = U_1 + U_2 + U_3`

i.e., where `U_1 , U_2, U_3` is stored energy in capacitor `C_1, C_2, C_3 ` and so on,

Total energy for the series combination = sum of individual energies stored in each capacitor

Energy for parallel combination

For parallel combination potential V will be same.

`C_p V = C_1V + C_2 V + C_3 V `

`1/2 C_p V^2 = 1/2 C_1V^2 + 1/2 C_2V^2 + 1/2 C_3 V^2`

`U_p = U_1 + U_2 + U_3`

i.e., where `U_1 , U_2, U_3` is stored energy in capacitor `C_1, C_2, C_3 ` and so on,

Total energy for the parallel combination = sum of individual energies stored in each capacitor