A = Plate area
Z = Plate spacing
Ke = Electric force constant = 8.9876e9 N m2 / C2
Q = Max charge on the plate (Coulombs)
Emax= Max electric field = 4 Pi Ke Q / A
V = Voltage between plates = E Z = 4 Pi Ke Q Z / A
En = Energy = .5 Q V = .5 A Z E2 / (4 π Ke)
e = Energy/Volume = E / A Z = .5 E2 / (4 π Ke)
q = Charge/Volume = Q / A / Z
C = Capacitance = Q/V = (4 Pi Ke)-1 A/Z (Farads)
c = Capacitance/Volume = C / A / Z = (4 Pi Ke)-1 Emax2 / V2
Eair= Max electric field in air= 3 MVolt/meter
k = Dielectric factor = Emax / Eair
Continuum Macroscopic
Energy/Volume = .5 E2 / (4 Pi Ke) <-> Energy = .5 C V2
= .5 q V = .5 Q V
c = (4 Pi Ke)-1 Emax2 / V2 <-> C = (4 Pi Ke)-1 A / Z
A capacitor can be specified by two parameters:
*) Maximum energy density or maximum electric field
*) Voltage between the plates
The maximum electric field is equal to the max field for air times
a dimensionless number characterizing the dielectric
Eair = Maximum electric field for air before electical breakdown
Emax = Maximum electric field in the capacitor
Rbohr= Bohr radius
= Characteristic size of atoms
= 5.2918e-11 m
= hbar2 / (ElectronMass*ElectronCharge2*Ke)
Ebohr= Bohr electric field
= Field generated by a proton at a distance of 1 Bohr radius
= 5.142e11 Volt/m
Maximum energy density = .5 * 8.854e-12 Emax2
Emax (MVolt/m) Energy density
(Joule/kg)
Al electrolyte capacitor 15.0 1000
Supercapacitor 90.2 36000
Bohr limit 510000 1.2e12 Capacitor with a Bohr electric field
