## Capacitance

• ### Units and Variables

• C - capacitance, in farads
• Q - charge, in coulombs
• V - electric potential, in volts
• ΔV - electric potential difference, in volts
• σ - surface charge density, in coulombs per meters squared
• d - distance between parallel plates, in meters
• A - area, in meters squared
• ε0 - Permittivity of free space, in coulomb squared per newton-meters squared
• U - electric potential energy, in joules
• uE - energy density
• κ - dielectric constant, in volts per meter
• ${\epsilon }_{0}=8.8542×{10}^{-12}\frac{{C}^{2}}{N{m}^{2}}$
• $C\equiv \frac{Q}{\mathrm{\Delta V}}$
• ### Sphere

• $C=4\pi {\epsilon }_{0}R$
• ### Cylindrical Capacitor (coaxial cable)

• $C=\frac{2\pi {\epsilon }_{0}𝓁}{\mathrm{ln}\left(\frac{\mathrm{outer radius}}{\mathrm{inner radius}}\right)}$
• ### Parallel Plate Capacitor

• $C=\frac{{\epsilon }_{0}A}{d}$
• $E=\frac{\sigma }{{\epsilon }_{0}}=\frac{Q}{{\epsilon }_{0}A}$
• $\mathrm{\Delta V}=Ed=\frac{Qd}{{\epsilon }_{0}A}$
• $C=\kappa \frac{{\epsilon }_{0}A}{d}$
• ### Parallel Combination

• ${Q}_{\mathrm{equivalent}}={Q}_{1}+{Q}_{2}+\dots$
• ${\mathrm{\Delta V}}_{1}={\mathrm{\Delta V}}_{2}={\mathrm{\Delta V}}_{3}=\dots$
• ${C}_{\mathrm{equivanent}}={C}_{1}+{C}_{2}+\dots$
• ### Series Combination

• ${Q}_{1}={Q}_{2}={Q}_{3}=\dots$
• ${\mathrm{\Delta V}}_{\mathrm{equivalent}}={\mathrm{\Delta V}}_{1}+{\mathrm{\Delta V}}_{2}+\dots$
• $\frac{1}{{C}_{\mathrm{equivalent}}}=\frac{1}{{C}_{1}}+\frac{1}{{C}_{2}}+\dots$
• $U=\frac{{Q}^{2}}{2C}=\frac{1}{2}Q\mathrm{\Delta V}=\frac{1}{2}C{\left(\mathrm{\Delta V}\right)}^{2}$
• ${u}_{E}=\frac{1}{2}{\epsilon }_{0}{E}^{2}$
• $C=\kappa {C}_{0}$