Capacitance

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
  • u_E - energy density
  • κ - dielectric constant, in volts per meter
• ${\epsilon }_0=8.8542\times {10}^{-12}\frac{C^2}{N{m}^2}$
• $C\equiv \frac{Q}{\mathrm{\Delta V}}$

• Sphere

  • $C=4\pi {\epsilon }_{0}R$

• Cylindrical Capacitor (coaxial cable)

  • $C=\frac{2\pi {\epsilon }_0𝓁}{\ln \left(\frac{\mathrm{outer\; radius}}{\mathrm{inner\; radius}}\right)}$

• 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

  • $Q_1=Q_2=Q_3=\dots$
  • ${\mathrm{\Delta V}}_{\mathrm{equivalent}}={\mathrm{\Delta V}}_1+{\mathrm{\Delta V}}_2+\dots$
  • $\frac{1}{C_{\mathrm{equivalent}}}=\frac{1}{C_1}+\frac{1}{C_2}+\dots$
• $U=\frac{Q^2}{2C}=\frac{1}{2}Q\mathrm{\Delta V}=\frac{1}{2}C{\left(\mathrm{\Delta V}\right)}^2$
• $u_E=\frac{1}{2}{\epsilon }_0{E}^2$
• $C=\kappa C_0$