Capacitance

• A capacitor (originally known as the condenser) is a device that can store electric charge, and normally consists of two condaucting objects (usually plates or sheets) placed near each other but not touching.

  • We say that a charge $Q$ (assume $Q$ is positive) is stored in the capacitor to mean that the conductor at higher potential has a charge $+Q$ and the conductor at lower potential has a charge $-Q$. (The net charge of the capacitor is zero)

• $C=\frac{Q}{V}$ is the capacitance (in $\mathrm{F}$, farads. defined as $1\mathrm{F}=1C/V$) of a capacitor storing a charge $Q$ when a voltage $V$ is applied across its plates.

• The total (or equivalent) capacitance of capacitors connected:

  • (series) $\frac{1}{C_{\text{total}}}=\frac{1}{C_1}+\frac{1}{C_2}+\dots +\frac{1}{C_n}$
  • (parallel) $C_{\text{total}}=C_1+C_2+\dots +C_n$

• $U=\frac{1}{2}C{V}^2$ is the electrostatic potential energy stored in a capacitor with capacitance $C$ charged to a voltage $V$

• $W=\frac{1}{C}{\int }_0^{Q}qdq=\frac{1}{2}\frac{Q^2}{C}$ is the work done to charge a capacitor to a charge $Q$

• $u=\frac{1}{2}ϵE^2$ is the electric energy density (in $J/{\mathrm{m}}^3$) stored in a region of space with electric field $E$

• (parallel-plate capacitor)

  • $C=ϵ\frac{A}{d}$
    • $A$ is the area of the plates (in ${\mathrm{m}}^2$)
    • $d$ is the separation between the plates (in $\mathrm{m}$)

• $ϵ$ is the permittivity (or absolute permittivity) of a dielectric material

  • $\mathbf{D}=ϵ\mathbf{E}$
  • ${ϵ}_0=8.85\times 10^{-12}{\mathsf{C}}^2/N\cdot {\mathsf{m}}^2$ is the permittivity of free space

• The relative permittivity (or dielectric constant) of a dielectric material is defined as ${ϵ}_r=\frac{ϵ}{{ϵ}_0}$ (dimensionless)

  • ${ϵ}_r=\frac{C}{C_0}=\frac{E_0}{E}=\frac{V_0}{V}$
  • (sometimes denoted by $K$ or $\kappa$)
  • $C_0$, $E_0$, and $V_0$ are the capacitance, electric field, and voltage when the space between the plates of a capacitor is vacuum. $C$, $E$, and $V$ are the capacitance, electric field, and voltage when the space between the plates of a capacitor is filled with a dielectric material