# OpenStax University Physics/E&M/Capacitance

## Chapter 8

#### Capacitance

${\displaystyle Q=CV}$ defines capacitance. For a parallel plate capacitor, ${\displaystyle C=\varepsilon _{0}{\tfrac {A}{d}}}$ where A is area and d is gap length.

▭ ${\displaystyle 4\pi \varepsilon _{0}{\tfrac {R_{1}R_{2}}{R_{2}-R_{1}}}}$ and ${\displaystyle {\tfrac {2\pi \varepsilon _{0}\ell }{\ln(R_{2}/R_{1})}}}$ for a spherical and cylindrical capacitor, respectively
▭ For capacitors in series (parallel) ${\displaystyle {\tfrac {1}{C_{S}}}=\sum {\tfrac {1}{C_{i}}}\left(C_{P}=\sum C_{i}\right)}$
▭  ${\displaystyle u={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}={\tfrac {1}{2C}}Q^{2}}$ ▭ Stored energy density ${\displaystyle u_{E}={\tfrac {1}{2}}\varepsilon _{0}E^{2}}$
▭ A dielectric with ${\displaystyle \kappa >1}$ will decrease the capacitor's electric field ${\displaystyle E={\tfrac {1}{\kappa }}E_{0}}$ and stored energy ${\displaystyle U={\tfrac {1}{\kappa }}U_{0}}$, but increase the capacitance ${\displaystyle C=\kappa C_{0}}$ due to the induced electric field ${\displaystyle {\vec {E}}_{i}=\left({\tfrac {1}{\kappa }}-1\right){\vec {E}}_{0}}$

#### For quiz at QB/d_cp2.8

${\displaystyle Q=CV}$ defines capacitance.

${\displaystyle C=\varepsilon _{0}{\tfrac {A}{d}}}$ where A is area and d<<A1/2 is gap length of parallel plate capacitor

${\displaystyle {\text{Series}}:\;{\tfrac {1}{C_{S}}}=\sum {\tfrac {1}{C_{i}}}.}$   ${\displaystyle {\text{ Parallel:}}\;C_{P}=\sum C_{i}.}$

${\displaystyle u={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}={\tfrac {1}{2C}}Q^{2}}$ = stored energy

${\displaystyle u_{E}={\tfrac {1}{2}}\varepsilon _{0}E^{2}}$ = energy density