Gauss' law

Charge and electric flux

Calculating e flux

\begin{equation} \Phi_E = \oint \vec{E} \cdot d\vec{A} = \oint E_{\perp} \, dA = \oint E \textnormal{ cos } \tau \, \, dA \end{equation}

Gauss' law

\begin{equation} \oint_S \vec{E} \cdot dA = \frac{Q_{enc}}{\epsilon_0} \end{equation}

Applying gauss law, cylindrical symmetry

Applying gauss law, planar symmetry

Applying gauss law, spherical symmetry

Charges on conductors

Electric field in various charge symmetries

Single point charge q

\begin{equation} E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \end{equation}

Infinite wire, charge per unit length \( \eta \)

\begin{equation} E = \frac{1}{2 \pi \epsilon_0} \frac{\eta}{r} \end{equation}

Infinite sheet of charge with charge per unit area \( \mu \)

\begin{equation} E = \frac{\mu}{2 \epsilon_0} \end{equation}

Two oppositely charged conducting plates with surface charge densities \( + \, \xi \) and \( - \, \xi \)

\begin{equation} E = \frac{\xi}{\epsilon_0} \end{equation}

Conducting shell with charge q spread uniformly on surface

Outside shell, r > R \begin{equation} E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \end{equation}

Inside shell, r < R \begin{equation} E = \, \, ... 0 \end{equation}

Conducting solid sphere with charge q spread uniformly through sphere

Outside sphere, r > R \begin{equation} E = \frac{1}{4 \pi \epsilon_0} \frac{q}{r^2} \end{equation}

Inside sphere, r < R \begin{equation} E = \frac{1}{4 \pi \epsilon_0} \frac{q \, r}{R^3} \end{equation}

Infinite conducting cylinder with radius R, charge per unit length \( \kappa \)

Outside cylinder, r > R \begin{equation} E = \frac{1}{2 \pi \epsilon_0} \frac{\kappa}{r} \end{equation}

Inside cylinder, r < R \begin{equation} E = \, \, ... 0 \end{equation}