Electrostatics

Coulomb's law and electric field

$F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}},$ $E = \frac{F}{q _{0}},$ $E = \frac{k q}{r ^{2}} (k = \frac{1}{4 π ε _{0}})$
Line of charge
$E = \frac{λ}{2 π ε _{0} r}$ (infinite)
Ring (on axis)
$E = \frac{k Q x}{( R ^{2} + x ^{2} ) ^{3/2}},$ $E_{m a x}$ at $x = \frac{R}{2}$
Sheet and plates
$E = \frac{σ}{2 ε _{0}}$ (sheet) $,$ $E = \frac{σ}{ε _{0}}$ (between oppositely charged plates)
Field of a sphere
conducting/shell: $0 (r < R), \frac{k Q}{r ^{2}} (r \geq R);$ solid (uniform): $\frac{k Q r}{R ^{3}} (r < R)$

$V = \frac{k q}{r},$ $E = - \nabla V,$ $E = - \frac{d V}{d r}$
Potential energy
$U = \frac{k q _{1} q _{2}}{r},$ $W = q Δ V$
Sphere
$V = \frac{k Q}{R}$ (inside a conductor) $;$ solid sphere centre $\frac{3 k Q}{2 R}$

$p = q d;$ $E_{axial} = \frac{2 k p}{r ^{3}},$ $E_{equatorial} = \frac{k p}{r ^{3}}$
$V = \frac{k p cos θ}{r ^{2}},$ $τ = p \times E,$ $U = - p \cdot E$

$E_{inside} = 0;$ charge sits on the surface $;$ $E_{surface} = \frac{σ}{ε _{0}}$
Connected spheres
$\frac{q _{1}}{q _{2}} = \frac{r _{1}}{r _{2}},$ $\frac{σ _{1}}{σ _{2}} = \frac{r _{2}}{r _{1}}$