Electrostatics

Coulomb's law
$ F=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r^{2}} $
Gauss law
$\phi \vec{E}. \vec{ds}=\frac{Q}{\epsilon_{o}}$
Surface charge density
$\sigma=\frac{Q}{Area (ds)}$
Linear charge density
$\lambda=\frac{Q}{length (l)}$
Electric field intensity due to uniformly charged spherical shell or hollow sphere.
$E= \frac{1}{4\pi\epsilon_{0}}\frac{Q}{r^{2}}$
In terms of surface charge density.
$E= \frac{\sigma R^{2}}{\epsilon r^{2} }$
If P point is Inside the circle (r<R).
$E=0$
If P point is on the surface (r=R).
$E=\frac{\sigma}{\epsilon}$
Electric field intensity due to an infinity long straight charged wire.
$E=\frac{Q}{2 \pi rl}$
In terms of Linear charge density.
$E=\frac{\lambda}{2 \pi r\epsilon}$
In terms of Surface charge density.
$E=\frac{\sigma R}{\epsilon r}$
Electric field intensity due to charged infinite Plane sheet.
$E=\frac{Q}{2A\epsilon}$
In terms of Surface charge density.
$E=\frac{\sigma}{2\epsilon}$
Potential
$V=\frac{1}{4\pi\epsilon_{0}}\left[\frac{q}{r}\right]$
Potential Energy
$U=V*q$
Expression for potential energy
$U=\frac{1}{4\pi\epsilon_{0}}\frac{q_{1}q_{2}}{r}$
Potential energy of system of 2 point charges.
$W_{2}=V_{1}*q_{2}$
$U=\frac{1}{4\pi\epsilon_{0}}\left[\frac{q_{1}q_{2}}{r_{12}}\right]$
Potential energy of system of 3 point charges.
$W_{3}=V_{2}*q_{3}$
$U=\frac{1}{4\pi\epsilon_{0}}~\left[\frac{q_{1}q_{2}}{r_{12}}~\frac{q_{1}q_{3}}{r_{13}}~\frac{q_{2}q_{3}}{r_{23}}\right]$
Potential energy of system of 4 point charges.
$W_{4}=V_{3}*q_{4}$
$U=\frac{1}{4\pi\epsilon_{0}}\left[\frac{q_{1}q_{2}}{r_{12}}\frac{q_{1}q_{3}}{r_{13}}\frac{q_{2}q_{3}}{r_{23}}\frac{q_{1}q_{4}}{r_{14}}\frac{q_{2}q_{4}}{r_{24}}\frac{q_{3}q_{4}}{r_{34}}\right]$
Potential energy of system of N point charges.
$W_{4}=V_{3}*q_{4}$
$U=\frac{1}{4\pi\epsilon_{0}}\sum_{all~pairs} \frac{q_{j}q_{k}}{r_{jk}}$
Potential energy of a single charge in an external electric field.
$W=q_{1}V+q_{2}V+\frac{Kq_{1}q_{2}}{r_{12}}$
Torque
$\tau=pE\sin\theta$
Force
$F=qE$
Work
$W=\tau\theta$
Potential energy of a dipole in a external electric field.
$W=pE\left[\cos\theta_{0}-\cos\theta\right]$
Case-2 ($\theta_{0}=\frac{\pi}{2}$)
$U=-pE\cos\theta$
Case-2 ($\theta=0$)
$U=pE(1-\cos\theta)$
Relation between electric field and electric potential.
$E=-\frac{dV}{dx}$
Electric potential due to point charge.
$V=0$
Electric potential due to an electric dipole.
$V=\frac{1}{4\pi\epsilon_{0}}\frac{p\cos\theta}{r^{2}}$
Potential at axial point, $\theta=0^\circ$
$V_{axial}=\frac{1}{4\pi\epsilon_{0}}\frac{p}{r^{2}}$
Potential at an equipotential point, $\theta=90^\circ$
$V=0$
For linear isotropic dielectric.
$P=\chi_{0}.E$
Capacitance
$C=\frac{Q}{V}$
Capacitance of a parallel plate capacitor without a dielectric.
$C=\frac{A\epsilon_{0}}{d}$
Capacitance of a parallel plate capacitor with a dielectric.
Case-1: If dielectric is completely filled. (d=t)
$C=\frac{A\epsilon_{0}K}{d}$
Case-2: If dielectric is partially fillded. (d>t)
$C=\frac{A\epsilon_{0}}{\left[d-t + \frac{t}{K}\right]}$
Case-3: Series arrangement.
$C=\frac{A\epsilon_{0}}{\left[ \frac{t_{1}}{k_{1}} + \frac{t_{2}}{k_{2}} + ... + \frac{t_{n}}{k_{n}} \right]}$
Case-4: Parallel arrangement.
$C=\frac{\epsilon_{0}}{d} \left(A_{1}K_{1}+A_{2}K_{2}+...+A_{n}K_{n} \right)$
Spherical capacitors
$C=4\pi\epsilon_{0} \left(\frac{kab}{b-a} \right)$
Cylindrical capacitors
$C=\frac{2\pi K\epsilon_{0}L}{ln \left( \frac{b}{a} \right)}$
Capacitors in series
$\frac{1}{C_{s}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}$
Case-1: For series combination of n capacitors.
$\frac{1}{C_{eq}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+...+\frac{1}{C_{n}}$
Case-2: If all capacitors have equal capacitance.
$\frac{1}{C_{eq}}=\frac{n}{C}$
Capacitors in Parallel
$C_{p}=C_{1}+C_{2}+C_{3}$
Case-1: For parallel combination of 'n' capacitors.
$C_{p}=C_{1}+C_{2}+C_{3}+...+C_{n}$
Case-2: If all capacitors have same capacitance.
$C_{p}=nC$
Energy stored in capacitor.
$U=\frac{1}{2} \frac{Q^{2}}{C}$
$U=\frac{1}{2}CV^{2}$
$U=\frac{1}{2}QV$
Energy density/Electrostatic pressure
$U=\frac{1}{2}\epsilon_{0}E^2$
Charging of capacitor
$V_{c}=E\left( 1-e^{-t/RC} \right)$
$i_{c}=\frac{E \ e^{-t/RC}}{R}$
Discharging of capacitors
$V_{c}=V_{0}\ e^{-t/RC}$
$Q=C\ V_{0}\ e^{-t/RC}$
$I=\frac{V_{0}}{R}\ e^{-t/RC}$