Materials Science and Engineering/Equations/Physics

Coulomb's Law

$F=k{\frac {Q_{1}Q_{2}}{r^{2}}}$ $F={\frac {1}{4\pi \epsilon _{0}}}{\frac {Q_{1}Q_{2}}{r^{2}}}$ The Electric Field

$\mathbf {E} ={\frac {\mathbf {F} }{q}}$ $dE={\frac {1}{4\pi \epsilon _{0}}}{\frac {dQ}{r^{2}}}$ Superposition Principle

$\mathbf {E} =\mathbf {E} _{1}+\mathbf {E} _{2}+\cdots$ Electric Flux

$\Phi _{E}=\int \mathbf {E} \cdot d\mathbf {A}$ Gauss's Law

$\oint \mathbf {\cdot } d\mathbf {A} ={\frac {Q_{encl}}{\epsilon _{0}}}$ Electrical Potential

$V_{a}={\frac {U_{a}}{q}}$ Relation between Electric Potential and Electric Field

$U_{b}-U_{a}=\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l}$ $V_{ba}=V_{b}-V_{a}=-\int _{a}^{b}\mathbf {E} \cdot d\mathbf {l}$ Electrical Potential Due to Point Charges

$E={\frac {1}{4\pi \epsilon }}{\frac {Q}{r^{2}}}$ $V={\frac {a}{4\pi \epsilon _{0}}}{\frac {Q}{r}}$ Potential Due to Charge Distributions

$V={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {dq}{r}}$ E Determined from V

$C={\frac {Q}{V_{ba}}}=\epsilon _{0}{\frac {A}{d}}$ Electrical Energy Storage

$W=\int _{0}^{Q}Vdq={\frac {1}{C}}\int _{0}^{Q}qdq={\frac {Q^{2}}{C}}$ $U={\frac {1}{2}}CV^{2}={\frac {1}{2}}\left({\frac {\epsilon _{0}A}{d}}\right)(E^{2}d^{2})$ $U={\frac {1}{2}}\epsilon _{0}E^{2}Ad$ $u={\mbox{energy density}}={\frac {1}{2}}\epsilon _{0}E^{2}$ Dielectrics

$C=KC_{0}\;$ $C=K\epsilon _{0}{\frac {A}{d}}$ $\epsilon =K\epsilon _{0}\;$ $C=\epsilon {\frac {A}{d}}$ $u={\frac {1}{2}}K\epsilon _{0}E^{2}={\frac {1}{2}}\epsilon E^{2}$ $Q=KQ_{0}\;$ $V={\frac {V_{0}}{K}}$ $E_{0}={\frac {V_{0}}{d}}$ $E=E_{D}={\frac {V}{d}}={\frac {V_{0}}{Kd}}$ $E_{D}={\frac {E_{0}}{K}}$ $E_{D}=E_{0}-E_{ind}\;$ $E_{D}={\frac {E_{0}}{K}}$ $E_{ind}=E_{0}\left(1-{\frac {1}{K}}\right)$ $E_{0}={\frac {\sigma }{\epsilon _{0}}}$ $\sigma ={\frac {Q}{A}}$ $E_{ind}={\frac {\sigma _{ind}}{\epsilon _{0}}}$ $E_{ind}=E_{0}\left(1-{\frac {1}{K}}\right)$ $\sigma _{ind}=\sigma \left(1-{\frac {1}{K}}\right)$ $Q_{ind}=Q\left(1-{\frac {1}{K}}\right)$ Electric Current

$I={\frac {dQ}{dt}}$ Ohm's Law

$R={\frac {V}{I}}$ Resistivity

$R=\rho {\frac {l}{A}}$ $\sigma ={\frac {1}{\rho }}$ Electric Power

$P={\frac {dU}{dt}}={\frac {dq}{dt}}V$ $P=IV\;$ Current Density and Drift Velocity

$I=\int \mathbf {j} \cdot d\mathbf {A}$ $\Delta Q=({\mbox{no. of charges}},N)\times ({\mbox{charge per particle}})$ $\Delta Q=(nV)(-e)\;$ $\Delta Q=-(nAv_{d}\Delta t)(e)\;$ $I={\frac {\Delta Q}{\Delta t}}=-neAv_{d}$ $\mathbf {j} =-ne\mathbf {v} _{d}$ $\mathbf {j} =\sum _{i}n_{i}q_{i}\mathbf {v} _{di}$ $I=\sum _{i}n_{i}q_{i}v_{di}A$ $\mathbf {j} =\sigma \mathbf {E} ={\frac {1}{\rho }}\mathbf {E}$ Force on Electric Current in a Magnetic Field

$\mathbf {F} =I\mathbf {l} \times \mathbf {B}$ $d\mathbf {F} =Id\mathbf {l} \times \mathbf {B}$ Force on Moving Charge in Magnetic Field

$\mathbf {F} =q\mathbf {v} \times \mathbf {B}$ Hall Effect

Electric field due to the separation of charge is the Hall field, $\mathbf {E} _{H}$ In equilibrium, the force from the electric field is balanced by the magnetic force $ev_{d}B$ $eE_{H}=ev_{d}B\;$ Magnetic Field Due to Straight Wire

$B={\frac {\mu _{0}}{2\pi }}{\frac {I}{r}}$ Force between Two Parallel Wires

$B_{1}={\frac {\mu _{0}}{2\pi }}{\frac {I_{1}}{d}}$ Ampere's Law

$\oint \mathbf {B} \cdot d\mathbf {l} =\mu _{0}I_{encl}$ Biot-Savart Law

$d\mathbf {B} ={\frac {\mu _{0}I}{4\pi }}{\frac {d\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$ Magnetic Fields in Magnetic Materials

$B=\mu nI\;$ Paramagnetism

Relative permeability:

$K_{m}={\frac {\mu }{\mu _{0}}}$ Magnetic susceptibility:

$\mathrm {X} _{m}=K_{m}-1\;$ Magnetization vector, M:

$\mathbf {M} ={\frac {\mathbf {\mu } }{V}}$ Curie's law:

$M=C{\frac {B}{T}}$ $E=-{\frac {d\Phi _{B}}{dt}}$ $\oint \mathbf {E} \cdot d\mathbf {l} =-{\frac {d\Phi _{B}}{dt}}$ Ampere's Law

$\oint \mathbf {B} \cdot d\mathbf {l} =\mu _{0}I_{encl}+\mu _{0}\epsilon _{0}{\frac {d\Phi _{E}}{dt}}$ Gauss's Law of Magnetism

$\Phi _{B}=\mathbf {B} \cdot d\mathbf {A}$ $\Phi _{B}=\oint \mathbf {B} \cdot d\mathbf {A}$ $\oint \mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\epsilon _{0}}}$ Maxwell's Equations

$\oint \mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\epsilon _{0}}}$ $\oint \mathbf {B} \cdot d\mathbf {A} =0$ $\oint \mathbf {E} \cdot d\mathbf {l} =-{\frac {d\Phi _{B}}{dt}}$ $\oint \mathbf {B} \cdot d\mathbf {I} =\mu _{0}I+\mu _{0}\epsilon _{0}{\frac {d\Phi _{E}}{dt}}$ Relation between Wavelength and Frequency

$\lambda ={\frac {c}{f}}$ Relation between Frequency and ω

$\omega =2\pi f\;$ Poynting Vector

$\mathbf {S} ={\frac {1}{\mu _{0}}}\left(\mathbf {E} \times \mathbf {B} \right)$ Index of Refraction

$n={\frac {c}{\nu }}$ Reflection: Snell's Law

$n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}\;$ Rayleigh Criterion

$\theta ={\frac {1.22\lambda }{D}}$ Empirical Formula Proposed by Max Planck

$I(\lambda ,T)={\frac {2\pi hc^{2}\lambda ^{-5}}{e^{hc/\lambda kT}-1}}$ Energy Emitted in Packets or Quanta

$E=hf\;$ Wave Nature of Matter

$\lambda ={\frac {h}{p}}$ Heisenberg Uncertainty Principle

$(\Delta x)(\Delta p_{x})\geq {\frac {h}{2\pi }}$ One-Dimensional Time-Independent Schrödinger Equation

$-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}\Psi (x)}{dx^{2}}}+U(x)\Psi (x)=E\Psi (x)$ Time-Dependent Schrödinger Equation

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi (x,t)}{\partial x^{2}}}+U(x)\Psi (x,t)=i\hbar {\frac {\partial \Psi (x,t)}{\partial t}}$ $\Psi (x,t)=\psi (x)e^{-i\left({\frac {E}{\hbar }}\right)t}$ Solution to Schrödinger Equation - Free Particle

$\psi =A\sin kx+B\cos kx\;$ $k={\sqrt {\frac {2mE}{\hbar ^{2}}}}$ Solution to Schrödinger Equation - Infinitely Deep Square Well

$\Psi _{n}={\sqrt {\frac {2}{L}}}\sin \left({\frac {n\pi }{L}}x\right)$ 