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:

 $M=C{\frac {B}{T}}$

Faraday's Law of Induction

 $E=-{\frac {d\Phi _{B}}{dt}}$
 $\oint \mathbf {E} \cdot d\mathbf {l} =-{\frac {d\Phi _{B}}{dt}}$

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

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