# Materials Science and Engineering/Equations/Physics

## Coulomb's Law

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


## The Electric Field

 ${\displaystyle \mathbf {E} ={\frac {\mathbf {F} }{q}}}$

 ${\displaystyle dE={\frac {1}{4\pi \epsilon _{0}}}{\frac {dQ}{r^{2}}}}$


### Superposition Principle

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


## Electric Flux

 ${\displaystyle \Phi _{E}=\int \mathbf {E} \cdot d\mathbf {A} }$


## Gauss's Law

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


## Electrical Potential

 ${\displaystyle V_{a}={\frac {U_{a}}{q}}}$



## Relation between Electric Potential and Electric Field

 ${\displaystyle U_{b}-U_{a}=\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} }$

 ${\displaystyle V_{ba}=V_{b}-V_{a}=-\int _{a}^{b}\mathbf {E} \cdot d\mathbf {l} }$


## Electrical Potential Due to Point Charges

 ${\displaystyle E={\frac {1}{4\pi \epsilon }}{\frac {Q}{r^{2}}}}$

 ${\displaystyle V={\frac {a}{4\pi \epsilon _{0}}}{\frac {Q}{r}}}$


## Potential Due to Charge Distributions

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


## E Determined from V

 ${\displaystyle C={\frac {Q}{V_{ba}}}=\epsilon _{0}{\frac {A}{d}}}$


## Electrical Energy Storage

 ${\displaystyle W=\int _{0}^{Q}Vdq={\frac {1}{C}}\int _{0}^{Q}qdq={\frac {Q^{2}}{C}}}$

 ${\displaystyle U={\frac {1}{2}}CV^{2}={\frac {1}{2}}\left({\frac {\epsilon _{0}A}{d}}\right)(E^{2}d^{2})}$
${\displaystyle U={\frac {1}{2}}\epsilon _{0}E^{2}Ad}$
${\displaystyle u={\mbox{energy density}}={\frac {1}{2}}\epsilon _{0}E^{2}}$


## Dielectrics

 ${\displaystyle C=KC_{0}\;}$
${\displaystyle C=K\epsilon _{0}{\frac {A}{d}}}$
${\displaystyle \epsilon =K\epsilon _{0}\;}$
${\displaystyle C=\epsilon {\frac {A}{d}}}$

 ${\displaystyle u={\frac {1}{2}}K\epsilon _{0}E^{2}={\frac {1}{2}}\epsilon E^{2}}$

 ${\displaystyle Q=KQ_{0}\;}$

 ${\displaystyle V={\frac {V_{0}}{K}}}$

 ${\displaystyle E_{0}={\frac {V_{0}}{d}}}$

 ${\displaystyle E=E_{D}={\frac {V}{d}}={\frac {V_{0}}{Kd}}}$

 ${\displaystyle E_{D}={\frac {E_{0}}{K}}}$

 ${\displaystyle E_{D}=E_{0}-E_{ind}\;}$

${\displaystyle E_{D}={\frac {E_{0}}{K}}}$

 ${\displaystyle E_{ind}=E_{0}\left(1-{\frac {1}{K}}\right)}$

 ${\displaystyle E_{0}={\frac {\sigma }{\epsilon _{0}}}}$

 ${\displaystyle \sigma ={\frac {Q}{A}}}$

 ${\displaystyle E_{ind}={\frac {\sigma _{ind}}{\epsilon _{0}}}}$

 ${\displaystyle E_{ind}=E_{0}\left(1-{\frac {1}{K}}\right)}$

 ${\displaystyle \sigma _{ind}=\sigma \left(1-{\frac {1}{K}}\right)}$

 ${\displaystyle Q_{ind}=Q\left(1-{\frac {1}{K}}\right)}$


## Electric Current

 ${\displaystyle I={\frac {dQ}{dt}}}$


## Ohm's Law

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


## Resistivity

 ${\displaystyle R=\rho {\frac {l}{A}}}$

 ${\displaystyle \sigma ={\frac {1}{\rho }}}$


## Electric Power

 ${\displaystyle P={\frac {dU}{dt}}={\frac {dq}{dt}}V}$

 ${\displaystyle P=IV\;}$


## Current Density and Drift Velocity

 ${\displaystyle I=\int \mathbf {j} \cdot d\mathbf {A} }$

 ${\displaystyle \Delta Q=({\mbox{no. of charges}},N)\times ({\mbox{charge per particle}})}$

 ${\displaystyle \Delta Q=(nV)(-e)\;}$

 ${\displaystyle \Delta Q=-(nAv_{d}\Delta t)(e)\;}$

 ${\displaystyle I={\frac {\Delta Q}{\Delta t}}=-neAv_{d}}$

 ${\displaystyle \mathbf {j} =-ne\mathbf {v} _{d}}$

 ${\displaystyle \mathbf {j} =\sum _{i}n_{i}q_{i}\mathbf {v} _{di}}$
${\displaystyle I=\sum _{i}n_{i}q_{i}v_{di}A}$

 ${\displaystyle \mathbf {j} =\sigma \mathbf {E} ={\frac {1}{\rho }}\mathbf {E} }$


## Force on Electric Current in a Magnetic Field

 ${\displaystyle \mathbf {F} =I\mathbf {l} \times \mathbf {B} }$

 ${\displaystyle d\mathbf {F} =Id\mathbf {l} \times \mathbf {B} }$


## Force on Moving Charge in Magnetic Field

 ${\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }$


## Hall Effect

Electric field due to the separation of charge is the Hall field, ${\displaystyle \mathbf {E} _{H}}$

In equilibrium, the force from the electric field is balanced by the magnetic force ${\displaystyle ev_{d}B}$

 ${\displaystyle eE_{H}=ev_{d}B\;}$


## Magnetic Field Due to Straight Wire

 ${\displaystyle B={\frac {\mu _{0}}{2\pi }}{\frac {I}{r}}}$


## Force between Two Parallel Wires

 ${\displaystyle B_{1}={\frac {\mu _{0}}{2\pi }}{\frac {I_{1}}{d}}}$


## Ampere's Law

 ${\displaystyle \oint \mathbf {B} \cdot d\mathbf {l} =\mu _{0}I_{encl}}$


## Biot-Savart Law

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


## Magnetic Fields in Magnetic Materials

 ${\displaystyle B=\mu nI\;}$


## Paramagnetism

Relative permeability:

 ${\displaystyle K_{m}={\frac {\mu }{\mu _{0}}}}$


Magnetic susceptibility:

 ${\displaystyle \mathrm {X} _{m}=K_{m}-1\;}$


Magnetization vector, M:

 ${\displaystyle \mathbf {M} ={\frac {\mathbf {\mu } }{V}}}$


Curie's law:

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


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


## Ampere's Law

 ${\displaystyle \oint \mathbf {B} \cdot d\mathbf {l} =\mu _{0}I_{encl}+\mu _{0}\epsilon _{0}{\frac {d\Phi _{E}}{dt}}}$


## Gauss's Law of Magnetism

 ${\displaystyle \Phi _{B}=\mathbf {B} \cdot d\mathbf {A} }$
${\displaystyle \Phi _{B}=\oint \mathbf {B} \cdot d\mathbf {A} }$
${\displaystyle \oint \mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\epsilon _{0}}}}$


## Maxwell's Equations

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


## Relation between Wavelength and Frequency

 ${\displaystyle \lambda ={\frac {c}{f}}}$


## Relation between Frequency and ω

${\displaystyle \omega =2\pi f\;}$


## Poynting Vector

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


## Index of Refraction

 ${\displaystyle n={\frac {c}{\nu }}}$


## Reflection: Snell's Law

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


## Rayleigh Criterion

 ${\displaystyle \theta ={\frac {1.22\lambda }{D}}}$


## Empirical Formula Proposed by Max Planck

 ${\displaystyle I(\lambda ,T)={\frac {2\pi hc^{2}\lambda ^{-5}}{e^{hc/\lambda kT}-1}}}$


## Energy Emitted in Packets or Quanta

 ${\displaystyle E=hf\;}$


## Wave Nature of Matter

 ${\displaystyle \lambda ={\frac {h}{p}}}$


## Heisenberg Uncertainty Principle

 ${\displaystyle (\Delta x)(\Delta p_{x})\geq {\frac {h}{2\pi }}}$


## One-Dimensional Time-Independent Schrödinger Equation

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


## Time-Dependent Schrödinger Equation

 ${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\Psi (x,t)}{\partial x^{2}}}+U(x)\Psi (x,t)=i\hbar {\frac {\partial \Psi (x,t)}{\partial t}}}$

 ${\displaystyle \Psi (x,t)=\psi (x)e^{-i\left({\frac {E}{\hbar }}\right)t}}$


## Solution to Schrödinger Equation - Free Particle

 ${\displaystyle \psi =A\sin kx+B\cos kx\;}$

 ${\displaystyle k={\sqrt {\frac {2mE}{\hbar ^{2}}}}}$


## Solution to Schrödinger Equation - Infinitely Deep Square Well

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