Materials Science and Engineering/Equations/Physics

Coulomb's Law

 $F=k\frac{Q_1{Q}_2}{r^2}$
 $F=\frac{1}{4\pi {ϵ}_0}\frac{Q_1{Q}_2}{r^2}$

The Electric Field

 $𝐄=\frac{𝐅}{q}$

 $dE=\frac{1}{4\pi {ϵ}_0}\frac{dQ}{r^2}$

Superposition Principle

 $𝐄={𝐄}_1+{𝐄}_2+\cdots$

Electric Flux

 ${\mathrm{\Phi }}_E=\int 𝐄\cdot d𝐀$

Gauss's Law

 $\oint \mathbf{\cdot }d𝐀=\frac{Q_{encl}}{{ϵ}_0}$

Electrical Potential

 $V_a=\frac{U_a}{q}$
 

Relation between Electric Potential and Electric Field

 $U_b-U_a=\underset{a}{\overset{b}{\int }}𝐅\cdot d𝐥$

 $V_{ba}=V_b-V_a=-\underset{a}{\overset{b}{\int }}𝐄\cdot d𝐥$

Electrical Potential Due to Point Charges

 $E=\frac{1}{4\pi ϵ}\frac{Q}{r^2}$

 $V=\frac{a}{4\pi {ϵ}_0}\frac{Q}{r}$

Potential Due to Charge Distributions

 $V=\frac{1}{4\pi {ϵ}_0}\int \frac{dq}{r}$

E Determined from V

 $C=\frac{Q}{V_{ba}}={ϵ}_0\frac{A}{d}$

Electrical Energy Storage

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

 $U=\frac{1}{2}C{V}^2=\frac{1}{2}\left(\frac{{ϵ}_{0}A}{d}\right)(E^2{d}^2)$
 $U=\frac{1}{2}{ϵ}_0{E}^{2}Ad$
 $u=\text{energy density}=\frac{1}{2}{ϵ}_0{E}^2$

Dielectrics

 $C=K{C}_0$
 $C=K{ϵ}_0\frac{A}{d}$
 $ϵ=K{ϵ}_0$
 $C=ϵ\frac{A}{d}$

 $u=\frac{1}{2}K{ϵ}_0{E}^2=\frac{1}{2}ϵE^2$

 $Q=K{Q}_0$

 $V=\frac{V_0}{K}$

 $E_0=\frac{V_0}{d}$

 $E=E_D=\frac{V}{d}=\frac{V_0}{Kd}$

 $E_D=\frac{E_0}{K}$

 $E_D=E_0-E_{ind}$

 $E_D=\frac{E_0}{K}$

 $E_{ind}=E_0(1-\frac{1}{K})$

 $E_0=\frac{\sigma }{{ϵ}_0}$

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

 $E_{ind}=\frac{{\sigma }_{ind}}{{ϵ}_0}$

 $E_{ind}=E_0(1-\frac{1}{K})$

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

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

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

 $I=\int 𝐣\cdot d𝐀$

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

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

 $\mathrm{\Delta }Q=-(nA{v}_d\mathrm{\Delta }t)(e)$

 $I=\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}=-neA{v}_d$

 $𝐣=-ne{𝐯}_d$

 $𝐣=\sum _i{n}_i{q}_i{𝐯}_{di}$
 $I=\sum _i{n}_i{q}_i{v}_{di}A$

 $𝐣=\sigma 𝐄=\frac{1}{\rho }𝐄$

Force on Electric Current in a Magnetic Field

 $𝐅=I𝐥\times 𝐁$

 $d𝐅=Id𝐥\times 𝐁$

Force on Moving Charge in Magnetic Field

 $𝐅=q𝐯\times 𝐁$

Hall Effect

Electric field due to the separation of charge is the Hall field, ${𝐄}_H$

In equilibrium, the force from the electric field is balanced by the magnetic force $e{v}_{d}B$

 $e{E}_H=e{v}_{d}B$

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

 $\oint 𝐁\cdot d𝐥={\mu }_0{I}_{encl}$

Biot-Savart Law

 $d𝐁=\frac{{\mu }_{0}I}{4\pi }\frac{d𝐥\times \widehat{𝐫}}{r^2}$

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:

 $𝐌=\frac{\mathit{\mu }}{V}$

Curie's law:

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

Faraday's Law of Induction

 $E=-\frac{d{\mathrm{\Phi }}_B}{dt}$
 $\oint 𝐄\cdot d𝐥=-\frac{d{\mathrm{\Phi }}_B}{dt}$

Ampere's Law

 $\oint 𝐁\cdot d𝐥={\mu }_0{I}_{encl}+{\mu }_0{ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}$

Gauss's Law of Magnetism

 ${\mathrm{\Phi }}_B=𝐁\cdot d𝐀$
 ${\mathrm{\Phi }}_B=\oint 𝐁\cdot d𝐀$
 $\oint 𝐄\cdot d𝐀=\frac{Q}{{ϵ}_0}$

Maxwell's Equations

 $\oint 𝐄\cdot d𝐀=\frac{Q}{{ϵ}_0}$
 $\oint 𝐁\cdot d𝐀=0$
 $\oint 𝐄\cdot d𝐥=-\frac{d{\mathrm{\Phi }}_B}{dt}$
 $\oint 𝐁\cdot d𝐈={\mu }_{0}I+{\mu }_0{ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}$

Relation between Wavelength and Frequency

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

Relation between Frequency and ω

$\omega =2\pi f$

Poynting Vector

 $𝐒=\frac{1}{{\mu }_0}(𝐄\times 𝐁)$

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

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

Energy Emitted in Packets or Quanta

 $E=hf$

Wave Nature of Matter

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

Heisenberg Uncertainty Principle

 $(\mathrm{\Delta }x)(\mathrm{\Delta }{p}_x)\ge \frac{h}{2\pi }$

One-Dimensional Time-Independent Schrödinger Equation

 $-\frac{{\hslash }^2}{2m}\frac{d^2\mathrm{\Psi }(x)}{d{x}^2}+U(x)\mathrm{\Psi }(x)=E\mathrm{\Psi }(x)$

Time-Dependent Schrödinger Equation

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

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

Solution to Schrödinger Equation - Free Particle

 $\psi =A\sin kx+B\cos kx$

 $k=\sqrt{\frac{2mE}{{\hslash }^2}}$

Solution to Schrödinger Equation - Infinitely Deep Square Well

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