Fundamental Physics/Electromagnetism/Electromagnetism Formulas

Electromagnetic Force

1. Electrostatic force . $F={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}\,r^{2}}}=K{\frac {q_{1}q_{2}}{r^{2}}}$
2. Electromotive force . $F_{E}=qE$
3. Electromagnetomotive force . $F_{B}=\pm qvB$
4. Electromagnetic force . $F=F_{E}+F_{B}=qE\pm qvB=q(E\pm vB)$

Electromagnetic Fields

Electric Field . ${\vec {E}}={\frac {\vec {F}}{q}}=K{\frac {q}{r}}={\frac {Kq}{r}}$

Electric Field Flux . $D=\epsilon {\vec {E}}=\epsilon {\frac {\vec {F}}{q}}=\epsilon K{\frac {q}{r}}=\epsilon {\frac {Kq}{r}}$

Magnetic Field . ${\vec {B}}={\frac {\vec {F}}{\pm vq}}=LI=K{\frac {qq}{\pm vqr^{2}}}=K{\frac {q}{vr^{2}}}={\frac {Kq}{v}}{\frac {1}{r^{2}}}$

Electric Field Flux . ${\vec {H}}={\frac {\vec {B}}{\mu }}=K{\frac {q}{\mu vr^{2}}}={\frac {Kq}{\mu v}}{\frac {1}{r^{2}}}$

Electromanet Magnetization

$B=LI$

$H={\frac {B}{\mu }}={\frac {LI}{\mu }}$

Permanent Electromagnet Magnetization vector equation

$\nabla \cdot D=\rho$

$\nabla \times E=\nabla B$

$\nabla \cdot B=0$

$\nabla \times H=J+\nabla B$

Electromagnetic Oscillation

Electromagnetic Oscillation

$\nabla \cdot E=0$

$\nabla \times E={\frac {1}{T}}E$

$\nabla \cdot B=0$

$\nabla \times B={\frac {1}{T}}B$

Electromagnetic Wave Equation

$\nabla ^{2}E=-\omega E$

$\nabla ^{2}B=-\omega B$

Electromagneti Wave Function

$E=ASin\omega t$

$B=ASin\omega t$

$\omega ={\sqrt {\frac {1}{T}}}=C=\lambda f$

$T=\mu \epsilon$

Electromagnetic Wave Radiation

$v=\omega {\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f$

$E=pv=pC=p\lambda f=hf$

$h=p\lambda$

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

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

Quantum Physics

Quanta

$h=p\lambda$

Particle-Wave Duality

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

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

Photon

$E_{h}=hf=h{\frac {\omega }{2\pi }}=\hbar \omega$

Photon's states

Radiant Photon . $E_{fo}=hf_{o}=h{\frac {\omega _{o}}{2\pi }}=\hbar \omega _{o}$

Electric Photon . $E_{f}=hf=h{\frac {\omega }{2\pi }}=\hbar \omega$

Electromagnetism Therems

Gauss' Law

Gauss's Law in integral form states that the electric flux, $\Phi$, through any closed surface is proportional to the amount of electric charge circumscribed by that surface. 

$\Phi =\oint _{S}\mathbf {E} \cdot d\mathbf {S} ={\frac {q_{in}}{\varepsilon _{0}}}$
$\Phi =\oint _{S}\mathbf {D} \cdot d\mathbf {S} =q_{in}$

$\Phi =EA={\frac {q_{in}}{\varepsilon _{0}}}$

$\Phi =DA=q_{in}$

Coulomb's Law

The magnitude of the electrostatic force between two point charges in vacuum is directly proportional to the magnitudes of each charge and inversely proportional to the square of the distance between the charges.

$F={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}\,r^{2}}}=K{\frac {q_{1}q_{2}}{r^{2}}}$

Ampere's Law

"Ampere's circuital law" (named after André-Marie Ampère, not directly named after the unit of current), gives the magnetic field in the vicinity of an infinitely long straight wire carrying an electric current. The magnetic field goes in circles around the wire, following a right-hand rule and calculated by

$B=LI$

For straight line conductor

$L={\frac {\mu \ }{2\pi R}}$

For circular loop conductor

$L={\frac {\mu \ }{2R}}$

For a coil of N circular loop conductor

$L={\frac {N\mu }{l}}$

Lenz's Law

Lenz's Law (pronounced /ˈlɛnts/), named after the physicist Emil Lenz who formulated it in 1834 .

The direction of current induced in a conductor by a changing magnetic field due to Faraday's law of induction will be such that it will create a magnetic field that opposes the change that produced it.

The current that profuces magnetic field is

$B=LI$

The induced current will create a magnetic field that opposes the change that produced it.

$\phi =-B=-LI$

Faraday's Electromagnetic Induction Law

Faraday's law states:

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit

$\epsilon =-{\frac {d\phi }{dt}}=-L{\frac {dI}{dt}}$

Faraday's Law, which states that the electromotive force around a closed path is equal to the negative of the time rate of change of magnetic flux enclosed by the path

Applications

Straight line conductor

$B=LI=B{\frac {\mu }{2\pi r}}$

Circular loop

$B=LI=B{\frac {\mu }{2r}}$

$V_{L}={\frac {dB}{dt}}=L{\frac {dI}{dt}}$

$QvB=m{\frac {v^{2}}{r}}$

$r={\frac {mv^{2}}{QvB}}={\frac {mv}{QB}}$

$v={\frac {Q}{m}}Br$

Coil of N circular loops

$B=LI=B{\frac {N\mu }{l}}$

$\phi =-NB=-NLI$

$H={\frac {B}{\mu }}={\frac {\phi }{N\mu }}$

$V_{L}={\frac {dB}{dt}}=L{\frac {dI}{dt}}$

$\epsilon =-{\frac {d\phi }{dt}}=-NL{\frac {dI}{dt}}$

$F=Bl$