# Fundamental Physics/Electromagnetism/Electromagnetism Formulaes

## Electric charge

 Electric charge Charge process Charge quantity Electric field Magnetic field Negative charge O + e -> - - Q -->E<-- B ↓ Positive charge O - e -> + + Q <--E--> B ↑

## Electric charge interaction force

 Electrostatic Force ${\displaystyle F_{q}=K{\frac {q_{+}q_{-}}{d^{2}}}}$ Electromotive Force ${\displaystyle F_{E}=qE}$ Electromagnetomotive Force ${\displaystyle F_{B}=\pm qvB}$ Electromagnetic Force ${\displaystyle F_{EB}=F_{E}+F_{B}=qE\pm qvB=q(E\pm B)}$

## Electromagnet

 Configuration Magnetic Field Magnetic field intensity For a straight wire circular magnetic field surrounds a point charge along the straight line ${\displaystyle B=LI={\frac {\mu }{2\pi r}}I}$ For a circular loop circular magnetic field surrounds a point charge along the circular loop ${\displaystyle B=LI={\frac {\mu }{2r}}I}$ For a coil of N circular loops lines of magnetic field runs from North pole (Positive polarity) to South pole (Negative polarity) ${\displaystyle B=LI={\frac {N\mu }{l}}I}$

## Electromagnetic induction

Electromagnetic induction takes place in a circular loop and a coil of N circular loops

${\displaystyle V={\frac {d}{dt}}B}$
${\displaystyle \epsilon =-{\frac {d}{dt}}\phi }$
 For a single circular loop ${\displaystyle B=LI={\frac {\mu }{2r}}I}$${\displaystyle V={\frac {dB}{dt}}=L{\frac {dI}{dt}}}$ Coil of N circular loops ${\displaystyle B=LI={\frac {N\mu }{l}}I}$${\displaystyle V={\frac {dB}{dt}}=L{\frac {dI}{dt}}}$${\displaystyle \phi =-NB=-NLI}$${\displaystyle \epsilon =-{\frac {d\phi }{dt}}=-N{\frac {dB}{dt}}=-NL{\frac {dI}{dt}}}$

## Electromagnetization

The way a coil of N circular loops turn metal inside the loops into a electromagnet describe by 4 vector equation below

${\displaystyle B=LI={\frac {N\mu }{l}}I}$
${\displaystyle H={\frac {B}{\mu }}={\frac {NI}{l}}}$

${\displaystyle \nabla \cdot D=\rho }$
${\displaystyle \nabla \times E=-\nabla B}$
${\displaystyle \nabla \cdot B=0}$
${\displaystyle \nabla \times H=J+\nabla B}$

## Electromagnetic oscillation wave

The way a coil of N circular loops generates oscillation of 2 fields , Electric filed and Electromagnetic field , perpendicular to each other which can be expressed mathematically by 4 vector equations below

### Electromagnetic oscillation

${\displaystyle \nabla \cdot E=0}$
${\displaystyle \nabla \times E={\frac {1}{T}}E}$
${\displaystyle \nabla \cdot B=0}$
${\displaystyle \nabla \times B={\frac {1}{T}}B}$

### Electromagnetic wave equation

Oscillation of Electromagnetic field and Electric field generates sinusoidal wave equations

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

### Electromagnetic wave function

Solving equations above gives Electromagnetic sinusoidal wave function

${\displaystyle E=ASin\omega t}$
${\displaystyle B=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f}$
${\displaystyle T=\mu \epsilon }$

${\displaystyle v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f}$
${\displaystyle E=pv=pC=p\lambda f=hf}$
${\displaystyle h=p\lambda }$
${\displaystyle p={\frac {h}{\lambda }}}$
${\displaystyle \lambda ={\frac {h}{p}}={\frac {C}{f}}}$

The following mathematical formulas can be derived

${\displaystyle E=hf=h{\frac {\omega }{2\pi }}=\hbar \omega }$
${\displaystyle p={\frac {h}{\lambda }}=h{\frac {k}{2\pi }}=\hbar k}$
${\displaystyle h=p\lambda ={\frac {E}{C}}\lambda =2\pi \hbar }$
${\displaystyle \omega ={\frac {E}{\hbar }}}$
${\displaystyle k={\frac {p}{\hbar }}}$
${\displaystyle \hbar ={\frac {h}{2\pi }}={\frac {E}{\omega }}={\frac {p}{k}}}$

${\displaystyle v=\omega _{o}={\sqrt {\frac {1}{\mu _{o}\epsilon _{o}}}}=C=\lambda _{o}f}$
${\displaystyle E=pv=pC=p\lambda _{o}f=hf_{o}}$
${\displaystyle h=p\lambda _{o}}$
${\displaystyle p={\frac {h}{\lambda _{o}}}}$
${\displaystyle \lambda _{o}={\frac {h}{p}}={\frac {C}{f_{o}}}}$

${\displaystyle v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f}$
${\displaystyle E=pv=pC=p\lambda f=hf}$
${\displaystyle h=p\lambda }$
${\displaystyle p={\frac {h}{\lambda }}}$
${\displaystyle \lambda ={\frac {h}{p}}={\frac {C}{f}}}$

Photon (Electromagnetic wave radiation) exists in 2 states at specific frequency . .

#### Heissenberg's uncertainty principle

States that

Photon cannot exist in 2 states at the same time


The chance to find one of its state (successful rate of finding photon) is 1/2 where h = p λ . h and p do not change, only wavelength changes with frequency

Mathematically, Uncertainty principle can be expressed as

${\displaystyle \Delta p\Delta \lambda ={\frac {1}{2}}{\frac {h}{2\pi }}={\frac {h}{4\pi }}={\frac {\hbar }{2}}}$