Physics/Synopsis

Force

Force	Symbol	Mathematical Formula
Opposing Force	$F_{-}$	$-F$
Gravity Force	$F_{g}$	$G{\frac {Mm}{r^{2}}}$
Motion Force	$F_{a}$	$ma=m{\frac {v}{t}}={\frac {p}{t}}$
Pressure Force	$F_{A}$	${\frac {F}{A}}$
Elastic Force	$F_{x}$	$-kx$
Circulation Force	$F_{r}$	$mvr=pr$
Centripetal Force	$F_{c}$	$m{\frac {v^{2}}{r}}$
Electrostatic Force	$F_{q}$	${\frac {q_{+}q_{-}}{r^{2}}}$
Electromotive force	$F_{E}$	$qE$
Electromagnetomotive Force	$F_{B}$	$\pm qvB$
Electromagnetic Force	$F_{EB}$	$q(E\pm vB)$

Motion

Linear Motion

O ----> O

Distance	$s$	$vt$
Time	$t$	$t$
Speed	$v$	${\frac {s}{t}}$
Accelleration	$a$	${\frac {v}{t}}$
Force	$F$	$m{\frac {v}{t}}$
Work	$W$	$Fs$
Energy	$E$	${\frac {W}{t}}$

O

↓

O

Distance	$s$	$h$
Time	$t$	$t$
Speed	$v$	${\frac {h}{t}}$
Accelleration	$a$	${\frac {h}{t^{2}}}$
Force	$F$	$mg$
Work	$W$	$mgh$
Energy	$E$	${\frac {mgh}{t}}$

• Non-Uniform linear motion

Distance	$s$	$(v_{o}+at)t$
Time	$t$	$t$
Speed	$v$	$v_{o}+at$
Accelleration	$a$	${\frac {\Delta v}{\Delta t}}$
Force	$F$	$m{\frac {\Delta v}{\Delta t}}$
Work	$W$	$Ft(v_{o}+at)$
Energy	$E$	$F(v_{o}+at)$

Non Linear Motion

Distance	$s(t)$	$\int v(t)dt$
Time	$t$	$t$
Speed	$v(t)$	$v(t)$
Acceleration	$a(t)$	${\frac {d}{dt}}v(t)$
Force	$F(t)$	$m{\frac {d}{dt}}v(t)$
Work	$W(t)$	$F\int v(t)dt$
Energy	$E(t)$	${\frac {F}{t}}\int v(t)dt$

Circular Motion

• Complete full circle

Distance	$s$	$2\pi r$
Time	$t$	$t$
Speed	$v$	$r{\frac {2\pi }{t}}=r\omega$
Acceleration	$a$	${\frac {r\omega }{t}}$
Angular spped	$\omega$	${\frac {2\pi }{t}}=2\pi f={\frac {v}{r}}$
Frequency	$f$	${\frac {1}{t}}$
Force	$F$	$m{\frac {r\omega }{t}}$
Work	$W$	$pr\omega$
Energy	$E$	${\frac {pr\omega }{t}}$

• Circle's arc

Distance	$s$	$r\theta$
Time	$t$	$t$
Speed	$v$	$r{\frac {\theta }{t}}$
Acceleration	$a$	${\frac {r\theta }{t^{2}}}$
Force	$F$	$m{\frac {r\theta }{t^{2}}}$
Work	$W$	$pr{\frac {\theta }{t}}$
Energy	$E$	${\frac {pr\theta }{t^{2}}}$

Momentum

Characteristics

$F=ma=m{\frac {v}{t}}={\frac {p}{t}}$

$p=mv=Ft$

Momentum of a mass

Speed	$v$	$v$
Mass	$m$	$m$
Momentum	$p$	$mv=Ft$
Force	$F$	$ma=m{\frac {v}{t}}={\frac {p}{t}}$
Work	$W$	$Fs=Fvt=pv$
Energy	$E$	$Fv=Fat=pa$

Momentum of a relativistic mass

Speed	$v$	$\gamma ={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}$
Mass	$m$	$m_{o}(\gamma -1)$
Momentum	$p$	$mv$
Energy	$E$	$pv$

Momentum of a massless quanta

Speed	$v$	$C=\lambda f$
Mass	$m$	$h=p\lambda$
Energy	$E$	$pv=pC=p\lambda f=hf$
Momentum	$p$	${\frac {h}{\lambda }}$
Wavelength	$\lambda$	${\frac {h}{p}}={\frac {C}{f}}$

Momentum of an electric charge in circle

Equilibrium	$QvB=p{\frac {v}{r}}$
Speed	$v={\frac {Q}{m}}Br$
Radius	$r={\frac {p}{QB}}$

Momentum of atom's free electron

Equilibrium	$hf=hf_{o}+{\frac {1}{2}}mv^{2}$
Speed	$v={\sqrt {{\frac {2}{m}}(hf-hf_{o})}}={\sqrt {{\frac {2}{m}}(nhf_{o})}}$ With $f>f_{o}=nf_{o}$ to have $v>0$

Momentum of a bind electron

Equilibrium	$nhf=mvr2\pi$
Speed	$v={\frac {1}{2\pi }}{\frac {nhf}{mr}}$
Radius	$r={\frac {1}{2\pi }}{\frac {nhf}{mv}}$
Potential Energy Level n	$n=2\pi {\frac {mv}{hf}}$

Wave

AC electrical sinusoidal wave generator

An interaction of 2 electromagnets creates an AC electricity that has amplitude varies sinusoidally

$v(t)=ASin(\omega t+\theta )$

LC sinusoidal wave generator

Series LC operates at equilibrium satisfy wave equation

${\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}i(t)$

that has root of a sinusoidal wave function

$i(t)=ASin\omega t$

$\omega ={\sqrt {\frac {1}{T}}}$

$T=LC$

Electromagnetic sinusoidal wave generator

A coil of N turns operates at equilibrium satisfy wave equation

$\nabla ^{2}E(t)=-\omega E(t)$

$\nabla ^{2}B(t)=-\omega B(t)$

that has root of a sinusoidal wave function

$E(t)=ASin\omega t$

$B(t)=ASin\omega t$

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

$T=\mu \epsilon$

Wave Characteristics

Distance	$s$	$\lambda$
Time	$t$	$t$
Speed	$v$	${\frac {\lambda }{t}}$
Angular speed	$\omega$	$2\pi f$
Frequency	$f$	${\frac {1}{t}}$
Sinusoidal wave equation	$f^{''}(t)$	$-\omega f(t)$
Sinusoidal wave function	$f(t)$	$ASin\omega t$

Wave types

Wave	2 dimensional sinusoidal wave	3 dimensional sinusoidal plane wave
Wave oscillation equation	${\frac {d^{2}}{dt^{2}}}f(t)=-\omega f(t)$
Wave function	$f(t)=ASin\omega t$	$E(t)=ASin\omega t$ $B(t)=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}=C=\lambda f$ $T=\mu \epsilon$

Electricity

Electricity Types

Electricity types	Definition	Mathematical Formula	Source
DC Electricity	Electricity that provides constant voltage over time	$v(t)=V$	Electrolysis, Electrochemcial Cell, PhotonVoltaic
AC Electricity	Electricity that provides sinusoidal changing voltage over changing time	$v(t)=VSin\omega t$	Electromagnetic induction

DC & AC Response

• Resistor circuit

Characteristics	DC	AC
Voltage	$V=IR$	$v=iR$
Current	$I={\frac {V}{R}}$	$i={\frac {v}{R}}$
Resistance	$R={\frac {V}{I}}$	$R={\frac {v}{i}}$
Power provided	$P_{V}=IV$	$P_{V}=iv$
Power Loss	$P_{R}=I^{2}R(T)={\frac {V^{2}}{R(T)}}$	$P_{R}=i^{2}R(T)={\frac {v^{2}}{R(T)}}$
Power delivered	$P=P_{V}-P_{R}$
Reactance		$X_{R}=0$
Impedance		$Z_{R}=X_{R}+R=R$
Phase		$0$

• Capacitor circuit

Characteristics	DC	AC
Voltage	$V=QC={\frac {W}{Q}}$	$v={\frac {1}{C}}\int idt$
Charge	$Q={\frac {V}{C}}$
Capacitance	$C={\frac {V}{Q}}$
Current	$I={\frac {Q}{t}}$	$i=C{\frac {dv}{dt}}$
Power provided	$P_{V}=IV=({\frac {Q}{t}})({\frac {W}{Q}})={\frac {W}{t}}$	$p={\frac {1}{2}}Cv^{2}$
Reactance		$X_{C}(t)={\frac {v}{i}}$ $X_{C}(j\omega )={\frac {1}{j\omega C}}$ $X_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90$
Impedance		$Z_{C}(t)=X_{C}+R_{C}$ $X_{C}(j\omega )={\frac {1}{j\omega C}}+R_{C}$ $X_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90+R\angle 0$
Phase		$Tan\theta ={\frac {1}{\omega T}}$
Time Constant		$T=CR_{C}$

• Inductor circuit

Characteristics	DC	AC
Magnetic Field Strength	$B=LI$
Current	$I={\frac {B}{L}}$
Inductance	$C={\frac {B}{I}}$
Current	$I={\frac {Q}{t}}$
Power provided	$P_{V}=IV=({\frac {B}{l}})({\frac {W}{Q}})={\frac {W}{t}}$
Reactance		$X_{L}(t)={\frac {v}{i}}$ $X_{L}(j\omega )=j\omega L$ $X_{L}(\omega \theta )=\omega L\angle 90$
Impedance		$Z_{C}(t)=X_{C}+R_{C}$ $X_{C}(j\omega )=j\omega L+R_{L}$ $X_{C}(\omega \theta )=\omega L\angle 90+R\angle 0$
Phase		$Tan\theta =\omega T$
Time Constant		$T={\frac {L}{R_{L}}}$

• Electric Decay

	${\frac {d}{dt}}i=-{\frac {1}{T}}i$	$i=Ae^{-{\frac {1}{T}}t}$		$T={\frac {L}{R}}$
	${\frac {d}{dt}}v=-{\frac {1}{T}}v$	$v=Ae^{-{\frac {1}{T}}t}$		$T=RC$

• Electric Oscillation

Modes of Oscillation	Oscillation equation	Wave Function	Angular Speed	Oscillation Time Constant	Oscilation Constant	Decay Constant '	Decay Time Constant
Electric decay current sinusoidal wave oscillation	${\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i$	$i=A(\alpha )Sin\omega t$	$\omega ={\sqrt {\beta -\alpha }}$	$T=LC$	$\beta ={\frac {1}{T}}$	$\alpha =\beta \gamma$	$\gamma =RC$
Electric peak current sinusoidal wave oscillation	$Z_{L}=-Z_{C}$ $Z_{t}=R$	$i(\omega =0)=0$ $i(\omega =\omega _{o})={\frac {v}{2}}$ $i(\omega =00)=0$	$\omega _{o}={\sqrt {\frac {1}{T}}}$	$T=LC$
Electric current sinusoidal wave oscillation	${\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i$	$i=ASin\omega t$	$\omega ={\sqrt {\frac {1}{T}}}$	$T=LC$
Electric current sinusoidal standing wave oscillation	$Z_{L}=-Z_{C}$	$V_{L}=-V_{C}$	$\omega _{o}={\sqrt {\frac {1}{T}}}$	$T=LC$

Electromagneticism

Electric charge

Charge acquired process	Electric charge	Charge quantity	Electric field	Magnetic field
Matter + e-	-	-Q	-->E<--	B ↓
Matter - e-	+	+Q	<--E-->	B ↑

Electromagnetic force

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

Electromagnetic Field

Configuration	Symbol	Mathematical Formulas
For any configuration	$B$	$B=LI$
For straight line conductor	$B$	$B=LI={\frac {\mu }{2\pi r}}I$	Circular B field
For circular loop conductor	$B$	$B=LI={\frac {\mu }{2r}}I$	Circular B field around a point charge
For coil of N circular loops conductor	$B$	$B=LI={\frac {N\mu }{l}}I$	Eleptic B field around the coil

Electromagnetic Principles

Electromagnetic field	$B$	$LI$
Induced electromagnetic field	$\phi$	$-NB=-NLI$
Electromagnetization field	$H$	${\frac {B}{\mu }}={\frac {\phi }{N\mu }}$
Eletromagnetization		$\nabla \cdot D=\rho$ $\nabla \times E=-\nabla B$ $\nabla \cdot B=0$ $\nabla \times H=J+\nabla B$

Coil's voltage induction intencity	$V$	$V={\frac {dB}{dt}}=L{\frac {dI}{dt}}$
Turn's induced voltage intencity	$\epsilon$	$-{\frac {d\phi }{dt}}=-N{\frac {dB}{dt}}=-NL{\frac {d}{dt}}I$

Electromagnetic oscillation		$\nabla \cdot E=0$ $\nabla \times E={\frac {1}{T}}$ $\nabla \cdot B=0$ $\nabla \times B={\frac {1}{T}}$
Electromagnetic wave		$\nabla ^{2}E=-\omega E$ $\nabla ^{2}B=-\omega B$ $E=ASin\omega t$ $B=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}=C=\lambda f$ $T=\mu \epsilon$
Electromagnetic wave radiation		$v=\omega ={\sqrt {\frac {1}{\mu \epsilon }}}=C=\lambda f$ $E=pv=pC=p\lambda f=hf$ $h=p\lambda$ $p={\frac {h}{\lambda }}$ $\lambda ={\frac {h}{p}}={\frac {C}{f}}$

Photon

Photon is defined as energy of a massless quanta travels as an electromagnetic oscillation wave radiation at speed equal to speed of light Mathematical formula of

$E=hf=\hbar \omega$

Photon exist in 2 states

Radiant Photon at threshold frequency fo

$E=hf_{o}=\hbar \omega _{o}$

Non Radiant Photon at frequency greater than threshold frequency fo

$E=hf=\hbar \omega$ . With f>fo

Photon can only exist in one state at a time . This is Heiseinberg's uncertainty principle the success rate of finding photon

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

Quanta of Photon

Quantity of photon's energy . Quanta is denoted as h measured in has a constant value h =

$h=p\lambda$

Quanta process Wave particle duality meaning

Sometimes it behaves like wave $\lambda ={\frac {h}{p}}$

Sometimes it behaves like particle $p={\frac {h}{\lambda }}$

Photon and matter

Photon and matter interacts to create Heat transfer through 3 phases Heat conduction, Heat convection and Heat radiation

Heat transfer
Heat conduction	Matter changes it's temperature while absorb heat energy	$\Delta T=T_{1}-T_{0}$ $E=mC\Delta T$
Heat convection	Matter conducts heat energy to the max at Threshold frequency fo	$f_{o}={\frac {C}{f_{o}}}$ $E=hf_{o}$
Heat radiation	Matter uses excess energy above maximum absorbing energy to release electron off atom	$hf-hf_{o}={\frac {1}{2}}mv^{2}$. When v>0 $v={\sqrt {{\frac {2}{m}}nhf_{o}}}$ with f > fo

Causes matter to decay through 3 kinds of decays

Mattter decay	Reaction
Alpha decay	Ur--> Th - He + Alpha radiation
Beta decay	C-->N + Beta radiation
Gamma decay	ee) + Gamma radiation

Experiment has shown that, electron of an atom can be freed or binded from absorbing or releasing photon's energy

Atom decay	Absorbing photon energy	Releasing photon energy
Equilibrium	$hf=hf_{o}+{\frac {1}{2}}mv^{2}$	$nhf=mvr2\pi$
v	${\sqrt {{\frac {2}{m}}(hf-hf_{o})}}={\sqrt {{\frac {2}{m}}(nf_{o})}}$	${\frac {1}{2\pi }}{\frac {nhf}{mr}}$
r		${\frac {1}{2\pi }}{\frac {nhf}{mv}}$
n		$2\pi {\frac {mv}{hf}}$