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	$a$	$\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$
Force	$F$	$m\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$
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	$hf=h{f}_o+\frac{1}{2}m{v}^2$
Speed	$v=\sqrt{\frac{2}{m}(hf-h{f}_o)}=\sqrt{\frac{2}{m}(nh{f}_o)}$ With $f>f_o=n{f}_o$ 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

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

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

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

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

that has root of a sinusoidal wave function

$E(t)=ASin\omega t$

$B(t)=ASin\omega t$

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

$T=\mu ϵ$

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	$f^{{\text{'}}^{\prime }}(t)$	$-\omega f(t)$
Sinusoidal wave function	$f(t)$	$ASin\omega t$

Wave types

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

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	$P_V=IV=(\frac{Q}{t})(\frac{W}{Q})=\frac{W}{t}$	$p=\frac{1}{2}C{v}^2$
Reactance		$X_C(t)=\frac{v}{i}$ $X_C(j\omega )=\frac{1}{j\omega C}$ $X_C(\omega \theta )=\frac{1}{\omega C}\mathrm{\angle }-90$
Impedance		$Z_C(t)=X_C+R_C$ $X_C(j\omega )=\frac{1}{j\omega C}+R_C$ $X_C(\omega \theta )=\frac{1}{\omega C}\mathrm{\angle }-90+R\mathrm{\angle }0$
Phase		$Tan\theta =\frac{1}{\omega T}$
Time Constant		$T=C{R}_C$

• 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		$X_L(t)=\frac{v}{i}$ $X_L(j\omega )=j\omega L$ $X_L(\omega \theta )=\omega L\mathrm{\angle }90$
Impedance		$Z_C(t)=X_C+R_C$ $X_C(j\omega )=j\omega L+R_L$ $X_C(\omega \theta )=\omega L\mathrm{\angle }90+R\mathrm{\angle }0$
Phase		$Tan\theta =\omega T$
Time Constant		$T=\frac{L}{R_L}$

• Electric Decay

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

• 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	$\frac{d^2}{d{t}^2}i=-2\alpha \frac{d}{dt}i-\beta i$	$i=A(\alpha )Sin\omega t$	$\omega =\sqrt{\beta -\alpha }$	$T=LC$	$\beta =\frac{1}{T}$	$\alpha =\beta \gamma$	$\gamma =RC$
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	$\frac{d^2}{d{t}^2}i=-\frac{1}{T}i$	$i=ASin\omega t$	$\omega =\sqrt{\frac{1}{T}}$	$T=LC$
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	$\varphi$	$-NB=-NLI$
Electromagnetization field	$H$	$\frac{B}{\mu }=\frac{\varphi }{N\mu }$
Eletromagnetization		$\mathrm{\nabla }\cdot D=\rho$ $\mathrm{\nabla }\times E=-\mathrm{\nabla }B$ $\mathrm{\nabla }\cdot B=0$ $\mathrm{\nabla }\times H=J+\mathrm{\nabla }B$

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

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

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

$E=hf=\hslash \omega$

Photon exist in 2 states

Radiant Photon at threshold frequency fo

$E=h{f}_o=\hslash {\omega }_o$

Non Radiant Photon at frequency greater than threshold frequency fo

$E=hf=\hslash \omega$ . 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

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

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	$\mathrm{\Delta }T=T_1-T_0$ $E=mC\mathrm{\Delta }T$
Heat convection	Matter conducts heat energy to the max at Threshold frequency fo	$f_o=\frac{C}{f_o}$ $E=h{f}_o$
Heat radiation	Matter uses excess energy above maximum absorbing energy to release electron off atom	$hf-h{f}_o=\frac{1}{2}m{v}^2$. When v>0 $v=\sqrt{\frac{2}{m}nh{f}_o}$ 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	$hf=h{f}_o+\frac{1}{2}m{v}^2$	$nhf=mvr2\pi$
v	$\sqrt{\frac{2}{m}(hf-h{f}_o)}=\sqrt{\frac{2}{m}(n{f}_o)}$	$\frac{1}{2\pi }\frac{nhf}{mr}$
r		$\frac{1}{2\pi }\frac{nhf}{mv}$
n		$2\pi \frac{mv}{hf}$