Fundamental Physics/Physics formulas

Matter

State	Solid . Đồng (Cu) , Sát(Fe) , ... Liquid . Nước(H₂O), ... Gas . Ôxygen (O₂)
Density	$D={\frac {M}{V}}$
Rutherford charged particle distribution model
Borh circular quantum energy model

Force

Force	Graph	Symbol	Mathematical formula
Motion force	-->O --> O	$F_{a}$	$ma$
Impulse	-->O -->	$F_{p}$	${\frac {p}{t}}$
Opposition force		$F_{-}$	$-F$
Presure force		$F_{A}$	${\frac {F}{A}}$
Elastic force		$F_{y}$ $F_{x}$	$-ky$ $-kx$
Friction force		$F_{\mu }$	$\mu N$
Circulation force		$F_{r}$	$m{\frac {\omega r}{t}}$
Centripetal force		$F_{a}$	$m{\frac {v^{2}}{r}}$
Electrostatic force		$F_{q}$	$K{\frac {q_{+}q_{-}}{r^{2}}}$
Electrodynamic force		$F_{E}$	$qE$
Electromomagnetomotive force		$F_{B}$	$\pm qvB$
Electromagnetic force		$F_{EB}$	$qE\pm qvB$

Motion

Momentum	O -->	$v=v$ $a={\frac {v}{t}}$ $s=vt$ $m=m$ $p=mv$ $F={\frac {p}{t}}$ $W=pv$ $E=pa$
Uniform horizontal linear motion	O --> O	$v=v$ $a={\frac {v}{t}}$ $s=vt$ $F=ma$ $W=Fs$ $E={\frac {W}{t}}$
Uniform vertical linear motion		$v={\frac {h}{t}}$ $a={\frac {h}{t^{2}}}$ $s=h$ $F=mg=m{\frac {GM}{h^{2}}}$ $W=mgh$ $E=mg{\frac {h}{t}}$
Uniform inclined linear motion		$v=v_{o}+a\Delta t$ $a={\frac {v-v_{o}}{t-t_{o}}}={\frac {\Delta v}{\Delta t}}$ $s=(v_{o}+a\Delta t)t$ $F=m{\frac {\Delta v}{\Delta t}}$ $W=F(v_{o}+a\Delta t)t$ $E=F(v_{o}+a\Delta t)$
Non uniform curve motion		$v=v(t)$ $a={\frac {d}{dt}}v(t)$ $s=\int v(t)dt$ $F=m{\frac {d}{dt}}v(t)$ $W=F\int v(t)dt$ $E={\frac {F}{t}}\int v(t)dt$
Uniform circular motion		$v={\frac {2\pi r}{t}}=\omega r$ $a={\frac {\omega r}{t}}$ $s=2\pi r$ $F=m{\frac {\omega r}{t}}$ $W=p\omega r$ $E=p{\frac {\omega r}{t}}$
Non uniform circular motion		$v={\frac {\theta r}{t}}$ $a={\frac {v^{2}}{r}}$ $s=\theta r$ $F=m{\frac {\theta r}{t}}$ $W=p{\frac {\theta r}{t}}$ $F=p{\frac {\theta r}{t^{2}}}$
Sinusodal wave motion		$v={\frac {\lambda }{t}}=\lambda f=\omega$ $a={\frac {\omega }{t}}$ $s=\lambda$ $F=m{\frac {\omega }{t}}$ $W=p{\frac {\omega }{t}}$ $E=p{\frac {\omega }{t^{2}}}$
Spring's vertical oscillation wave		$\omega ={\sqrt {\frac {k}{m}}}$ $y^{''}(t)=-\omega y(t)$ $y(t)=A\sin \omega t$
Spring's horizontal oscillation wave		$\omega ={\sqrt {\frac {k}{m}}}$ $x^{''}(t)=-\omega x(t)$ $x(t)=A\sin \omega t$
Pendulum's vertical oscillation wave		$\omega ={\sqrt {\frac {l}{g}}}$ $y^{''}(t)=-\omega y(t)$ $y(t)=A\sin \omega t$
Electric current oscillation wave		$\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$ $i^{''}(t)=-\omega i(t)$ $i(t)=A\sin \omega t$
/Electric current decay oscillation wave/		$\omega ={\sqrt {\beta -\alpha }}$ $T=LC$ $i^{''}(t)=-2\alpha i^{'}(t)-\beta i(t)$ $i(t)=A(\alpha )\sin \omega t$
Electromagnetic oscillation wave		$\omega ={\sqrt {\frac {1}{T}}}$ $T=\mu \epsilon$ $E^{''}(t)=-\omega E(t)$ $B^{''}(t)=-\omega B(t)$ $B(t)=A\sin \omega t$ $B(t)=A\sin \omega t$
Relative motion		$v=\omega =\lambda f=C$ $E=pv=pC=p\lambda f=C$ $h=p\lambda$ $p={\frac {h}{\lambda }}$ $h={\frac {h}{p}}$ $\gamma ={\sqrt {1-{\frac {v^{2}}{C^{2}}}}}$ $E=M\gamma ^{2}$ $M=m_{o}(\gamma -1)$ $p=M\gamma$

Electricity

Electricity types and electricity sources

Electricity	Mathematical formula	Electricity source
DC electricity	provides constant voltage over time $v(t)=V$	Electrolysis Electrovoltaic Photonvoltaic
AC electricity	provides time varying sinusoidal voltage over time $v(t)=V\sin \omega t$	Electromagnetic induction

DC and AC response of electric conductors

Electricity	DC	AC
Resistor	$V=IR$ $I={\frac {V}{R}}$ $P_{V}=IV$ $R={\frac {V}{I}}={\frac {1}{G}}=\rho {\frac {l}{A}}$ $G={\frac {I}{V}}={\frac {1}{R}}=\sigma {\frac {A}{l}}$ $B=IL=I{\frac {\mu }{2\pi r}}$ $P_{R}=I^{2}R(T)$ $R(T)=R_{o}+nT$ $R(T)=R_{o}e^{nT}$ $P=P_{V}-P_{R}$	$v_{R}(t)=i(t)Z_{R}$ $i_{R}(t)={\frac {v(t)}{Z_{R}}}$ $P_{R}(t)=i(t)^{2}Z_{R}={\frac {v^{2}(t)}{Z_{R}}}$ $Z_{R}={\frac {v_{R}(t)}{i_{R}(t)}}=R+X_{R}=R\angle 0=R$ $X_{R}=0$
Capacitor	$V={\frac {W}{Q}}={\frac {Q}{C}}$ $I={\frac {Q}{t}}$ $P={\frac {W}{t}}={\frac {W}{Q}}{\frac {Q}{t}}=IV$ $C={\frac {Q}{V}}=\epsilon {\frac {A}{l}}$ $E={\frac {V}{d}}$	$v_{C}(t)={\frac {1}{C}}\int i(t)$ $i_{C}(t)=C{\frac {dv(t)}{dt}}$ $P={\frac {1}{2}}Cv^{2}(t)$ $Z_{C}=R_{C}+X_{C}=R+{\frac {1}{\omega C}}\angle -90^{0}=R_{C}+{\frac {1}{j\omega C}}=R_{C}+{\frac {1}{sC}}$ $X_{C}={\frac {1}{\omega C}}\angle -90={\frac {1}{j\omega C}}={\frac {1}{sC}}$ $Tan\theta ={\frac {1}{\omega T}}$ $T=RC$
Inductor	$B=IL$ $I={\frac {B}{L}}$ $L={\frac {B}{I}}=\mu N^{2}{\frac {l}{A}}$	$v_{L}(t)=L{\frac {di(t)}{dt}}$ $i_{L}(t)={\frac {1}{L}}\int v(t)dt$ $P={\frac {1}{2}}Li^{2}(t)$ $Z_{L}=R_{L}+X_{L}=R+\omega L\angle 90^{o}=R+j\omega L=R+sL$ $X_{L}=\omega L\angle 90^{o}=j\omega L=sL$ $Tan\theta =\omega T$ $T={\frac {L}{R}}$

Electric devices

Electric exponential decay	RC series	$C{\frac {dv(t)}{dt}}+v(t)R=0$ ${\frac {dv(t)}{dt}}=-{\frac {1}{T}}v(t)R$ ${\frac {dv(t)}{v(t)}}=-{\frac {1}{T}}dt$ $\int {\frac {dv(t)}{v(t)}}=-{\frac {1}{T}}\int dt$ $Lnv(t)=-{\frac {1}{T}}t+c$ $v(t)=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}$ $T=RC$
Electric exponential decay	RL series	$V_{L}+V_{R}=0$ $L{\frac {di(t)}{dt}}+i(t)R=0$ ${\frac {di(t)}{dt}}=-{\frac {1}{T}}i(t)$ $\int {\frac {di(t)}{i(t)}}=-{\frac {1}{T}}\int dt$ $Lni(t)=-{\frac {1}{T}}+c$ $i(t)=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}$ $T={\frac {L}{R}}$
Electric current oscillation		$L{\frac {di}{dt}}+{\frac {1}{C}}\int vdt=0$ ${\frac {d^{2}i}{dt}}+{\frac {1}{T}}=0$ ${\frac {d^{2}i}{dt^{2}}}=-{\frac {1}{T}}i$ $i(t)=e^{\pm j{\sqrt {\frac {1}{T}}}t}=e^{\pm j\omega t}=A\sin \omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$
Electric voltage standing wave oscillation		$Z_{L}=-Z_{C}$ $v_{C}=-v_{L}$ $\omega _{o}=\pm j{\sqrt {\frac {1}{LC}}}$ $V_{C}=-V_{L}$ $V(\theta )=A\sin(\omega _{o}t+2\pi )-A\sin(\omega _{o}t-2\pi )$
Electric current decay oscillation		$L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0$ ${\frac {d^{2}i}{dt}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0$ ${\frac {d^{2}i}{dt}}=-2\alpha {\frac {di}{dt}}-\beta i$ $\beta ={\frac {1}{T}}={\frac {1}{LC}}$ $\alpha =\beta \gamma ={\frac {R}{2L}}$ $T=LC$ $\gamma =RC$ Phương trình trên có nghiệm như sau $\alpha =\beta$ . 1 nghiệm thực $i=Ae^{-\alpha t}=A(\alpha )$ $\alpha >\beta$ . 2 nghiệm thực $i=Ae^{-\alpha \pm {\sqrt {\alpha -\beta }}t}$ $\alpha <\beta$ . 2 nghiệm phức $i=Ae^{-\alpha \pm j{\sqrt {\beta -\alpha }}t}$ $i=Ae^{-\alpha t}e^{\pm j{\sqrt {\beta -\alpha }}t}$ $i=A(\alpha )Sin\omega t$ $A(\alpha )=Ae^{-\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$
Electric peak current oscillation		$Z_{L}=-Z_{C}$ $Z_{t}=R$ $\omega _{o}={\sqrt {\frac {1}{T}}}$ $T=LC$ $Z_{t}=R$ $i={\frac {v}{R}}$ $i(\omega =0)=0$ $i(\omega =\omega _{o})={\frac {v}{R}}$ $i(\omega =00)=0$
Electromagnetic wave oscillation		Phương trình vector dao động điện từ $\nabla \cdot E=0$ $\nabla \times E=-{\frac {1}{T}}E$ $\nabla \cdot B=0$ $\nabla \times B=-{\frac {1}{T}}B$ Phương trình sóng $\nabla ^{2}E=-\omega E$ $\nabla ^{2}B=-\omega B$ Hàm số sóng $E=A\sin \omega t$ $B=A\sin \omega t$ $\omega ={\sqrt {\frac {1}{T}}}=C=\lambda f$ $T=\mu \epsilon$ Sóng điện từ
Low frequency voltage stabilizer		${\frac {v_{o}}{v_{2}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{1+j\omega T}}$ $T=RC$ $\omega _{o}={\frac {1}{T}}={\frac {1}{RC}}$ $v_{o}(\omega =0)=v_{i}$ $v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}$ $v_{o}(\omega =00)=0$
Low frequency voltage stabilizer		${\frac {v_{o}}{v_{2}}}={\frac {R}{R=j\omega L}}={\frac {1}{1+j\omega T}}$ $T={\frac {L}{R}}$ $\omega _{o}={\frac {1}{T}}={\frac {R}{L}}$ $v_{o}(\omega =0)=v_{i}$ $v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}$ $v_{o}(\omega =00)=0$
High frequency voltage stabilizer		${\frac {v_{o}}{v_{2}}}={\frac {j\omega T}{1+j\omega T}}$ $T=RC$ $\omega _{o}={\frac {1}{T}}={\frac {1}{RC}}$ $v_{o}(\omega =0)=0$ $v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}$ $v_{o}(\omega =00)=v_{i}$
High frequency voltage stabilizer		${\frac {v_{o}}{v_{2}}}={\frac {j\omega T}{1+j\omega T}}$ $T={\frac {L}{R}}$ $\omega _{o}={\frac {1}{T}}={\frac {R}{L}}$ $v_{o}(\omega =0)=0$ $v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}$ $v_{o}(\omega =00)=v_{i}$
Band pass voltage stabilizer		${\frac {v_{o}}{v_{i}}}=({\frac {1}{1+j\omega T_{L}}})({\frac {j\omega T_{H}}{1+j\omega _{H}}})$ $T_{L}={\frac {L}{R}}$ $T_{H}=RC$ $\omega _{L}-\omega _{H}={\frac {R}{L}}-{\frac {1}{RC}}$
Band pass voltage stabilizer		${\frac {v_{o}}{v_{i}}}=({\frac {1}{1+j\omega T_{L}}})({\frac {j\omega T_{H}}{1+j\omega _{H}}})$ $T_{L}=RC$ $T_{H}={\frac {L}{R}}$ $\omega _{L}-\omega _{H}={\frac {1}{RC}}-{\frac {R}{L}}$

Electromagneticism

Electromagnetic fields

A straight line conductor		$v=iR$ $i={\frac {v}{R}}$ $R={\frac {v}{i}}=\rho {\frac {l}{A}}$ $G={\frac {i}{v}}={\frac {1}{\rho }}{\frac {A}{l}}$ $B={\frac {\mu }{2\pi r}}i$ $R(T)=R_{o}+nT$ $R(T)=R_{o}e^{nT}$ $E_{R}=i^{2}R(T)=mC\Delta T$ $m={\frac {i^{2}R(T)}{C\Delta T}}$ $C={\frac {i^{2}R(T)}{m\Delta T}}$
A circular loop conductor		$B={\frac {\mu }{2r}}i$ $V={\frac {d}{dt}}B=L{\frac {d}{dt}}i$ $m{\frac {v^{2}}{r}}=QvB$ $r={\frac {mv^{2}}{QB}}$ $v={\frac {Q}{m}}Br$
A coil of N circular loop conductor		$B={\frac {N}{l}}\mu i$ $-\phi =-NB=-NLi$ $H={\frac {B}{\mu }}={\frac {N\mu }{l}}$ $V={\frac {d}{dt}}B=L{\frac {d}{dt}}i$ $-\epsilon =-{\frac {d}{dt}}\phi =NL{\frac {d}{dt}}i$ $F=Bl=N\mu i$ $+=N$ $-=S$ $B=N-->S$ $\phi =N<--S$ $H=N-->S$
Electromagnetization		$\nabla \cdot D=\rho$ $\nabla \times E=-\nabla B$ $\nabla \cdot B=0$ $\nabla \times H=J+\nabla B$
Electromagnetic oscillation		$\nabla \cdot E=0$ $\nabla \times E=-{\frac {1}{T}}E$ $\nabla \cdot B=0$ $\nabla \times B=-{\frac {1}{T}}$
Electromagnetic wave		$\nabla ^{2}E=-\omega E$ $\nabla ^{2}B=-\omega B$ $E=A\sin \omega t$ $B=A\sin \omega t$ $\omega =\lambda f={\sqrt {\frac {1}{T}}}=C$ $T=\mu \epsilon$
Electromagnetic wave radiation		$v=\omega =\lambda f={\sqrt {\frac {1}{\mu \epsilon }}}=C$ $E=pv=pC=p\lambda f=hf$ $h=p\lambda$ $p={\frac {h}{\lambda }}$ $\lambda ={\frac {h}{p}}={\frac {C}{f}}$

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