Fundamental Physics/Physics formulas

Matter

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

Force

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

Motion

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

Electricity

Electricity types and electricity sources
 Electricity Mathematical formula Electricity source DC electricity provides constant voltage over time ${\displaystyle v(t)=V}$ Electrolysis  Electrovoltaic  Photonvoltaic AC electricity provides time varying sinusoidal voltage over time ${\displaystyle v(t)=V\sin \omega t}$ Electromagnetic induction
DC and AC response of electric conductors
 Electricity DC AC Resistor ${\displaystyle V=IR}$ ${\displaystyle I={\frac {V}{R}}}$ ${\displaystyle P_{V}=IV}$ ${\displaystyle R={\frac {V}{I}}={\frac {1}{G}}=\rho {\frac {l}{A}}}$ ${\displaystyle G={\frac {I}{V}}={\frac {1}{R}}=\sigma {\frac {A}{l}}}$ ${\displaystyle B=IL=I{\frac {\mu }{2\pi r}}}$ ${\displaystyle P_{R}=I^{2}R(T)}$ ${\displaystyle R(T)=R_{o}+nT}$ ${\displaystyle R(T)=R_{o}e^{nT}}$ ${\displaystyle P=P_{V}-P_{R}}$ ${\displaystyle v_{R}(t)=i(t)Z_{R}}$ ${\displaystyle i_{R}(t)={\frac {v(t)}{Z_{R}}}}$  ${\displaystyle P_{R}(t)=i(t)^{2}Z_{R}={\frac {v^{2}(t)}{Z_{R}}}}$ ${\displaystyle Z_{R}={\frac {v_{R}(t)}{i_{R}(t)}}=R+X_{R}=R\angle 0=R}$ ${\displaystyle X_{R}=0}$ Capacitor ${\displaystyle V={\frac {W}{Q}}={\frac {Q}{C}}}$ ${\displaystyle I={\frac {Q}{t}}}$ ${\displaystyle P={\frac {W}{t}}={\frac {W}{Q}}{\frac {Q}{t}}=IV}$ ${\displaystyle C={\frac {Q}{V}}=\epsilon {\frac {A}{l}}}$ ${\displaystyle E={\frac {V}{d}}}$ ${\displaystyle v_{C}(t)={\frac {1}{C}}\int i(t)}$  ${\displaystyle i_{C}(t)=C{\frac {dv(t)}{dt}}}$ ${\displaystyle P={\frac {1}{2}}Cv^{2}(t)}$ ${\displaystyle Z_{C}=R_{C}+X_{C}=R+{\frac {1}{\omega C}}\angle -90^{0}=R_{C}+{\frac {1}{j\omega C}}=R_{C}+{\frac {1}{sC}}}$  ${\displaystyle X_{C}={\frac {1}{\omega C}}\angle -90={\frac {1}{j\omega C}}={\frac {1}{sC}}}$ ${\displaystyle Tan\theta ={\frac {1}{\omega T}}}$ ${\displaystyle T=RC}$ Inductor ${\displaystyle B=IL}$ ${\displaystyle I={\frac {B}{L}}}$ ${\displaystyle L={\frac {B}{I}}=\mu N^{2}{\frac {l}{A}}}$ ${\displaystyle v_{L}(t)=L{\frac {di(t)}{dt}}}$ ${\displaystyle i_{L}(t)={\frac {1}{L}}\int v(t)dt}$ ${\displaystyle P={\frac {1}{2}}Li^{2}(t)}$ ${\displaystyle Z_{L}=R_{L}+X_{L}=R+\omega L\angle 90^{o}=R+j\omega L=R+sL}$ ${\displaystyle X_{L}=\omega L\angle 90^{o}=j\omega L=sL}$ ${\displaystyle Tan\theta =\omega T}$ ${\displaystyle T={\frac {L}{R}}}$
Electric devices
 Electric exponential decay RC series ${\displaystyle C{\frac {dv(t)}{dt}}+v(t)R=0}$  ${\displaystyle {\frac {dv(t)}{dt}}=-{\frac {1}{T}}v(t)R}$  ${\displaystyle {\frac {dv(t)}{v(t)}}=-{\frac {1}{T}}dt}$  ${\displaystyle \int {\frac {dv(t)}{v(t)}}=-{\frac {1}{T}}\int dt}$  ${\displaystyle Lnv(t)=-{\frac {1}{T}}t+c}$  ${\displaystyle v(t)=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}}$  ${\displaystyle T=RC}$ Electric exponential decay RL series ${\displaystyle V_{L}+V_{R}=0}$  ${\displaystyle L{\frac {di(t)}{dt}}+i(t)R=0}$  ${\displaystyle {\frac {di(t)}{dt}}=-{\frac {1}{T}}i(t)}$  ${\displaystyle \int {\frac {di(t)}{i(t)}}=-{\frac {1}{T}}\int dt}$  ${\displaystyle Lni(t)=-{\frac {1}{T}}+c}$  ${\displaystyle i(t)=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}}$  ${\displaystyle T={\frac {L}{R}}}$ Electric current oscillation ${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int vdt=0}$  ${\displaystyle {\frac {d^{2}i}{dt}}+{\frac {1}{T}}=0}$  ${\displaystyle {\frac {d^{2}i}{dt^{2}}}=-{\frac {1}{T}}i}$  ${\displaystyle i(t)=e^{\pm j{\sqrt {\frac {1}{T}}}t}=e^{\pm j\omega t}=A\sin \omega t}$  ${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ Electric voltage standing wave oscillation ${\displaystyle Z_{L}=-Z_{C}}$ ${\displaystyle v_{C}=-v_{L}}$  ${\displaystyle \omega _{o}=\pm j{\sqrt {\frac {1}{LC}}}}$ ${\displaystyle V_{C}=-V_{L}}$ ${\displaystyle V(\theta )=A\sin(\omega _{o}t+2\pi )-A\sin(\omega _{o}t-2\pi )}$ Electric current decay oscillation ${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0}$ ${\displaystyle {\frac {d^{2}i}{dt}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0}$ ${\displaystyle {\frac {d^{2}i}{dt}}=-2\alpha {\frac {di}{dt}}-\beta i}$ ${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$ ${\displaystyle \alpha =\beta \gamma ={\frac {R}{2L}}}$ ${\displaystyle T=LC}$  ${\displaystyle \gamma =RC}$ Phương trình trên có nghiệm như sau${\displaystyle \alpha =\beta }$  . 1 nghiệm thực${\displaystyle i=Ae^{-\alpha t}=A(\alpha )}$  ${\displaystyle \alpha >\beta }$  . 2 nghiệm thực${\displaystyle i=Ae^{-\alpha \pm {\sqrt {\alpha -\beta }}t}}$  ${\displaystyle \alpha <\beta }$  . 2 nghiệm phức${\displaystyle i=Ae^{-\alpha \pm j{\sqrt {\beta -\alpha }}t}}$  ${\displaystyle i=Ae^{-\alpha t}e^{\pm j{\sqrt {\beta -\alpha }}t}}$ ${\displaystyle i=A(\alpha )Sin\omega t}$ ${\displaystyle A(\alpha )=Ae^{-\alpha t}}$ ${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$ Electric peak current oscillation ${\displaystyle Z_{L}=-Z_{C}}$ ${\displaystyle Z_{t}=R}$ ${\displaystyle \omega _{o}={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ ${\displaystyle Z_{t}=R}$ ${\displaystyle i={\frac {v}{R}}}$ ${\displaystyle i(\omega =0)=0}$  ${\displaystyle i(\omega =\omega _{o})={\frac {v}{R}}}$  ${\displaystyle i(\omega =00)=0}$ Electromagnetic wave oscillation Phương trình vector dao động điện từ ${\displaystyle \nabla \cdot E=0}$ ${\displaystyle \nabla \times E=-{\frac {1}{T}}E}$ ${\displaystyle \nabla \cdot B=0}$ ${\displaystyle \nabla \times B=-{\frac {1}{T}}B}$ Phương trình sóng ${\displaystyle \nabla ^{2}E=-\omega E}$ ${\displaystyle \nabla ^{2}B=-\omega B}$  Hàm số sóng ${\displaystyle E=A\sin \omega t}$ ${\displaystyle B=A\sin \omega t}$  ${\displaystyle \omega ={\sqrt {\frac {1}{T}}}=C=\lambda f}$  ${\displaystyle T=\mu \epsilon }$ Sóng điện từ Low frequency voltage stabilizer ${\displaystyle {\frac {v_{o}}{v_{2}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{1+j\omega T}}}$ ${\displaystyle T=RC}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {1}{RC}}}$  ${\displaystyle v_{o}(\omega =0)=v_{i}}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=0}$ Low frequency voltage stabilizer ${\displaystyle {\frac {v_{o}}{v_{2}}}={\frac {R}{R=j\omega L}}={\frac {1}{1+j\omega T}}}$ ${\displaystyle T={\frac {L}{R}}}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {R}{L}}}$  ${\displaystyle v_{o}(\omega =0)=v_{i}}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=0}$ High frequency voltage stabilizer ${\displaystyle {\frac {v_{o}}{v_{2}}}={\frac {j\omega T}{1+j\omega T}}}$ ${\displaystyle T=RC}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {1}{RC}}}$  ${\displaystyle v_{o}(\omega =0)=0}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=v_{i}}$ High frequency voltage stabilizer ${\displaystyle {\frac {v_{o}}{v_{2}}}={\frac {j\omega T}{1+j\omega T}}}$ ${\displaystyle T={\frac {L}{R}}}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {R}{L}}}$  ${\displaystyle v_{o}(\omega =0)=0}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=v_{i}}$ Band pass voltage stabilizer ${\displaystyle {\frac {v_{o}}{v_{i}}}=({\frac {1}{1+j\omega T_{L}}})({\frac {j\omega T_{H}}{1+j\omega _{H}}})}$  ${\displaystyle T_{L}={\frac {L}{R}}}$ ${\displaystyle T_{H}=RC}$  ${\displaystyle \omega _{L}-\omega _{H}={\frac {R}{L}}-{\frac {1}{RC}}}$ Band pass voltage stabilizer ${\displaystyle {\frac {v_{o}}{v_{i}}}=({\frac {1}{1+j\omega T_{L}}})({\frac {j\omega T_{H}}{1+j\omega _{H}}})}$  ${\displaystyle T_{L}=RC}$ ${\displaystyle T_{H}={\frac {L}{R}}}$  ${\displaystyle \omega _{L}-\omega _{H}={\frac {1}{RC}}-{\frac {R}{L}}}$

Electromagneticism

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