# OpenStax College Physics/Equations

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#### Oscillatory Motion and Waves

* $F=-kx$, is Hooke's law, where F is the restoring force, k is the force constant, and x is the displacement from equilibrium.
* $PE_{el}={\tfrac {1}{2}}kx^{2}$ is the potential energy stored in the deformation of a spring or elastic system.
* $fT=1$ where f is the frequency and T is period. It is convenient to define omega as, $\omega \equiv 2\pi /T={\sqrt {k/m}}$, for a spring-mass system. For a simple pendulum, $\omega ={\sqrt {g/L}}$.
* $x=X\cos(\omega t),\quad v=-\omega X\sin(\omega t),\quad a=-\omega ^{2}X\cos(\omega t)$ describe the position, velocity, and acceleration of simple harmonic motion, respectively.
* $f\lambda =v_{w}$ is the wave velocity, where λ is wavelength.
* The node is a point of zero motion and the antinode is a point of maximum motion of a standing wave.
* $f_{B}=|f_{1}-f_{2}|$ is the beat frequency when two waves of slightly different frequency are superimposed.

##### Physics of Hearing

* $v_{w}=f\lambda \approx ({\text{331m/s}}){\sqrt {T/{\text{273K}}}}$ is the speed of sound in air, where T is the absolute temperature.
* $I=P/A=(\Delta p)^{2}/(2\rho v_{w})$ is intensity of sound, where P is the power passing through an area A, Δp is the pressure amplitude, and ρ is the density of the medium.
* $\beta {\text{(dB)}}=10\log _{10}(I/I_{0})$ is the intensity level in decibels, where I$_0$ = 10$^{-12}$W/m$^2$ is the threshold level of hearing.
* $f_{obs}=f_{s}\left({\frac {v_{w}}{v_{w}\pm v_{s}}}\right)$ is the observed Doppler shifted frequency for a stationary observer when the source of frequency f$_s$ is moving at speed $v_{s}$. The (+/-) sign refers to motion (away/towards) the observer.
* $f_{obs}=f_{s}\left({\frac {v_{w}\pm v_{obs}}{v_{w}}}\right)$ is the frequency perceived by an observer moving at speed $v_{s}$ with respect to a stationary source of frequencyf$_s$. The (+/-) sign refers to motion (towards/away) from the source.
* $f_{n}=n{\frac {v_{w}}{4L}}$ is the resonant frequency of the n-th mode of a standing sound wave in a tube that is closed at one end. The mode numbers are $n=1,3,5...$.
* $f_{n}=n{\frac {v_{w}}{2L}}$ with $n=1,2,3...$ are the resonant modes for a tube open at both ends.
* $a=(Z_{2}-Z_{1})^{2}/(Z_{1}+Z_{2})^{2}$ is the intensity reflection coefficient, which is the ratio of the intensity of the reflected wave to that of the incident wave at a boundary between two media. The acoustical impedance is $Z=\rho v_{w}$.

##### Electric Charge and Electric Field

The charge of the electron is q$_e$ = −e, where the fundamental charge is e≈1.6 x 10$^{−19}$ C. (C≡Coulombs is the SI unit of charge).
* $F=k{\frac {|q_{1}q_{2}|}{r^{2}}}$ is Coulomb's law for the force between two point charges q$_1$ and q$_2$ separated by a distance r. Coulomb's constant is k ≈ 8.99x10$^9$ N m$^2$ C$^{−2}$.
* ${\vec {F}}=q_{t}{\vec {E}}$ if the force on a test charge q$_t$ due to the electric field ${\vec {E}}=\Sigma {\vec {E}}_{j}$ at the location of the test charge. The vector sum is over $n=1\rightarrow N$ source charges, and the magnitude of each contribution to the field is $|{\vec {E}}_{n}|=kq_{n}/r_{nt}^{2}$ where q$_n$ is the source charge and r$_{nt}$ is the distance from the source point to the test charge.

##### Electric Potential and Electric Field

* $\Delta {\text{PE}}=q\Delta V=q(V_{B}-V_{A})$ is the potential energy associated with moving a charge q from a point where the electric potential is V$_A$ to a point where it is V$_B$.
* $1eV\approx 1.6\times 10^{-19}{\text{ Joules}}$
* $E=-\Delta V/\Delta s$ is the component of the electric field parallel to a small displacement $\Delta {\vec {s}}$ (the electric field points from high to low voltage, and ΔV=Ed if one moves a distance d along a uniform electric field E.)
* $V=kQ/r$ is the electric potential at a distance r from a point charge Q.
* $Q=CV$ is the charge stored by a capacitor of capacitance C with a voltage drop V across its terminals. For a parallel plate capacitor, $C=\varepsilon _{0}A/d$ for plates of area A separated by a (small) distance d. The permittivity of free space is ε$_0$ = 8.85×10$^{−12}$F/m. If a dielectric fills the gap, replace ε$_0$ by κε$_0$
* ${\frac {1}{C_{S}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+{\frac {1}{C_{3}}}+\dots$ is the total capacitance in series.
* $C_{p}=C_{1}+C_{2}+C_{3}+\dots$ is the total capacitance in parallel.
* $E_{cap}={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}=Q^{2}/(2C)$ is the energy stored in a capacitor.

##### Electric Current, Resistance, and Ohm's Law

The electric current, in amperes (A≡C/s) is the rate at which charge Q flows: $I=\Delta Q/\Delta t$
* $I=nqAv_{d}$ relates the current through the wire to the density of carriers n, charge of each carrier q, cross-sectional area A, and carrier drift velocity v$_d$.
* $V=IR$ is w:Ohm's law, where 1Ω=1V/A. $R=\rho L/A$ where ρ is resistivity, L is length, and A is area. For changes in temperature T, both R and ρ typically increase linearly with ΔT, as $\rho =\rho _{0}+\alpha \Delta T$, where α is the temperature coefficient of resistivity.
* $P=IV=I^{2}R=V^{2}/R$ is electrical power (energy divided by time). For alternating current, the average power is $P_{\text{ave}}=I_{\text{rms}}V_{\text{rms}}=I_{\text{rms}}^{2}R=V_{\text{rms}}^{2}/R$, where rms denotes root mean square). If $X(t)=X_{0}\sin(2\pi ft)$, $X_{\text{rms}}={\tfrac {1}{\sqrt {2}}}X_{0}$ where X$_0$ is the peak value and f is frequency.

##### Circuits and DC Instruments

If resistors are in series, the equivalent resistance is $R_{s}=R_{1}+R_{2}+R_{3}+\ldots$. If they are in parallel, $R_{eq}^{-1}=R_{1}^{-1}+R_{2}^{-1}+R_{3}^{-1}+\ldots$
* $V_{\text{terminal}}={\text{emf}}-Ir$ relates the terminal voltage to current I, internal resistance r and emf is the current, and r is the internal resistance.
* Kirchoff's node rule is $\Sigma I_{\text{out}}=\Sigma I_{\text{in}}$. The loop rules is $\Sigma \Delta V=0$. The voltage drop $\Delta V>0$ if the path crosses through a voltage source from the negative to the positive terminals, or if it travels opposite the current when passing through a resistor.
* $\tau =RC$ is the decay constant for a resistor and capacitor.
* While charging, $V={\text{emf}}\,(1-e^{-t/RC})$, so that if the voltage is $V$, it will rise by $0.631({\text{emf}}-V)$ in the next RC time.
* While discharging, the voltage will drop from $V$ to $0.368V$ in one RC time.

##### Magnetism

$F=qvB\sin \theta$
* $r=mv/qB$ is the orbital radius of a charged particle in a magnetic field (m,v,q are mass, speed, and charge, respectively.)
* $\varepsilon =vB\ell$ is the Hall emf across a distance ℓ, for moving charged particles (v B, ℓ are mutually perpendicular vectors.)
* $F=I\ell B\sin \theta$ is the force on a wire of length ℓ, current I, in the presence of uniform magnetic field. The direction follows RHR-1.
* $\tau =NIAB\sin \theta$ is the torque on N turns of wire around an area A, where θ is the angle between the uniform magnetic field and the perpendicular to the loop.
* $B={\frac {\mu _{0}I}{2\pi r}}$ is the magnetic field at a distance r from a long straight wire, where μ$_0$ = 4π × 10$^{−7}$ T·m/A is the permeability of free space.
* $B={\frac {\mu _{0}I}{2R}}$ is the magnetic field at the center of a loop of radius R.
* $B=\mu _{0}nI$ is the magnetic field inside a long, thin solenoid, where n=N/ℓ is the number of turns per unit length.
* Parallel wires attract (repel) if the currents are parallel (antiparallel), with $F/\ell =\mu _{0}{\frac {I_{1}I_{2}}{2\pi r}}$.
* The net force on any charged particle is zero if $v=E/B$ and the velocity, magnetic, and electric fields are mutually perpendicular (see Wein filter.)

##### Electromagnetic Induction, AC Circuits, and Electrical Technologies

Magnetic flux is $\Phi =AB\sin \theta$ where A is area and θ is the angle between the magnetic field, B, and the normal to the area.
* If a coil has N turns the induced emf is $emf=-\Delta \Phi /\Delta t$
* The motional $emf=vB\ell$ if a wire of length $\ell$, its velocity and the magnetic field are mutually perpendicular.
* If a coil is rotating at angular speed ω in a magnetic field, $emf=NAB\omega \sin \omega t$, with the peak $emf_{0}=NAB\omega$
* In an ideal transformer, the primary and secondary voltages and currents are related by ${\frac {V_{s}}{V_{p}}}={\frac {N_{s}}{N_{p}}}$ and $I_{s}V_{s}=I_{p}V_{p}$.
* Mutual induction $emf_{2}=-M{\frac {\Delta I_{1}}{\Delta t}},\;emf_{1}=-M{\frac {\Delta I_{2}}{\Delta t}}$
* Self inductance $L={\frac {\Delta \Phi }{\Delta I}}$ where N is the number of turns and $I$ is the current in one turn.
* Self inductance of a long thin solenoid $emf=-L{\frac {\Delta I}{\Delta t}}$, and $L=\mu _{0}N^{2}A/\ell$ where $N$ is the number of turns, $A$ is area and $\ell$ is length, where μ$_0$ = 4π×10$^{−7}$ H/m ≈ 1.2566...×10$^{−6}$ N/A$^2$ or T⋅m/A or Wb/(A⋅m) or V⋅s/(A⋅m).
* $E_{ind}={\tfrac {1}{2}}LI^{2}$ is the energy stored in an inductor.

##### Electromagnetic Waves

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##### Geometric Optics

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##### Vision and Optical Instruments

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##### Wave Optics

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##### Special Relativity

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##### Introduction to Quantum Physics

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##### Atomic Physics

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##### Radioactivity and Nuclear Physics

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##### Medical Applications of Nuclear Physics

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##### Particle Physics

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##### Frontiers of Physics

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##### Footnotes

* ↑ frequency is revolutions per second and period is seconds per revolution: (rev/sec)(sec/rev)=1
* ↑ Note that v$_{max}$=ωX and a$_{max}$=ω$^2$X.
* ↑ Destructive interference occurs when two wave are out of phase; constructive interference occurs the waves are in phase and add to maximum amplitude.
* ↑ Audible frequencies typically range between 20 and 20,000 Hz. Infrasound (ultrasound) is below (above) that range.
* ↑ a=Γ$^2$ where &Gamma is defined using Wikipedia:reflection coefficient.
* ↑ For more information about this sum, see w:Special:Permalink/791000226#Superposition_principle.
* ↑ Here the area is assumed to be normal to the flow of current. Also, I=JA where J is current density.
* ↑ the latter follows from energy conservation