AcademyEquation Sheet

Academy

Equation Sheet

A curated Level 1 - Physics formula reference with a print-ready PDF view.

Course Formula Reference

Curated equations grouped by section and linked to their source topics.

Curated from the displayed equations in Level 1 - Physics topic content. Derivation-only steps, narrow example substitutions, naming-only expressions, and repeated rearrangements are intentionally omitted.

43 sections306 equationsDownload PDF

Measurement and Vectors

7 equations

Reported value

Measurement Uncertainty

\[x=x_{\mathrm{best}}\pm\Delta x\]

Fractional uncertainty

Measurement Uncertainty

\[\mathrm{fractional}=\frac{\Delta x}{|x_{\mathrm{best}}|}\]

Vector components

Resolving Vectors into Components

\[a_x=a\cos\theta,\qquad a_y=a\sin\theta\]

Vector magnitude

Resolving Vectors into Components

\[a=\sqrt{a_x^2+a_y^2}\]

Component form

Unit Vector Notation

\[\vec a=a_x\hat{\imath}+a_y\hat{\jmath}+a_z\hat{k}\]

Dot product

Dot and Cross Products

\[\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta\]

Cross product magnitude

Dot and Cross Products

\[|\vec a\times\vec b|=|\vec a||\vec b|\sin\theta\]

One-Dimensional Motion

7 equations

Average velocity

Average Velocity

\[\bar v_x=\frac{\Delta x}{\Delta t}\]

Instantaneous velocity

Instantaneous Velocity

\[v_x=\frac{dx}{dt}\]

Acceleration

Acceleration as a Rate of Change

\[a_x=\frac{dv_x}{dt}=\frac{d^2x}{dt^2}\]

Constant acceleration

Constant-Acceleration Models

\[v_x=v_{0x}+a_xt,\qquad x=x_0+v_{0x}t+\frac12a_xt^2\]

No-time form

Constant-Acceleration Models

\[v_x^2=v_{0x}^2+2a_x(x-x_0)\]

Free fall

Free-Fall Motion

\[a_y=-g,\qquad y=y_0+v_{0y}t-\frac12gt^2\]

Integral relationships

Position and Velocity from Calculus

\[\Delta x=\int_{t_i}^{t_f}v_x(t)\,dt,\qquad \Delta v_x=\int_{t_i}^{t_f}a_x(t)\,dt\]

Motion in Space

6 equations

Position and velocity

Position and Velocity Vectors

\[\vec r=x\hat{\imath}+y\hat{\jmath}+z\hat{k},\qquad \vec v=\frac{d\vec r}{dt}\]

Speed

Position and Velocity Vectors

\[v=|\vec v|=\sqrt{v_x^2+v_y^2+v_z^2}\]

Projectile position

Projectile Models

\[x=x_0+v_0\cos\theta\,t,\qquad y=y_0+v_0\sin\theta\,t-\frac12gt^2\]

Level-ground range

Projectile Models

\[R=\frac{v_0^2\sin 2\theta}{g}\]

Circular kinematics

Circular Motion Kinematics

\[v=R|\omega|,\qquad a_c=R\omega^2=\frac{v^2}{R}\]

Relative motion

Relative Motion

\[\vec r_{PA}=\vec r_{PB}+\vec r_{BA},\qquad \vec v_{PA}=\vec v_{PB}+\vec v_{BA}\]

Forces and Newton's Laws

5 equations

Newton's second law

Force, Mass, and Acceleration

\[\sum\vec F=m\vec a\]

Component law

Force, Mass, and Acceleration

\[\sum F_x=ma_x,\qquad \sum F_y=ma_y\]

Weight

Weight Versus Mass

\[\vec W=m\vec g,\qquad W=mg\]

Newton's third law

Action-Reaction Pairs

\[\vec F_{AB}=-\vec F_{BA}\]

Incline components

Drawing Free-Body Diagrams

\[W_{\parallel}=mg\sin\theta,\qquad W_{\perp}=mg\cos\theta\]

Applying Force Models

5 equations

Force equilibrium

Equilibrium with Newton's First Law

\[\sum\vec F=\vec 0\]

Friction

Static and Kinetic Friction

\[0\le f_s\le\mu_sN,\qquad f_k=\mu_kN\]

Radial dynamics

Dynamics of Circular Motion

\[\sum F_r=m\frac{v^2}{r}=mr\omega^2\]

Flat curve limit

Dynamics of Circular Motion

\[v_{\max}=\sqrt{\mu_sgr}\]

Banked curve

Dynamics of Circular Motion

\[\tan\theta=\frac{v^2}{rg}\]

Work, Energy, and Power

5 equations

Constant-force work

Work by a Force

\[W=\vec F\cdot\Delta\vec r=F\Delta r\cos\theta\]

Variable-force work

Energy Transfer with Variable Forces

\[W=\int_{\vec r_i}^{\vec r_f}\vec F\cdot d\vec r\]

Kinetic energy

Kinetic Energy and the Work-Energy Principle

\[K=\frac12mv^2\]

Work-energy principle

Kinetic Energy and the Work-Energy Principle

\[W_{\mathrm{net}}=\Delta K=K_f-K_i\]

Power

Power and Rate of Work

\[P=\frac{dW}{dt}=\vec F\cdot\vec v\]

Potential Energy and Conservation

5 equations

Near-Earth gravitational energy

Gravitational Potential Energy

\[U_g=mgy,\qquad \Delta U_g=mg\Delta y\]

Spring energy

Elastic Potential Energy

\[F_s=-kx,\qquad U_s=\frac12kx^2\]

Mechanical energy accounting

Conservative Forces

\[K_i+U_i+W_{nc}=K_f+U_f\]

Force from potential

Force from Potential Energy

\[F_x=-\frac{dU}{dx},\qquad \vec F=-\nabla U\]

Speed from energy

Reading Energy Diagrams

\[v=\sqrt{\frac{2(E-U)}{m}}\]

Momentum Systems

7 equations

Momentum

Momentum and Impulse

\[\vec p=m\vec v\]

Impulse-momentum

Momentum and Impulse

\[\vec J=\int_{t_1}^{t_2}\vec F\,dt=\Delta\vec p\]

Momentum conservation

Momentum Conservation

\[\vec P_f=\vec P_i\quad(\vec J_{\mathrm{ext}}=\vec 0)\]

Center of mass

Center of Mass

\[\vec r_{\mathrm{cm}}=\frac{1}{M}\sum_i m_i\vec r_i\]

Center-of-mass dynamics

Center of Mass

\[\sum\vec F_{\mathrm{ext}}=M\vec a_{\mathrm{cm}}\]

Elastic collision

Elastic Collision Models

\[m_1u_1+m_2u_2=m_1v_1+m_2v_2,\qquad K_i=K_f\]

Rocket equation

Rocket Motion

\[\Delta v=v_e\ln\left(\frac{m_0}{m_f}\right)\]

Rigid-Body Rotation

10 equations

Angular velocity

Angular Velocity

\[\omega=\frac{d\theta}{dt}=2\pi f=\frac{2\pi}{T}\]

Angular acceleration

Angular Acceleration

\[\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}\]

Constant angular acceleration

Constant Angular Acceleration

\[\omega=\omega_0+\alpha t,\qquad \theta=\theta_0+\omega_0t+\frac12\alpha t^2\]

No-time angular form

Constant Angular Acceleration

\[\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)\]

Linear-angular links

Connecting Linear and Angular Motion

\[s=r\theta,\qquad v_t=r\omega,\qquad a_t=r\alpha\]

Centripetal acceleration

Connecting Linear and Angular Motion

\[a_c=r\omega^2=\frac{v_t^2}{r}\]

Moment of inertia

Calculating Moments of Inertia

\[I=\sum_i m_ir_{\perp i}^2,\qquad I=\int r_\perp^2\,dm\]

Parallel-axis theorem

Parallel-Axis Theorem

\[I=I_{\mathrm{cm}}+Md^2\]

Rotational kinetic energy

Rotational Kinetic Energy

\[K_{\mathrm{rot}}=\frac12I\omega^2\]

Rolling kinetic energy

Rotational Kinetic Energy

\[K=\frac12Mv_{\mathrm{cm}}^2+\frac12I_{\mathrm{cm}}\omega^2\]

Rotational Dynamics

7 equations

Torque

\[\vec\tau=\vec r\times\vec F,\qquad \tau=rF\sin\phi\]

Rotational law

Torque and Angular Acceleration

\[\sum\tau_z=I\alpha\]

Rotational work and power

Work and Power in Rotation

\[W=\int_{\theta_i}^{\theta_f}\tau\,d\theta,\qquad P=\tau\omega\]

Angular momentum

Angular Momentum

\[\vec L=\vec r\times\vec p,\qquad L=I\omega\]

Torque and angular momentum

Angular Momentum

\[\vec\tau_{\mathrm{ext}}=\frac{d\vec L}{dt}\]

Angular momentum conservation

Conserving Angular Momentum

\[\tau_{\mathrm{ext}}=0\Rightarrow L_i=L_f\]

Slow precession

Gyroscopes and Precession

\[\Omega=\frac{\tau}{L}=\frac{Mgd}{I_s\omega_s}\]

Equilibrium and Materials

7 equations

Static equilibrium

Conditions for Static Equilibrium

\[\sum\vec F=\vec 0,\qquad \sum\vec\tau=\vec 0\]

Center of gravity

Center of Gravity

\[x_{\mathrm{cg}}=\frac{\sum_i m_ix_i}{\sum_i m_i}\]

Normal stress and strain

Stress and Strain

\[\sigma=\frac{F_\perp}{A},\qquad \epsilon=\frac{\Delta L}{L_0}\]

Shear stress and strain

Stress and Strain

\[\tau=\frac{F_\parallel}{A},\qquad \gamma=\frac{\Delta x}{h}\]

Young modulus

Elastic Moduli

\[Y=\frac{\sigma}{\epsilon}=\frac{FL_0}{A\Delta L}\]

Bulk modulus

Elastic Moduli

\[B=-\frac{\Delta p}{\Delta V/V}\]

Elastic energy density

Elastic Versus Plastic Behavior

\[u=\frac12\sigma\epsilon\]

Fluids

8 equations

Density

Density in Gases and Liquids

\[\rho=\frac{m}{V}\]

Pressure

Pressure in Fluids

\[p=\frac{F_\perp}{A}\]

Hydrostatic pressure

Pressure in Fluids

\[p=p_0+\rho gh\]

Buoyant force

Buoyancy

\[F_B=\rho_fgV_{\mathrm{disp}}\]

Continuity

Fluid Flow

\[\rho_1A_1v_1=\rho_2A_2v_2,\qquad A_1v_1=A_2v_2\ \text{if incompressible}\]

Bernoulli equation

Bernoulli Models

\[p+\frac12\rho v^2+\rho gy=\mathrm{constant}\]

Viscous law

Viscosity and Turbulence

\[\tau=\eta\frac{dv}{dy}\]

Reynolds number

Viscosity and Turbulence

\[\mathrm{Re}=\frac{\rho vL}{\eta}\]

Gravitation

6 equations

Universal gravitation

Universal Gravitation

\[F_g=G\frac{m_1m_2}{r^2}\]

Gravitational field

Universal Gravitation

\[\vec g=-G\frac{M}{r^2}\hat e_r,\qquad \vec F_g=m\vec g\]

Circular orbit

Satellite Motion

\[v=\sqrt{\frac{GM}{r}},\qquad T=2\pi\sqrt{\frac{r^3}{GM}}\]

Kepler third law

Kepler Models of Orbits

\[T^2=\frac{4\pi^2}{GM}a^3\]

Escape speed

Black Holes

\[v_{\mathrm{esc}}=\sqrt{\frac{2GM}{r}}\]

Schwarzschild radius

Black Holes

\[r_s=\frac{2GM}{c^2}\]

Oscillations

8 equations

Frequency and angular frequency

Describing Oscillation

\[f=\frac1T,\qquad \omega=2\pi f=\frac{2\pi}{T}\]

SHM condition

Simple Harmonic Motion

\[a=-\omega^2x\]

Mass-spring oscillator

Simple Harmonic Motion

\[\omega=\sqrt{\frac{k}{m}},\qquad T=2\pi\sqrt{\frac{m}{k}}\]

SHM displacement

Simple Harmonic Motion

\[x=A\cos(\omega t+\phi)\]

Oscillator energy

Energy in Oscillators

\[E=\frac12mv^2+\frac12kx^2=\frac12kA^2\]

Simple pendulum

Simple Pendulums

\[\omega=\sqrt{\frac{g}{L}},\qquad T=2\pi\sqrt{\frac{L}{g}}\]

Physical pendulum

Physical Pendulums

\[T=2\pi\sqrt{\frac{I}{mgd}}\]

Damped frequency

Damping

\[\omega_d=\sqrt{\omega_0^2-\beta^2}\]

Mechanical Waves

8 equations

Wave speed

Periodic Wave Patterns

\[v_w=f\lambda=\frac{\lambda}{T}\]

Sinusoidal wave

Mathematical Wave Descriptions

\[y(x,t)=A\cos(kx-\omega t+\phi)\]

Wave number and angular frequency

Mathematical Wave Descriptions

\[k=\frac{2\pi}{\lambda},\qquad \omega=\frac{2\pi}{T}\]

Phase speed

Mathematical Wave Descriptions

\[v_w=\frac{\omega}{k}\]

String wave speed

Speed of Transverse Waves

\[v_w=\sqrt{\frac{T}{\mu}}\]

Average wave power

Energy Transport in Waves

\[\langle P\rangle=\frac12\mu A^2\omega^2v_w\]

String normal modes

String Normal Modes

\[\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nv_w}{2L}\]

Superposition

\[y=y_1+y_2\]

Sound

9 equations

Sound wave speed

Sound as a Wave

\[v=f\lambda=\frac{\omega}{k}\]

Speed of sound

Speed of Sound

\[v=\sqrt{\frac{B}{\rho}},\qquad v=\sqrt{\frac{\gamma RT}{M}}\]

Intensity

Sound Intensity

\[I=\frac{P}{A},\qquad I=\frac{P}{4\pi r^2}\]

Decibel level

Sound Intensity

\[\beta=10\log_{10}\left(\frac{I}{I_0}\right)\]

Open-open pipe

Standing Sound Waves

\[\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nv}{2L}\]

Open-closed pipe

Standing Sound Waves

\[\lambda_n=\frac{4L}{2n-1},\qquad f_n=\frac{(2n-1)v}{4L}\]

Beat frequency

Beats

\[f_{\mathrm{beat}}=|f_1-f_2|\]

Sound Doppler shift

Doppler Shifts

\[f_L=f_S\frac{v+v_L}{v-v_S}\]

Mach number

Shock Waves

\[M=\frac{v_S}{v},\qquad \sin\theta=\frac1M\]

Temperature and Heat

7 equations

Kelvin and Celsius

Gas Thermometers and Absolute Temperature

\[T=T_C+273.15\]

Temperature-change heat

Heat as Energy Transfer

\[Q=mc\Delta T\]

Phase-change heat

Calorimetry and Phase Change

\[Q=\pm mL\]

Calorimetry

Calorimetry and Phase Change

\[\sum_iQ_i=0\]

Thermal expansion

Thermal Expansion

\[\Delta L=\alpha L_0\Delta T,\qquad \Delta V=\beta V_0\Delta T\]

Conduction

Heat Transfer Mechanisms

\[H=kA\frac{T_H-T_C}{L}\]

Thermal radiation

Heat Transfer Mechanisms

\[H=e\sigma AT^4,\qquad H_{\mathrm{net}}=e\sigma A(T^4-T_s^4)\]

Matter at Thermal Scale

7 equations

Ideal gas

Equations of State

\[pV=nRT=Nk_BT\]

Molecules and moles

Molecular Properties of Matter

\[N=nN_A,\qquad m=nM\]

Heat capacity

Heat Capacity

\[Q=mc\Delta T=nC_m\Delta T\]

Molecular pressure

Ideal Gas Model

\[pV=\frac13Nm_0v_{\mathrm{rms}}^2\]

Average molecular kinetic energy

Ideal Gas Model

\[\langle K_{\mathrm{tr}}\rangle=\frac32k_BT\]

RMS speed

Molecular Speeds

\[v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}\]

Clapeyron relation

Phases of Matter

\[\frac{dp}{dT}=\frac{L}{T\Delta v}\]

First-Law Thermodynamics

8 equations

Ideal gas state

Thermodynamic Systems

\[pV=nRT\]

Boundary work

Work During Volume Changes

\[W=\int_{V_i}^{V_f}p\,dV\]

First law

Internal Energy and the First Law

\[\Delta U=Q-W\]

Ideal-gas internal energy

Internal Energy of Ideal Gases

\[\Delta U=nC_V\Delta T,\qquad U=\frac{f}{2}nRT\]

Ideal-gas heat capacities

Heat Capacity of Ideal Gases

\[Q_V=nC_V\Delta T,\qquad Q_P=nC_P\Delta T\]

Mayer relation

Heat Capacity of Ideal Gases

\[C_P-C_V=R,\qquad \gamma=\frac{C_P}{C_V}\]

Adiabatic ideal gas

Adiabatic Ideal-Gas Processes

\[pV^\gamma=\mathrm{constant},\qquad TV^{\gamma-1}=\mathrm{constant}\]

Adiabatic work

Adiabatic Ideal-Gas Processes

\[W=nC_V(T_i-T_f)\]

Entropy and Heat Engines

5 equations

Entropy change

Entropy

\[\Delta S=\int_i^f\frac{\delta Q_{\mathrm{rev}}}{T}\]

Constant-temperature entropy

Entropy

\[\Delta S=\frac{Q_{\mathrm{rev}}}{T}\]

Phase-change entropy

Entropy

\[\Delta S=\frac{mL}{T}\]

Ideal-gas entropy

Entropy

\[\Delta S=nC_V\ln\left(\frac{T_f}{T_i}\right)+nR\ln\left(\frac{V_f}{V_i}\right)\]

Second-law test

Entropy

\[\Delta S_{\mathrm{univ}}=\Delta S_{\mathrm{sys}}+\Delta S_{\mathrm{surr}}\ge0\]

Electric Charge and Fields

7 equations

Quantized charge

Electric Charge

\[q=Ne,\qquad e=1.60\times10^{-19}\,\mathrm C\]

Coulomb force

Coulomb Forces

\[F=k\frac{|q_1q_2|}{r^2},\qquad k=\frac{1}{4\pi\epsilon_0}\]

Electric field and force

Electric Fields and Forces

\[\vec E=\frac{\vec F_0}{q_0},\qquad \vec F=q\vec E\]

Point-charge field

Calculating Electric Fields

\[\vec E=k\frac{Q}{R^2}\hat R\]

Field superposition

Calculating Electric Fields

\[\vec E_{\mathrm{net}}=\sum_i\vec E_i\]

Electric dipole

Electric Dipoles

\[\vec p=q\vec d,\qquad \vec\tau=\vec p\times\vec E\]

Dipole energy

Electric Dipoles

\[U=-\vec p\cdot\vec E\]

Gauss's Law

6 equations

Electric flux

Charge and Electric Flux

\[\Phi_E=\int_S\vec E\cdot d\vec A\]

Gauss's law

Gauss's Law

\[\oint_S\vec E\cdot d\vec A=\frac{q_{\mathrm{enc}}}{\epsilon_0}\]

Conductor electrostatics

Charge on Conductors

\[\vec E=0\ \text{inside a conductor},\qquad \vec E_{\mathrm{outside}}=\frac{\sigma}{\epsilon_0}\hat n\]

Spherical symmetry

Symmetry Applications of Gauss's Law

\[E=\frac{q_{\mathrm{enc}}}{4\pi\epsilon_0r^2}\]

Line charge

Symmetry Applications of Gauss's Law

\[E=\frac{\lambda}{2\pi\epsilon_0r}\]

Infinite sheet

Symmetry Applications of Gauss's Law

\[E=\frac{\sigma}{2\epsilon_0}\]

Electric Potential

6 equations

Electric potential

Electric Potential

\[V=\frac{U}{q},\qquad \Delta U=q\Delta V\]

Electric work

Electric Potential

\[W_{\mathrm{elec}}=-q\Delta V\]

Point-charge potential energy

Electric Potential Energy

\[U=k\frac{q_1q_2}{r}\]

Point-source potential

Calculating Potential

\[V=k\frac{Q}{r}\]

Potential superposition

Calculating Potential

\[V=k\sum_i\frac{Q_i}{R_i}\]

Field from potential

Potential Gradients

\[E_x=-\frac{dV}{dx},\qquad \vec E=-\nabla V\]

Capacitors and Dielectrics

8 equations

Capacitance

\[C=\frac{Q}{\Delta V}\]

Parallel-plate capacitor

Capacitance

\[C=\epsilon_0\frac{A}{d}\]

Capacitor networks

Series and Parallel Capacitors

\[C_{\mathrm{eq}}=\sum_iC_i\ \text{(parallel)},\qquad \frac{1}{C_{\mathrm{eq}}}=\sum_i\frac{1}{C_i}\ \text{(series)}\]

Capacitor energy

Stored Energy and Field Energy

\[U=\frac{Q^2}{2C}=\frac12CV^2=\frac12QV\]

Electric field energy density

Stored Energy and Field Energy

\[u_E=\frac12\epsilon_0E^2\]

Dielectric capacitance

Dielectrics

\[C=\kappa C_0,\qquad \epsilon=\kappa\epsilon_0\]

Induced dipoles

Molecular View of Induced Charge

\[p=q\ell,\qquad U=-\vec p\cdot\vec E\]

Material Gauss law

Gauss's Law in Materials

\[\vec D=\epsilon_0\vec E+\vec P,\qquad \oint\vec D\cdot d\vec A=q_{\mathrm f}\]

Current and Resistance

8 equations

Current

Electric Current

\[I=\frac{dQ}{dt},\qquad J=\frac{I}{A}\]

Resistance and Ohm's law

Resistance

\[R=\frac{V}{I},\qquad V=IR\]

Uniform wire

Resistance

\[R=\rho\frac{L}{A}\]

Resistivity and conductivity

Resistivity

\[\vec E=\rho\vec J,\qquad \vec J=\sigma\vec E\]

Temperature model

Resistivity

\[\rho(T)=\rho_0[1+\alpha(T-T_0)]\]

Electric power

Electric Power

\[P=IV=I^2R=\frac{V^2}{R}\]

Source with internal resistance

emf and Circuit Models

\[V_{\mathrm{term}}=\mathcal E-Ir,\qquad I=\frac{\mathcal E}{R+r}\]

Drift current

Microscopic Model of Conduction

\[I=n|q|Av_d,\qquad \vec J=nq\vec v_d\]

DC Circuits

7 equations

Resistor networks

Series and Parallel Resistor Networks

\[R_{\mathrm{eq}}=R_1+R_2+\cdots,\qquad \frac{1}{R_{\mathrm{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots\]

Two parallel resistors

Series and Parallel Resistor Networks

\[R_{\mathrm{eq}}=\frac{R_1R_2}{R_1+R_2}\]

Kirchhoff laws

Kirchhoff Analysis

\[\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}},\qquad \sum_{\mathrm{loop}}\Delta V=0\]

RC time constant

RC Transients

\[\tau=RC\]

RC charging

RC Transients

\[V_C(t)=\mathcal E(1-e^{-t/\tau}),\qquad I(t)=\frac{\mathcal E}{R}e^{-t/\tau}\]

RC discharging

RC Transients

\[V_C(t)=V_0e^{-t/\tau}\]

Line loss and efficiency

Power Distribution

\[P_{\mathrm{loss}}=I^2R_{\mathrm{line}},\qquad \eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\]

Magnetic Fields and Forces

8 equations

Magnetic force on charge

Magnetic Fields

\[\vec F_B=q\vec v\times\vec B,\qquad F_B=|q|vB\sin\theta\]

Magnetic flux

Magnetic Flux

\[\Phi_B=BA\cos\theta=\int\vec B\cdot d\vec A\]

Circular magnetic motion

Charged Particles in Magnetic Fields

\[r=\frac{mv_\perp}{|q|B},\qquad \omega=\frac{|q|B}{m}\]

Force on current

Forces on Current-Carrying Conductors

\[\vec F=I\vec L\times\vec B,\qquad F=ILB\sin\theta\]

Current-loop torque

Torque on Current Loops

\[\mu=NIA,\qquad \vec\tau=\vec\mu\times\vec B,\qquad \tau=NIAB\sin\theta\]

Magnetic dipole energy

Torque on Current Loops

\[U=-\vec\mu\cdot\vec B\]

Hall voltage

Hall Effect

\[V_H=\frac{IB}{nqt}\]

Velocity selector and rigidity

Particle Motion Applications

\[v=\frac{E}{B},\qquad p=|q|Br\]

Sources of Magnetic Fields

7 equations

Moving charge field

Fields from Moving Charges

\[\vec B=\frac{\mu_0}{4\pi}\frac{q\vec v\times\hat r}{r^2}\]

Biot-Savart law

Biot-Savart for Current Elements

\[d\vec B=\frac{\mu_0}{4\pi}\frac{I\,d\vec l\times\hat r}{r^2}\]

Long straight wire

Fields Around Straight Currents

\[B=\frac{\mu_0I}{2\pi r}\]

Current loop field

Fields from Current Loops

\[B(x)=\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}},\qquad B_{\mathrm{center}}=\frac{\mu_0I}{2R}\]

Parallel currents

Forces Between Parallel Currents

\[\frac{F}{L}=\frac{\mu_0I_1I_2}{2\pi d}\]

Ampere's law

Ampere's Law

\[\oint\vec B\cdot d\vec l=\mu_0I_{\mathrm{enc}}\]

Solenoid and toroid

Symmetry Applications of Ampere's Law

\[B=\mu_0nI\ \text{(solenoid)},\qquad B=\frac{\mu_0NI}{2\pi r}\ \text{(toroid)}\]

Electromagnetic Induction

8 equations

Magnetic flux

Induction Observations

\[\Phi_B=\int\vec B\cdot d\vec A\]

Faraday's law

Faraday's Law

\[\mathcal E=-N\frac{d\Phi_B}{dt}\]

Motional emf

\[|\mathcal E|=B\ell v\]

Field form of Faraday's law

Induced Electric Fields

\[\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}\]

Displacement current

Displacement Current

\[I_d=\epsilon_0\frac{d\Phi_E}{dt}\]

Maxwell equations

Maxwell's Equations

\[\oint\vec E\cdot d\vec A=\frac{q_{\mathrm{enc}}}{\epsilon_0},\quad \oint\vec B\cdot d\vec A=0,\quad \oint\vec B\cdot d\vec\ell=\mu_0I_{\mathrm{enc}}+\mu_0\epsilon_0\frac{d\Phi_E}{dt}\]

Electromagnetic wave speed

Maxwell's Equations

\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]

Flux quantum

Superconductivity

\[\Phi_0=\frac{h}{2e},\qquad \Phi=n\Phi_0\]

Inductance

8 equations

Self-inductance

Self-Inductance and Inductors

\[L=\frac{N\Phi_B}{I},\qquad \mathcal E_L=-L\frac{dI}{dt}\]

Solenoid inductance

Self-Inductance and Inductors

\[L=\mu\frac{N^2A}{l}\]

Inductor energy

Magnetic Field Energy

\[U_B=\frac12LI^2,\qquad u_B=\frac{B^2}{2\mu}\]

RL time constant

RL Transients

\[\tau=\frac{L}{R}\]

RL transients

RL Transients

\[I(t)=\frac{\mathcal E}{R}(1-e^{-tR/L}),\qquad I(t)=I_0e^{-tR/L}\]

LC oscillations

LC Oscillations

\[\omega_0=\frac{1}{\sqrt{LC}},\qquad T=2\pi\sqrt{LC}\]

LC energy

LC Oscillations

\[\frac{q^2}{2C}+\frac12LI^2=\mathrm{constant}\]

Mutual inductance

Mutual Inductance

\[M=\frac{N_2\Phi_{21}}{I_1},\qquad \mathcal E_2=-M\frac{dI_1}{dt}\]

Alternating Current

8 equations

AC frequency and rms

Phasors and AC Signals

\[\omega=2\pi f,\qquad V_{\mathrm{rms}}=\frac{V_0}{\sqrt2},\qquad I_{\mathrm{rms}}=\frac{I_0}{\sqrt2}\]

Reactance

Resistance and Reactance

\[X_C=\frac{1}{\omega C},\qquad X_L=\omega L\]

Series LRC impedance

LRC Series Circuits in AC

\[Z=\sqrt{R^2+(X_L-X_C)^2}\]

Series LRC current and phase

LRC Series Circuits in AC

\[I_{\mathrm{rms}}=\frac{V_{\mathrm{rms}}}{Z},\qquad \tan\phi=\frac{X_L-X_C}{R}\]

Average AC power

Power in AC Circuits

\[P_{\mathrm{avg}}=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi=I_{\mathrm{rms}}^2R\]

AC resonance

Resonance in AC Circuits

\[X_L=X_C,\qquad \omega_0=\frac{1}{\sqrt{LC}},\qquad f_0=\frac{1}{2\pi\sqrt{LC}}\]

Quality factor

Resonance in AC Circuits

\[Q=\frac{\omega_0L}{R}\]

Ideal transformer

Transformers

\[\frac{V_s}{V_p}=\frac{N_s}{N_p},\qquad V_pI_p=V_sI_s,\qquad \frac{I_s}{I_p}=\frac{N_p}{N_s}\]

Electromagnetic Waves

7 equations

Vacuum wave speed

Maxwell's Equations to Waves

\[c=\frac{1}{\sqrt{\mu_0\epsilon_0}}\]

Field ratio

Maxwell's Equations to Waves

\[\frac{E}{B}=c\]

Wave relations

Plane Waves and Light Speed

\[c=f\lambda=\frac{\omega}{k},\qquad n=\frac{c}{v}\]

Poynting vector

Energy and Momentum in Electromagnetic Waves

\[\vec S=\frac{1}{\mu_0}\vec E\times\vec B\]

Average intensity

Energy and Momentum in Electromagnetic Waves

\[I=\frac12c\epsilon_0E_0^2\]

Radiation pressure

Energy and Momentum in Electromagnetic Waves

\[p_{\mathrm{rad}}=\frac{I}{c}\ \text{or}\ \frac{2I}{c}\]

Standing EM modes

Standing Electromagnetic Waves

\[\lambda_n=\frac{2L}{n},\qquad f_n=\frac{nc}{2L}\]

Light Propagation

6 equations

Light wave relation

Nature of Light

\[v=f\lambda,\qquad c=3.00\times10^8\,\mathrm{m\,s^{-1}}\]

Refractive index

Nature of Light

\[n=\frac{c}{v},\qquad \lambda=\frac{\lambda_0}{n}\]

Reflection and refraction

Reflection and Refraction

\[\theta_r=\theta_i,\qquad n_1\sin\theta_1=n_2\sin\theta_2\]

Critical angle

Total Internal Reflection

\[\theta_c=\sin^{-1}\left(\frac{n_2}{n_1}\right)\quad(n_1>n_2)\]

Polarization

\[I=I_0\cos^2\theta,\qquad \tan\theta_B=\frac{n_2}{n_1}\]

Rayleigh scattering

Scattering

\[I_{\mathrm{scattered}}\propto\lambda^{-4}\]

Geometric Optics

8 equations

Plane mirror

Plane-Surface Reflection

\[\theta_r=\theta_i,\qquad s'=-s,\qquad m=+1\]

Spherical mirror

Spherical Mirrors

\[f=\frac{R}{2},\qquad \frac{1}{s}+\frac{1}{s'}=\frac{1}{f},\qquad m=-\frac{s'}{s}\]

Spherical refraction

Spherical Refraction

\[\frac{n_1}{s}+\frac{n_2}{s'}=\frac{n_2-n_1}{R}\]

Thin lens

Thin Lenses

\[\frac{1}{s}+\frac{1}{s'}=\frac{1}{f},\qquad m=-\frac{s'}{s}\]

Lens power and lensmaker

Thin Lenses

\[P=\frac1f,\qquad \frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)\]

Magnifier

Magnifiers

\[M=\frac{N}{f}\ \text{(relaxed)},\qquad M=1+\frac{N}{f}\ \text{(near point)}\]

Microscope

Microscopes

\[M\approx-\frac{LN}{f_of_e}\]

Telescope

Telescopes

\[M=-\frac{f_o}{f_e},\qquad L=f_o+f_e\]

Interference

6 equations

Path and phase difference

Coherence and Interference

\[\Delta\phi=\frac{2\pi\Delta r}{\lambda}\]

Interference conditions

Coherence and Interference

\[\Delta r=m\lambda\ \text{(constructive)},\qquad \Delta r=\left(m+\frac12\right)\lambda\ \text{(destructive)}\]

Double-slit fringes

Two-Source Light Interference

\[d\sin\theta=m\lambda,\qquad y_m\approx\frac{m\lambda L}{d},\qquad \Delta y\approx\frac{\lambda L}{d}\]

Two-beam intensity

Intensity in Interference Patterns

\[I=4I_0\cos^2\left(\frac{\delta}{2}\right)\]

Thin-film reflection

Thin-Film Interference

\[2nt=\left(m+\frac12\right)\lambda_0\ \text{(constructive)},\qquad 2nt=m\lambda_0\ \text{(destructive)}\]

Michelson displacement

Michelson Interferometer

\[N=\frac{2\Delta L}{\lambda},\qquad \Delta L=\frac{N\lambda}{2}\]

Diffraction

7 equations

Fresnel number

Fresnel and Fraunhofer Regimes

\[N_F=\frac{a^2}{\lambda L}\]

Single-slit minima

Single-Slit Diffraction

\[a\sin\theta_m=m\lambda,\qquad y_m\approx\frac{m\lambda L}{a}\]

Single-slit intensity

Intensity in Single-Slit Patterns

\[I=I_0\left(\frac{\sin\beta}{\beta}\right)^2,\qquad \beta=\frac{\pi a\sin\theta}{\lambda}\]

Diffraction grating

Diffraction Gratings

\[d\sin\theta_m=m\lambda,\qquad d=\frac1{n_L}\]

Grating resolving power

Diffraction Gratings

\[R=mN\]

Bragg law

X-Ray Diffraction

\[2d\sin\theta=m\lambda\]

Rayleigh criterion

Circular Apertures and Resolution

\[\theta_R\approx1.22\frac{\lambda}{D},\qquad s_{\min}\approx1.22\frac{\lambda L}{D}\]

Relativity

9 equations

Lorentz factor

Invariance of Physical Laws

\[\beta=\frac{v}{c},\qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}\]

Time dilation

Time Dilation

\[\Delta t=\gamma\Delta\tau\]

Length contraction

Length Contraction

\[L=\frac{L_0}{\gamma}\]

Lorentz transformation

Lorentz Transformations

\[x'=\gamma(x-vt),\qquad t'=\gamma\left(t-\frac{vx}{c^2}\right)\]

Velocity addition

Lorentz Transformations

\[u_x'=\frac{u_x-v}{1-u_xv/c^2}\]

Relativistic Doppler effect

Relativistic Doppler Effect

\[f_o=f_s\sqrt{\frac{1+\beta}{1-\beta}}\ \text{(approaching)},\qquad f_o=f_s\sqrt{\frac{1-\beta}{1+\beta}}\ \text{(receding)}\]

Relativistic momentum

Relativistic Momentum

\[\vec p=\gamma m\vec v\]

Relativistic energy

Relativistic Work and Energy

\[E=\gamma mc^2,\qquad E_0=mc^2,\qquad K=(\gamma-1)mc^2\]

Energy-momentum relation

Relativistic Work and Energy

\[E^2=(pc)^2+(mc^2)^2\]

Photons

8 equations

Photon energy and momentum

Wave-Particle Duality

\[E=hf=\frac{hc}{\lambda},\qquad p=\frac{h}{\lambda}\]

Photoelectric effect

Photoelectric Effect

\[K_{\max}=hf-\phi,\qquad eV_s=K_{\max}\]

Photoelectric thresholds

Photoelectric Effect

\[f_0=\frac{\phi}{h},\qquad \lambda_0=\frac{hc}{\phi}\]

X-ray cutoff

X-Ray Production

\[K=eV,\qquad \lambda_{\min}=\frac{hc}{eV}\]

Compton shift

Compton Scattering

\[\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta)\]

Pair production threshold

Pair Production

\[E_{\mathrm{th}}=2m_ec^2=1.022\,\mathrm{MeV}\]

Uncertainty principle

Uncertainty Principle

\[\Delta x\,\Delta p\ge\frac{\hbar}{2},\qquad \Delta E\,\Delta t\gtrsim\frac{\hbar}{2}\]

de Broglie wavelength

Wave-Particle Duality

\[\lambda=\frac{h}{p}\]

Matter Waves

7 equations

Electron de Broglie wavelength

Electron Waves

\[\lambda=\frac{h}{p},\qquad \lambda=\frac{h}{\sqrt{2m_eeV}}\]

Atomic photon energy

Atomic Spectra

\[E_\gamma=hf=\frac{hc}{\lambda}=E_i-E_f\]

Bohr model

Bohr Energy Levels

\[m_evr_n=n\hbar,\qquad r_n=n^2a_0\]

Hydrogen energies

Bohr Energy Levels

\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2}\]

Wien law

Continuous Spectra

\[\lambda_{\max}T=b\]

Closest approach

Nuclear Atom

\[r_{\min}=\frac{2kZe^2}{K}\]

Confinement estimate

Uncertainty Revisited

\[K\sim\frac{\hbar^2}{2mL^2}\]

Quantum Wave Functions

8 equations

Normalization and probability

Wave Functions

\[\int_{-\infty}^{\infty}|\psi|^2\,dx=1,\qquad P_{[a,b]}=\int_a^b|\psi|^2\,dx\]

Expectation value

Wave Functions

\[\langle x\rangle=\int_{-\infty}^{\infty}x|\psi|^2\,dx\]

One-dimensional Schrodinger equation

One-Dimensional Schrodinger Equation

\[i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)\right]\psi\]

Stationary Schrodinger equation

One-Dimensional Schrodinger Equation

\[-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2}+V(x)\phi=E\phi\]

Particle in a box

Particle in a Box

\[\psi_n(x)=\sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}\]

Tunneling estimate

Potential Barriers and Tunneling

\[\kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar},\qquad T\sim e^{-2\kappa a}\]

Quantum harmonic oscillator

Harmonic Oscillator

\[V(x)=\frac12m\omega^2x^2,\qquad E_n=\left(n+\frac12\right)\hbar\omega\]

Quantum measurement

Measurement in Quantum Mechanics

Atomic Quantum Structure

8 equations

Three-dimensional Schrodinger equation

Three-Dimensional Schrodinger Equation

\[i\hbar\frac{\partial\psi}{\partial t}=\left[-\frac{\hbar^2}{2m}\nabla^2+U(\vec r)\right]\psi\]

Three-dimensional normalization

Three-Dimensional Schrodinger Equation

\[\int|\psi|^2\,dV=1\]

3D box energy

Particle in a Three-Dimensional Box

\[E=\frac{h^2}{8m}\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right)\]

Hydrogen atom

Hydrogen Atom

\[E_n=-\frac{13.6\,\mathrm{eV}}{n^2},\qquad L=\sqrt{\ell(\ell+1)}\hbar,\qquad L_z=m_\ell\hbar\]

Electron spin

Electron Spin

\[s=\frac12,\qquad S=\sqrt{s(s+1)}\hbar,\qquad S_z=m_s\hbar\]

Pauli capacities

Exclusion Principle

\[N_{\mathrm{orbital}}=2,\qquad N_e=2n^2\]

Zeeman effect

Zeeman Effect

\[\mu_B=\frac{e\hbar}{2m_e},\qquad \Delta E=m_\ell\mu_BB,\qquad \Delta f=\frac{\Delta E}{h}\]

Characteristic x-ray

X-Ray Spectra

\[E_\gamma=E_i-E_f,\qquad \lambda=\frac{hc}{E_\gamma}\]

Nuclear Physics

8 equations

Nuclear composition and radius

Nuclear Properties

\[A=Z+N,\qquad R=R_0A^{1/3}\]

Binding energy

Binding Energy

\[\Delta m=Zm_{\mathrm p}+Nm_{\mathrm n}-M_{\mathrm{nucleus}},\qquad B=\Delta mc^2\]

Radioactive decay

Radioactivity

\[N=N_0e^{-\lambda t},\qquad A=\lambda N,\qquad t_{1/2}=\frac{\ln2}{\lambda}\]

Radiometric dating

Activity and Half-Life

\[t=-\frac1\lambda\ln\left(\frac{N}{N_0}\right)\]

Radiation dose

Biological Effects of Radiation

\[D=\frac{E}{m},\qquad H=w_RD\]

Nuclear reaction conservation

Nuclear Reactions

\[\sum A_i=\sum A_f,\qquad \sum Z_i=\sum Z_f\]

Nuclear Q value

Nuclear Reactions

\[Q=(m_i-m_f)c^2\]

Coulomb barrier scale

Fusion

\[U_C\approx\frac{1}{4\pi\epsilon_0}\frac{Z_1Z_2e^2}{r}\]

Particles and Cosmology

6 equations

Accelerator energy and track radius

Accelerators and Detectors

\[\Delta K=qV,\qquad p=|q|Br\]

Rest energy

History of Fundamental Particles

\[E_0=mc^2\]

Particle conservation

Particle Families

\[\sum Q_i=\sum Q_f,\qquad \sum B_i=\sum B_f\]

Quark charges

Quarks and Gluons

\[Q_u=+\frac23,\qquad Q_d=Q_s=-\frac13\]

Hubble law and redshift

Expanding Universe

\[v=H_0d,\qquad 1+z=\frac{\lambda_{\mathrm{obs}}}{\lambda_{\mathrm{emit}}}=\frac{a_0}{a_{\mathrm{emit}}}\]

Early-universe scaling

Early Universe

\[T(z)=T_0(1+z),\qquad E\sim k_BT\]

Flashcards

Physics Models