AP Physics B Modern Physics

AP Physics B: Modern Physics

EQUATIONS

INFORMATION

Velocity Coefficient

$Velocity Coefficient$

$\\ \bold{\beta} = velocity \hspace{4} coefficient \\ \bold{v} = velocity \hspace{4} (m \hspace{2} s^{-1}) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \hspace{4} (3.00 \hspace{2} x \hspace{2} 10^8 \hspace{2} m \hspace{2} s^{-1})$

Lorentz Factor

$Lorentz Factor$

$\\ \bold{\gamma} = Lorentz \hspace{4} factor \\ \bold{\beta} = velocity \hspace{4} coefficient$

Lorentz Transformations

$Lorentz Transformations$

$\\ \bold{t | t'} = time \hspace{4} measured \hspace{4} by \\ \indent \hspace{14} observer | observer' \hspace{4} (s) \\ \bold{\gamma} = Lorentz \hspace{4} factor \\ \bold{v} = measured \hspace{4} scalar \hspace{4} velocity \hspace{4} (m \hspace{2} s^{-1}) \\ \bold{x|x'} = distance \hspace{4} in \hspace{4} x-direction \hspace{4} as \\ \indent \hspace{18} measured \hspace{4} by \hspace{4} observer \hspace{4} / \\ \indent \hspace{17} observer' \hspace{4} (m) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \hspace{4} (3.00 \hspace{2} x \hspace{2} 10^8 \hspace{2} m \hspace{2} s^{-1}) \\ \bold{y|y'} = distance \hspace{4} in \hspace{4} y-direction \hspace{4} as \\ \indent \hspace{18} measured \hspace{4} by \hspace{4} observer \hspace{4} / \\ \indent \hspace{16} observer' \hspace{4} (m) \\ \bold{z|z'} = distance \hspace{4} in \hspace{4} z-direction \hspace{4} as \\ \indent \hspace{18} measured \hspace{4} by \hspace{4} observer \hspace{4} / \\ \indent \hspace{16} observer' \hspace{4} (m) \\$

Time Dilation

$Time Dilation$

$\\ \bold{\Delta t} = time \hspace{4} between \hspace{4} two \hspace{4} co-local \hspace{4} events \\ \indent \hspace{14} for \hspace{4} an \hspace{4} observer \hspace{4} in \hspace{4} some \hspace{4} intertial \\ \indent \hspace{14} frame \hspace{4}(s)\\ \bold{\Delta t'} = time \hspace{4} between \hspace{4} two \hspace{4} co-local \\ \indent \hspace{14} events \hspace{4} for \hspace{4} another, \hspace{4} inertially \\ \indent \hspace{14} moving \hspace{4} with velocity \hspace{4} \bold{v}, w.r.t \hspace{4} the \\ \indent \hspace{11} \hspace{4} former \hspace{4} observer \hspace{4} (s)\\ \bold{\gamma} = Lorentz \hspace{4} factor\\$

Length Contraction

$Length Contraction$

$\\ \bold{L} = length \hspace{4} measured \hspace{4} by \hspace{4} observer \\ \indent \hspace{6} travelling \hspace{4} at \hspace{4} velocity \hspace{4}\bold{v} \hspace{4} (m) \\ \bold{L_0} = proper \hspace{4} length \hspace{4} (m) \\ \bold{\gamma} = Lorentz \hspace{4} factor$

Relativistic Linear Momentum

$Relativistic Linear Momentum$

$\\ \vec{p} = linear \hspace{4} momentum \hspace{4} (kg \hspace{2} m \hspace{2} s^{-1}) \\ \bold{\gamma} = Lorentz \hspace{4} factor \\ \bold{m} = mass \hspace{4} of \hspace{4} particle \hspace{4} (kg) \\ \vec{u} = velocity \hspace{4} of \hspace{4} the \hspace{4} particle \hspace{4} (m \hspace{2} s^{-1})$

Relativistic Kinetic Energy

$Relativistic Kinetic Energy$

$\\ \bold{K} = relativistic \hspace{4} kinetic \hspace{4} energy \hspace{4} (J) \\ \bold{\gamma} = Lorentz \hspace{4} factor \\ \bold{m} = mass \hspace{4} of \hspace{4} the \hspace{4} particle \hspace{4} (kg) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \\ \indent \hspace{3} (3.00 \times 10^8 m \hspace{2} s^{-1})$

Rest Energy

$Rest Energy$

$\\ \bold{E_0} = rest \hspace{4} energy \hspace{4} (J) \\ \bold{m} = mass \hspace{4} of \hspace{4} the \hspace{4} particle \hspace{4} (kg) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \\ \indent \hspace{3} (3.00 \times 10^8 m \hspace{2} s^{-1})$

Total Energy

$Total Energy$

$\\ \bold{E} = total \hspace{4} energy \hspace{4} (J) \\ \bold{\gamma} = Lorentz \hspace{4} factor \\ \bold{m} = mass \hspace{4} of \hspace{4} the \hspace{4} particle \hspace{4} (kg) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \\ \indent \hspace{3} (3.00 \times 10^8 m \hspace{2} s^{-1})$

Total Energy - Relativistic Linear Momentum Relationship

$Total Energy - Relativistic Linear Momentum Relationship$

$\\ \bold{E} = total \hspace{4} energy \hspace{4} (J) \\ \bold{p} = linear \hspace{4} momentum \hspace{4} (kg \hspace{2} m \hspace{2} s^{-1}) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \\ \indent \hspace{3} (3.00 \times 10^8 m \hspace{2} s^{-1}) \\ \bold{m} = mass \hspace{4} of \hspace{4} the \hspace{4} particle \hspace{4} (kg)$

Energy of a Photon

$Energy of a Photon$

$\\ \bold{E} = energy \hspace{4} of \hspace{4} a \hspace{4} photon \hspace{4} (J) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.626 \hspace{2}x \hspace{2} 10^{-34} J \hspace{2} s ) \\ \bold{\nu} = frequency \hspace{4} of \hspace{4} the \hspace{4} light \hspace{4}(Hz) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \hspace{4} (3.0 \hspace{2} x \hspace{2} 10^8 \hspace{4} m \hspace{2} s^{-1}) \\ \bold{\lambda} = wavelength \hspace{4} (m)$

Photoelectric Effect

$Photoelectric Effect$

$\\ \bold{K_{max}} = maximum \hspace{4} kinetic \hspace{4} energy \hspace{4} (J) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.626 \times 10^{-34} \hspace{2} J \hspace{2} s) \\ \bold{\nu} = frequency \hspace{4} of \hspace{4} light \hspace{4} (s^{-1}) \\ \bold{\phi} = work \hspace{4} function \hspace{4} (J)$

Compton Effect

$Compton Effect$

$\\ \bold{\lambda} = initial \hspace{4} wavelength \hspace{4} (m) \\ \bold{\lambda'} = wavelength \hspace{4} post - scattering \hspace{4} (m) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.626 \times 10^{-34} \hspace{2} J \hspace{2} s) \\ \bold{m_e} = rest \hspace{4} mass \hspace{4} of \hspace{4} electron \hspace{4} (kg) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \hspace{4} (3.00 \times 10^8 \hspace{2} m \hspace{2} s^{-1}) \\ \bold{\theta} = scattering \hspace{4} angle \hspace{4} (deg)$

De Broglie Relations

$De Broglie Relations$

$\\ \bold{E} = energy \hspace{4} of \hspace{4} a \hspace{4} photon \hspace{4} (J) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.625 \hspace{2}x \hspace{2} 10^{-34} J \hspace{2} s ) \\ \bold{\nu} = frequency \hspace{4} of \hspace{4} the \hspace{4} light \hspace{4}(Hz) \\ \bold{p} = momentum \hspace{4} (kg \hspace{2} m \hspace{2} s^{-1}) \\ \bold{\lambda} = wavelength \hspace{4} (m)$

Phase Speed

$Phase Speed$

$\\ \bold{v_{phase}} = phase \hspace{4} speed \hspace{4} (m \hspace{2} s^{-1}) \\ \bold{\omega} = angular \hspace{4} frequency \hspace{4} (s^{-1}) \\ \bold{k} = wavenumber \hspace{4} (m^{-1})$

Reduced Planck's Constant

$Reduced Planck's Constant$

$\\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{4} (1.055 \times 10^{-34} \hspace{2} J \hspace{2} s) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.626 \times 10^{-34} \hspace{2} J \hspace{2} s)$

Heisenberg's Uncertainty Principle

$Heisenberg's Uncertainty Principle$

$\\ \bold{\Delta x} = uncertainty \hspace{4} in \hspace{4} position \hspace{4} (m) \\ \bold{\Delta p} = uncertainty \hspace{4} in \hspace{4} momentum \hspace{4} (kg \hspace{2} m \hspace{2} s^{-1}) \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{5} (1.055 \times 10^{-34} \hspace{4} J \hspace{2} s)$

Heisenberg's Uncertainty Principle (Energy - Time)

$Heisenberg's Uncertainty Principle (Energy - Time)$

$\\ \bold{\Delta E} = uncertainty \hspace{4} in \hspace{4} energy \hspace{4} (J) \\ \bold{\Delta t} = uncertainty \hspace{4} in \hspace{4} time \hspace{4} (s) \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{5} (1.055 \times 10^{-34} \hspace{4} J \hspace{2} s)$

Balmer's Formula

$Balmer's Formula$

$\\ \bold{\lambda} = wavelength \hspace{4} (m) \\ \bold{R} = Rydberg \hspace{4} constant \hspace{4} (1.097 \hspace{2} x \hspace{2} 10^7 \hspace{2} m^{-1}) \\ \bold{n} = number \hspace{4} of \hspace{4} outer \hspace{4} orbit$

Bohr Formula

$Bohr Formula$

$\\ \bold{\lambda} = wavelength \hspace{4} (m) \\ \bold{R} = Rydberg \hspace{4} constant \hspace{4} (1.097 \hspace{2} x \hspace{2} 10^7 \hspace{2} m^{-1}) \\ \bold{N} = Number \hspace{4} of \hspace{4} inner \hspace{4} orbit \\ \bold{n} = number \hspace{4} of \hspace{4} outer \hspace{4} orbit$

Energies of Quantum States for Hydrogen

$Energies of Quantum States for Hydrogen$

$\\ \bold{E_n} = energy \hspace{4} of \hspace{4} quantum \hspace{4} state \hspace{4} n \hspace{4} (J) \\ \bold{k} = Coulomb \hspace{4} constant \\ \indent \hspace{4} (8.988 \times 10^9 \hspace{2} N \hspace{2} m^2 \hspace{4} C^{-2}) \\ \bold{e} = electronic \hspace{4} charge \hspace{4} (1.602 \times 10^{-19} \hspace{2} C) \\ \bold{a_0} = Bohr \hspace{4} radius \hspace{4} (5.292 \times 10^{-11} \hspace{2} m) \\ \bold{n} = quantum \hspace{4} number$

Atom's Orbital Angular Momentum

$Atom's Orbital Angular Momentum$

$\\ \bold{L} = orbital \hspace{4} angular \hspace{4} momentum \hspace{4} (kg \hspace{2} m^2 \hspace{2} s^{-1}) \\ \bold{n} = principal \hspace{4} quantum \hspace{4} number \\ \bold{l} = orbital \hspace{4}quantum \hspace{4} number \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{4} (1.055 \times 10^{-34} \hspace{2} J \hspace{2} s) \\$

Space Quantization

$Space Quantization$

$\\ \bold{L_z} = space \hspace{4} quantization \hspace{4} (kg \hspace{2} m^2 \hspace{2} s^{-1}) \\ \bold{m_l} = orbital \hspace{4} magnetic \hspace{4} quantum \hspace{4} number \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{4} (1.055 \times 10^{-34} \hspace{2} J \hspace{2} s) \\$

Spin Angular Momentum

$Spin Angular Momentum$

$\\ \bold{S} = spin \hspace{4} angular \hspace{4} momentum \\ \indent \hspace{4} (kg \hspace{2} m^2 \hspace{2} s^{-1}) \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{4} (1.055 \times 10^{-34} \hspace{2} J \hspace{2} s) \\$

Spin Angular Momentum (Z Component)

$Spin Angular Momentum (Z Component)$

$\\ \bold{S_z} = z-component \hspace{4} of \hspace{4} the \hspace{4} spin \\ \indent \hspace{10} angular \hspace{4} momentum \hspace{4} (kg \hspace{2} m^2 \hspace{2} s^{-1}) \\ \bold{m_s} = spin \hspace{4} magnetic \hspace{4} quantum \hspace{4} number \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{4} (1.055 \times 10^{-34} \hspace{2} J \hspace{2} s) \\$

Rotational Energy of a Diatomic Molecule

$Rotational Energy of a Diatomic Molecule$

$\\ \bold{E_{rot}} = rotational \hspace{4} energy \hspace{4} (J) \\ \bold{E_J} = rotational \hspace{4} energy \hspace{4} of \hspace{4} the \hspace{4} J^{th} \\ \indent \hspace{12} energy \hspace{4} state \hspace{4} (s) \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{5} (1.055 \times 10^{-34} \hspace{4} J \hspace{2} s) \\ \bold{I} = moment \hspace{4} of \hspace{4} intertia \hspace{4} (kg \hspace{2} m^2) \\ \bold{J} = rotational \hspace{4} quantum \hspace{4} number$

Vibrational Energy of a Diatomic Molecule

$Vibrational Energy of a Diatomic Molecule$

$\\ \bold{E_{vib}} = vibrational \hspace{4} energy \hspace{4} (J) \\ \bold{v} = vibrational \hspace{4} quantum \hspace{4} number \\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{5} (1.055 \times 10^{-34} \hspace{4} J \hspace{2} s) \\ \bold{k} = force \hspace{4} constant \hspace{4} (kg \hspace{2} s^{-2}) \\ \bold{\mu} = reduced \hspace{4} mass \hspace{4} (kg)$

Free - Electron Theory of Metals

$Free - Electron Theory of Metals$

$\\ \bold{N(E)} = number \hspace{4} of \hspace{4} states \hspace{4} (m^{-3}) \\ \bold{m_e} = mass \hspace{4} of \hspace{4} an \hspace{4} electron \hspace{4} (kg) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.626 \times 10^{-34} \hspace{4} J \hspace{2} s) \\ \bold{E} = electron \hspace{4} energy \hspace{4} level \hspace{4}(J) \\ \bold{E_F} = Fermi \hspace{4} energy \hspace{4} (J)\\ \bold{k} = Boltzmann \hspace{4} constant \\ \indent \hspace{4} (1.381 \times 10^{-23} \hspace{2} m^2 \hspace{2} kg \hspace{2} s^{-2} \hspace{2} K^{-1}) \\ \bold{T} = temperature \hspace{4} (K)$

Fermi Energy at Absolute Zero

$Fermi Energy at Absolute Zero$

$\\\bold{E_F} = Fermi \hspace{4} energy \hspace{4} (J) \\ \bold{h} = Planck's \hspace{4} constant \hspace{4} (6.626 \times 10^{-34} \hspace{4} J \hspace{2} s) \\ \bold{m_e} = mass \hspace{4} of \hspace{4} an \hspace{4} electron \hspace{4} (kg) \\ \bold{n_e} = total \hspace{4} number \hspace{4} of \hspace{4} conduction \\ \indent \hspace{10} electrons \hspace{4} per \hspace{4} unit \hspace{4} volume \hspace{4}(m^{-3})$

Nuclear Magneton

$Nuclear Magneton$

$\\ \bold{\mu_n} = nuclear \hspace{4} magneton \hspace{4} (5.05 \times 10^{-27} \hspace{2} J \hspace{2} T^{-1}) \\ \bold{e} = elementary \hspace{4} charge \hspace{4} (1.602 \times 10^{-19} \hspace{2} C)\\ \bold{\hbar} = reduced \hspace{4} Planck's \hspace{4} constant \\ \indent \hspace{4} (6.626 \times 10^{-34} \hspace{4} J \hspace{2} s) \\ \bold{m_p} = mass \hspace{4} of \hspace{4} a \hspace{4} proton \hspace{4} (1.673 \times 10^{-27} \hspace{2} kg)$

Nuclear Radius (Spherical)

$Nuclear Radius (Spherical)$

$\\ \bold{r} = nuclear \hspace{4} radius \hspace{4} (m) \\ \bold{A} = atomic \hspace{4} mass \hspace{4} number$

Nuclear Binding Energy

$Nuclear Binding Energy$

$\\ \bold{E_b} = binding \hspace{4} energy \hspace{4} (MeV) \\ \bold{Z} = number \hspace{4} of \hspace{4} electrons\\ \bold{M(H)} = atomic \hspace{4} mass \hspace{4} of \hspace{4} neutral \\ \indent \hspace{30} hydrogen \hspace{4} (u) \\ \bold{N} = number \hspace{4} of \hspace{4} particles\\ \bold{m_n} = mass \hspace{4} of \hspace{4} the \hspace{4} neutron \hspace{4} (u) \\ \bold{M(^A_ZX)} = atomic \hspace{4} mass \hspace{4} of \hspace{4} ^A_ZX \\ \indent \hspace{37} isotope \hspace{4} (u)$

Semiempirical Binding - Energy Formula

$Semiempirical Binding - Energy Formula$

$\\ \bold{E_b} = binding \hspace{4} energy \hspace{4} (MeV) \\ \bold{C_\&hash;} = coefficients \\ \bold{A} = total \hspace{4} number \hspace{4} of \hspace{4} nucleons \\ \bold{Z} = number \hspace{4} of \hspace{4} electrons\\ \bold{N} = number \hspace{4} of \hspace{4} particles\\$

Nuclear Decay

$Nuclear Decay$

$\\ \bold{N} = number \hspace{4} of \hspace{4} nuclei \hspace{4} remaining \\ \bold{N_0} = initial \hspace{4} number \hspace{4} of \hspace{4} nuclei \\ \bold{\lambda} = decay \hspace{4} constant (s^{-1}) \\ \bold{t} = time \hspace{4} (s)$

Half - Life

$Half - Life$

$\\ \bold{T_{1/2}} = half - life \hspace{4} (s) \\ \bold{\lambda} = decay \hspace{4} constant \hspace{4} (s^{-1}) \\ \bold{\tau} = average \hspace{4} lifetime \hspace{4} (s)$

Reaction Energy

$Reaction Energy$

$\\ for \hspace{4} reaction \hspace{4} of \hspace{4} type: \\ a + X \rightarrow Y + b \\\\ \bold{M_\&hash;} = mass \hspace{4} of \hspace{4} \&hash; \hspace{4} (kg) \\ \bold{c} = speed \hspace{4} of \hspace{4} light \hspace{4} (3.00 \times 10^8 \hspace{2} m \hspace{2} s^{-1})$

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