⌘K

The full library

266 formulas, every one with a live simulation. Search, filter, dive in.

Mechanics

F = ma

Newton's Second Law

Force equals mass times acceleration: heavier objects need more push.

elasticity

F = -kx

Hooke's Law

A spring pushes back proportionally to how far you stretch it.

gravitation

F = \frac{GMm}{r^2}

Newton's Law of Universal Gravitation

Every mass attracts every other mass; force drops with the square of distance.

energy

KE = \frac{1}{2}mv^2

Kinetic Energy

Energy of motion: doubles with mass, quadruples with speed.

circular motion

a_c = \frac{v^2}{r}

Centripetal Acceleration

Moving in a circle requires constant inward acceleration; faster or tighter = more.

kinematics

R = \frac{v^2 \sin(2\theta)}{g}

Projectile Range

Launch angle of 45° maximizes range; steeper or flatter angles fall shorter.

energy

W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2

Work-Energy Theorem

Net work on an object equals the change in its kinetic energy.

oscillations

T = 2\pi\sqrt{\frac{m}{k}}

Simple Harmonic Motion Period

Heavier masses oscillate slower; stiffer springs oscillate faster.

dynamics

p = mv

Linear Momentum

Momentum measures how hard it is to stop a moving object — it scales with both mass and speed.

dynamics

J = F \Delta t = \Delta p

Impulse-Momentum Theorem

Stopping the same momentum over a longer time means smaller force — the whole point of airbags, crumple zones, and bending your knees.

energy

U = mgh

Gravitational Potential Energy

Lifting something stores energy in the gravitational field — drop it and the energy reappears as motion.

energy

P = \frac{W}{t} = Fv

Mechanical Power

Power is how fast you spend energy. Same work, less time = more power.

rotation

\tau = rF\sin\theta

Torque

Torque is twist. The farther you push from the pivot, and the more perpendicular your push, the more spin you create.

rotation

L = I\omega

Angular Momentum

Angular momentum is rotational inertia times spin speed. Squeeze inward and you must spin faster to keep it conserved.

rotation

KE_{\text{rot}} = \tfrac{1}{2} I \omega^2

Rotational Kinetic Energy

Spinning things store energy just like moving things — and a small radius increase pays off big because I scales with r².

kinematics

v = \sqrt{2gh}

Free Fall Velocity

Drop something from height h in vacuum — by the time it hits, all the gravitational potential mgh has turned into kinetic ½mv².

oscillations

T = 2\pi \sqrt{\frac{L}{g}}

Simple Pendulum Period

Longer pendulums swing slower. The period depends only on length and gravity — not mass and (almost) not on amplitude.

rotational dynamics

I = mr^2

Moment of Inertia (Point Mass)

Mass placed far from the rotation axis resists rotation much more than the same mass placed close in.

gravitation

v_{\text{esc}} = \sqrt{\frac{2GM}{R}}

Escape Velocity

The speed at which kinetic energy exactly equals the gravitational binding energy — no faster, you fly free.

gravitation

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

Kepler's Third Law

Farther orbits take longer — and not linearly: the period grows as the 3/2 power of the orbital radius.

dynamics

F_k = \mu_k N

Kinetic Friction Force

Once an object is already sliding, friction resists motion with a force proportional to how hard the surfaces are pressed together.

dynamics

F_{s,\max} = \mu_s N

Maximum Static Friction

Static friction adjusts itself to match whatever you push with — up to a hard ceiling set by μ_s N. Cross that line and it gives way.

dynamics

N = m g \cos\theta

Normal Force on an Incline

The surface only needs to support the part of gravity pointing into it — cos θ of the weight.

dynamics

F_\parallel = m g \sin\theta

Gravity Component Along an Incline

Only the sine-component of gravity accelerates an object down the slope — the cosine-component is canceled by the normal force.

dynamics

T = \frac{2 m_1 m_2 g}{m_1 + m_2}

Atwood Machine Tension

The tension is twice the harmonic mean of the two weights — equal masses give T = mg (no motion), unequal masses give something in between.

dynamics

v_1' = \frac{(m_1 - m_2) v_1 + 2 m_2 v_2}{m_1 + m_2}

1D Elastic Collision (Final Velocities)

Conserve both momentum and kinetic energy and the velocities of the two bodies swap (when masses are equal) or rearrange linearly in mass ratios.

dynamics

v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}

Perfectly Inelastic Collision (1D)

Momentum is conserved, but kinetic energy is not — the bodies stick together and share a single final velocity equal to the system center-of-mass velocity.

dynamics

e = -\frac{v_1' - v_2'}{v_1 - v_2}

Coefficient of Restitution

A single dimensionless number, 0 ≤ e ≤ 1, that says how much of the relative speed survives a collision. e=1 is perfectly elastic; e=0 is perfectly inelastic.

dynamics

x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}

Center of Mass (Two Particles)

A mass-weighted average of positions — the single point that behaves like the whole system for Newton's second law.

rotational dynamics

I = I_{cm} + m d^2

Parallel Axis Theorem

Spinning about a shifted axis costs extra rotational inertia equal to m·d² — because every particle now traces a larger circle.

rotational dynamics

v = R \omega

Rolling Without Slipping

The contact point of a rolling wheel is instantaneously at rest — so the translation speed of the center equals the tangential speed at the rim.

rotational dynamics

I_1 \omega_1 = I_2 \omega_2

Conservation of Angular Momentum

With no external torque, the product Iω is fixed: pull mass closer to the axis and you must spin faster to compensate.

analytical mechanics

S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t)\, dt, \qquad \delta S = 0

Principle of Stationary Action

Of all imaginable paths between two events, nature takes the one whose action doesn't change under small wiggles.

analytical mechanics

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0

Euler–Lagrange Equation

Pick any coordinates you like, write energy in, equations of motion come out — no force diagrams needed.

analytical mechanics

p_i = \frac{\partial L}{\partial \dot{q}_i}

Generalized (Conjugate) Momentum

Each coordinate gets its own momentum — differentiate L by the velocity of that coordinate.

analytical mechanics

H(q, p, t) = \sum_i p_i \dot{q}_i - L

The Hamiltonian (Legendre Transform)

Swap each velocity for its momentum; what's left is (usually) the total energy as a function of position and momentum.

analytical mechanics

\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}

Hamilton's Canonical Equations

H is a landscape over phase space; states flow along its contour lines, position fed by momentum and momentum drained by force.

analytical mechanics

\{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i} \right)

Poisson Bracket & Liouville Flow

The bracket measures how two phase-space functions 'stir' each other; pairing anything with H gives its time evolution.

analytical mechanics

\frac{\partial L}{\partial q_i} = 0 \;\Rightarrow\; \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = 0

Noether's Theorem

Every continuous symmetry of the action hands you a conserved quantity — no exceptions, no extra work.

analytical mechanics

\mathcal{L} = \tfrac{1}{2}(m_1{+}m_2)l_1^2\dot\theta_1^2 + \tfrac{1}{2}m_2 l_2^2\dot\theta_2^2 + m_2 l_1 l_2 \dot\theta_1\dot\theta_2\cos(\theta_1{-}\theta_2) + (m_1{+}m_2)gl_1\cos\theta_1 + m_2 gl_2\cos\theta_2

Double Pendulum (Lagrangian Chaos)

Two pendulums chained together: the Lagrangian is easy to write, the motion is impossible to predict long-term.

gravitation

F_{\text{tidal}} \approx \frac{2GMmR}{d^{3}}

Gravity weakens with distance, so the Moon pulls Earth's near side harder than its center, and the center harder than the far side. In Earth's frame that difference stretches the planet along the Moon line and squeezes it sideways — two ocean bulges, two high tides a day. Because the effect goes as 1/d³ (not 1/d²), the nearby Moon out-tides the enormous Sun.

collisions

\sum_i m_i \vec{v}_i = \text{constant} \quad\Longleftrightarrow\quad m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2

Conservation of Linear Momentum

With no outside push, the total momentum of a system can't change — collisions only trade it between parts.

dynamics with drag

v_t = \sqrt{\dfrac{2 m g}{\rho\, C_d\, A}}

Terminal Velocity

You stop accelerating when quadratic air drag grows to exactly match your weight.

rotating frames

\vec{a}_{\text{Cor}} = -2\,\vec{\omega}\times\vec{v}

Coriolis Deflection

In a rotating frame, anything that moves gets pushed sideways — the frame, not a real force, does the deflecting.

rigid body dynamics

\vec{\tau} = \frac{d\vec{L}}{dt}, \qquad \Omega_p = \frac{\tau}{L} = \frac{\tau}{I\omega}

Gyroscopic Precession

A torque changes angular momentum's direction, not the spin's size, so the axle swings sideways instead of falling.

rigid body dynamics

\vec{L} = I_{\text{front}}\,\vec{\omega}_{\text{front}} + I_{\text{back}}\,\vec{\omega}_{\text{back}} = 0

The Falling-Cat Problem

Change your shape in a cycle and you can reorient with zero angular momentum — geometry rotates you, not spin.

ballistics

R = \frac{v^2 \sin 2\theta}{g}, \qquad v = \sqrt{\frac{2\,\eta\, M g h}{m}}

Physics of the Trebuchet

A heavy counterweight's drop becomes a light stone's speed; range then follows the projectile formula.

oscillations instability

\ddot{\theta} + 2\zeta_{\text{net}}\,\omega_n\,\dot{\theta} + \omega_n^2\,\theta = 0, \qquad \zeta_{\text{net}} = \zeta_{\text{struct}} - \zeta_{\text{aero}}(U)

Tacoma Narrows: Aeroelastic Flutter

Past a critical wind, the airflow pumps energy in faster than damping drains it, so oscillations blow up.

celestial mechanics

\Delta\phi = \frac{6\pi G M}{a\,(1-e^2)\,c^2} \quad \text{per orbit}

Perihelion Precession of Mercury

Relativity bends spacetime just enough that the orbit's closest point creeps forward each lap.

astrophysics

M_{\text{Ch}} \approx \frac{\omega_3^0\,\sqrt{3\pi}}{2}\left(\frac{\hbar c}{G}\right)^{3/2}\frac{1}{(\mu_e m_H)^2} \approx 1.44\left(\frac{2}{\mu_e}\right)^2 M_\odot

The Chandrasekhar Limit

Relativistic electron pressure scales like gravity, so above ~1.4 solar masses gravity always wins.

Electromagnetism

electrostatics

F = k_e \frac{|q_1 q_2|}{r^2}

Coulomb's Law

Electric force between two charges falls off as the square of the distance between them.

electrostatics

\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}

Electric Field of a Point Charge

Each charge creates a field that tells other charges how much force they would feel.

electrostatics

\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

Gauss's Law

The total electric flux through a closed surface equals the enclosed charge divided by ε₀.

electrostatics

V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}

Electric Potential (Point Charge)

Potential is the energy per unit charge — it falls off as 1/r, not 1/r².

electrostatics

C = \varepsilon_0 \frac{A}{d}

Parallel Plate Capacitor

Bigger plates and smaller gaps store more charge per volt.

circuits

V = IR

Ohm's Law

Voltage is the push, resistance is the friction, current is how much flows.

magnetostatics

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

Biot–Savart Law

Each bit of current creates a magnetic field perpendicular to both the current and distance.

magnetostatics

\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

Ampère's Law (with Maxwell's correction)

Magnetic field loops around currents; total circulation equals enclosed current times μ₀.

electromagnetic induction

\mathcal{E} = -\frac{d\Phi_B}{dt}

Faraday's Law of Induction

A changing magnetic flux through a loop induces a voltage that opposes the change.

electromagnetic dynamics

\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

Lorentz Force Law

Electric fields push charges; magnetic fields deflect moving charges sideways.

magnetostatics

\oint \vec{B} \cdot d\vec{A} = 0

Gauss's Law for Magnetism

Magnetic field lines always close on themselves — no isolated north or south poles.

electrostatics

U = \tfrac{1}{2} C V^2

Energy Stored in a Capacitor

To charge a capacitor you must push charge against a growing voltage — the work done is stored as electric field energy between the plates.

circuits

V(t) = V_0 \left(1 - e^{-t/\tau}\right), \quad \tau = RC

RC Circuit Charging

Current is largest at t = 0 when the capacitor is empty; as voltage builds up, current decays, asymptotically filling the capacitor to V₀.

magnetostatics

\vec{F} = I \vec{L} \times \vec{B}

Magnetic Force on a Current-Carrying Wire

Each moving charge in the wire feels a Lorentz force; summed over all of them, the wire experiences a net push perpendicular to both the current and the field.

magnetostatics

B = \mu_0 n I

Magnetic Field Inside a Solenoid

Each turn contributes a circular field; many tightly packed turns add up inside the tube and cancel outside — leaving a near-uniform interior field.

magnetostatics

L = \mu_0 n^2 V = \mu_0 \frac{N^2 A}{\ell}

Self-Inductance of a Solenoid

When current changes, flux through every loop changes, inducing an EMF that opposes the change. The geometric constant tying flux to current is the inductance.

magnetostatics

U = \tfrac{1}{2} L I^2

Energy Stored in an Inductor

To establish a current in an inductor you must do work against the back-EMF. That work survives as magnetic field energy stored in the coil's interior.

circuits

f = \frac{1}{2\pi \sqrt{LC}}

LC Oscillation Frequency

Energy sloshes between the capacitor's electric field and the inductor's magnetic field — a pure electromagnetic version of a mass-on-a-spring.

electromagnetic waves

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

Poynting Vector

Wherever both E and B exist, energy flows perpendicular to both. The Poynting vector tells you how much energy crosses a unit area per second, and in what direction.

electromagnetic induction

\varepsilon = B L v

Motional EMF

Free charges inside the moving rod feel a magnetic force qv×B that pushes them along the rod — positive charge piles up at one end until the resulting electric field balances the push. That charge separation is a battery made of motion.

electromagnetic dynamics

f_c = \frac{q B}{2\pi m}

Cyclotron Frequency

The magnetic force qvB always points to the center, bending the path into a circle. Faster particles ride bigger circles, but the extra path length exactly cancels the extra speed — every lap takes the same time.

circuits

I(t) = \frac{V_0}{R}\left(1 - e^{-t/\tau}\right), \quad \tau = \frac{L}{R}

RL Circuit Current Growth

An inductor hates changes in current: its back-EMF L·dI/dt initially cancels the battery entirely, so current starts at zero and creeps up. As the current settles, the back-EMF dies away and the resistor takes over.

electromagnetic induction

\frac{V_s}{V_p} = \frac{N_s}{N_p}

Ideal Transformer Equation

Both coils share the same changing flux through the iron core, so each turn of wire sees the same induced voltage. Voltage per coil is just volts-per-turn × turns — the ratio of turns is the ratio of voltages.

electromagnetic waves

c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

Speed of Electromagnetic Waves

A changing electric field makes a magnetic field (Ampère–Maxwell); a changing magnetic field makes an electric field (Faraday). The two regenerate each other in a self-sustaining ripple whose speed is fixed by how strongly each effect couples — ε₀ and μ₀.

electromagnetic dynamics

\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I + \varepsilon_0 \frac{d\Phi_E}{dt} \right)

Ampère–Maxwell Law

Maxwell noticed Ampère's law contradicts itself at a charging capacitor: a loop around the wire sees current, but the same loop with its surface bulged through the gap sees none. His fix: a changing electric field acts exactly like a current — the displacement current — and creates magnetic field just the same.

electromagnetic waves

P = \frac{q^2 a^2}{6 \pi \varepsilon_0 c^3}

Larmor Formula

A charge at rest has radial field lines. Accelerate it, and the news of its new motion spreads outward at c — the field lines develop a traveling kink. That kink is a transverse field pulse carrying energy: radiation. No acceleration, no kink, no radiation.

circuits

v_d = \frac{I}{n A q}

Drift Velocity

Electrons in a wire already zip around at ~10⁶ m/s thermally, but in random directions that cancel. An applied field adds a tiny net drift on top — a slow river current under a churning sea. The signal is fast (a field change at near light speed) but the carriers themselves barely move.

circuits

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

Temperature Dependence of Resistivity

Resistance comes from electrons scattering off the lattice. Heat makes the ions vibrate harder, so electrons collide more often and drift less freely — resistivity climbs nearly linearly over ordinary temperature ranges.

magnetostatics

\tau = \mu B \sin\theta,\quad \mu = N I A

Magnetic Dipole Moment & Torque

A current loop is a tiny bar magnet: its moment μ points along the loop's axis. A uniform field can't push it (forces cancel) but it twists it, trying to align μ with B — exactly how a compass needle swings north.

electromagnetic waves

\nabla\!\cdot\!\vec{E}=\tfrac{\rho}{\varepsilon_0},\ \nabla\!\cdot\!\vec{B}=0,\ \nabla\!\times\!\vec{E}=-\partial_t\vec{B},\ \nabla\!\times\!\vec{B}=\mu_0\vec{J}+\mu_0\varepsilon_0\partial_t\vec{E}

Maxwell's Equations (Unified)

Two of the four equations say charges make diverging E-fields and that there are no magnetic charges; the other two say a changing B makes a curling E and a changing E (or a current) makes a curling B. That mutual feedback lets the fields regenerate each other and sail off as light — no charges required.

circuits

\Delta V = I\,R_{seg} = I\,\rho\,\frac{L}{A}

Why Birds on Power Lines Don't Get Shocked

Current follows the path of least resistance. The bird's body (tens of kΩ) sits in parallel with a few centimeters of thick wire (micro-ohms). The wire is millions of times more conductive, so essentially all the current stays in the metal and the bird carries a vanishing trickle. Shock needs a voltage difference across you — and across two nearby points on one wire that difference is microscopic.

magnetostatics

V_H = \frac{I B}{n q t}

Hall Effect

Moving carriers in a perpendicular field feel a sideways Lorentz push and pile up on one edge. The charge buildup creates a transverse electric field until it exactly cancels the magnetic deflection. The leftover voltage across the strip reveals both the sign and the density of the carriers — a direct count of charge carriers.

electrostatics

\vec{E}_{inside} = 0

Why a Faraday Cage Shields You

A conductor's free charges move until they kill any field inside the metal. Put a cavity inside and those charges arrange on the outer surface to cancel the external field throughout the hollow — leaving the interior field-free. For oscillating waves a mesh works too, as long as the holes are far smaller than the wavelength, because the induced surface currents re-radiate a canceling field.

applications

f = \frac{\gamma B_0}{2\pi}

How MRI Imaging Works (Larmor)

Every proton is a tiny spinning magnet. In a strong field B₀ it doesn't just align — it precesses like a wobbling top, at a frequency set only by the field. Hit it with a radio pulse at exactly that frequency and it absorbs energy, tips over, then relaxes back while broadcasting a faint radio signal. A field gradient makes the frequency a position label — turning the echo into an image.

ac circuits

\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f \mu \sigma}}

Skin Effect in Conductors

An alternating current sets up a changing magnetic field inside the conductor, which by Faraday's law drives eddy currents that oppose the flow in the core and reinforce it near the surface. The faster the oscillation, the more the current is squeezed into a thin surface layer of thickness δ — the skin depth.

plasma physics

\frac{\sin^2\alpha_1}{B_1} = \frac{\sin^2\alpha_2}{B_2}

Physics of the Aurora

A charged particle spirals around a magnetic field line. As it moves toward a pole the field gets stronger and tighter, and conservation of the magnetic moment (the first adiabatic invariant) forces its spiral to widen in pitch until it stops and reflects — a magnetic mirror. Particles with small enough pitch angle slip through the mirror and slam into the upper atmosphere, exciting oxygen and nitrogen that glow green and red.

electromagnetic waves

f_c = \frac{c}{2a}\quad(\text{TE}_{10})

Waveguide Cutoff Frequency

A guided wave can be pictured as a plane wave zig-zagging between the conducting walls. To satisfy the boundary conditions, exactly a half-wavelength (for TE₁₀) must fit across the width a. If the free-space wavelength is too long — frequency too low — it can't fit at any bounce angle, so the wave can't propagate and instead decays exponentially: it's evanescent.

theoretical

e\,g = 2\pi n \hbar = n h \;\Rightarrow\; g = \frac{n h}{e}

Dirac Magnetic Monopole Quantization

Wrap an electron's quantum wavefunction around a monopole and its phase must come back to itself — single-valuedness. The phase picked up is set by the magnetic flux from the monopole; demanding it be a multiple of 2π forces the product of electric and magnetic charge to be quantized. Turn it around: even one monopole makes every electric charge a multiple of a basic unit.

Thermodynamics

gas laws

PV = nRT

Ideal Gas Law

Pressure times volume is proportional to temperature for a fixed amount of gas.

energy conservation

\Delta U = Q - W

First Law of Thermodynamics

Energy in (heat) minus energy out (work) equals the change stored inside.

heat transfer

q = -k \frac{dT}{dx}

Fourier's Law of Heat Conduction

Heat flows from hot to cold, faster through better conductors and steeper gradients.

thermal properties

\Delta L = \alpha L_0 \Delta T

Linear Thermal Expansion

Materials grow longer when heated — by an amount proportional to their length and temperature rise.

heat engines

\eta = 1 - \frac{T_C}{T_H}

Carnot Efficiency

No engine can beat the efficiency set by the ratio of its cold and hot reservoir temperatures.

radiation

P = \sigma A T^4

Stefan-Boltzmann Law

Hot objects radiate energy as light — and the power skyrockets with temperature (fourth power!).

entropy

\Delta S = \int \frac{\delta Q_{\text{rev}}}{T}

Entropy Change

Entropy measures how much energy has spread out — it always increases in the universe overall.

statistical mechanics

f(v) = 4\pi n \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 e^{-\frac{mv^2}{2k_BT}}

Maxwell-Boltzmann Speed Distribution

Gas molecules have a spread of speeds — most cluster near a peak, with a long tail of fast outliers.

statistical mechanics

S = k_B \ln W

Boltzmann Entropy

Entropy counts the number of microscopic ways a macroscopic state can be realized — more ways means higher entropy.

phase transitions

\frac{dP}{dT} = \frac{L}{T \Delta v}

Clausius-Clapeyron Equation

Steeper vapor-pressure curve where latent heat is high — phase boundary tilt is governed by entropy of vaporization.

calorimetry

C_v = \left(\frac{\partial U}{\partial T}\right)_V

Heat Capacity at Constant Volume

Heat capacity at constant volume measures the energy needed to raise temperature when no work is done — pure internal energy storage.

calorimetry

C_p = \left(\frac{\partial H}{\partial T}\right)_P

Heat Capacity at Constant Pressure

At constant P, some heat goes into work done expanding the gas — so C_p is always larger than C_v by exactly R (for ideal gases).

thermodynamic potentials

G = H - TS

Gibbs Free Energy

Gibbs free energy is the maximum non-PV work extractable from a system at constant T and P — and it must decrease for spontaneous processes.

thermodynamic potentials

F = U - TS

Helmholtz Free Energy

Helmholtz energy is the maximum work extractable at constant T and V. Like Gibbs but without the PV term — appropriate for sealed rigid containers.

blackbody radiation

\lambda_{\max} T = b

Wien's Displacement Law

Hotter objects emit shorter-wavelength radiation; the peak wavelength is inversely proportional to temperature.

blackbody radiation

B_\lambda(T) = \frac{2 h c^2}{\lambda^5} \frac{1}{e^{hc/\lambda k_B T} - 1}

Planck Radiation Law

Quantized photon energies cap the high-frequency emission of a blackbody, solving the ultraviolet catastrophe.

Statistical Mechanics

Z = \sum_i e^{-\beta E_i}, \quad \beta = \frac{1}{k_B T}

Canonical Partition Function

Z counts microstates weighted by their Boltzmann suppression; hotter systems explore more states.

Quantum Statistical Mechanics

f(E) = \frac{1}{e^{(E-\mu)/(k_B T)} + 1}

Fermi-Dirac Distribution

Each quantum state can hold at most one fermion; the chemical potential μ acts as a 'cut-off' — states below are occupied, above are empty.

Quantum Statistical Mechanics

\langle n_i \rangle = \frac{1}{e^{\beta(\varepsilon_i - \mu)} - 1}

Bose-Einstein Distribution

Bosons actively prefer to share states — the more particles already in a state, the more likely the next one joins (stimulated emission).

Statistical Mechanics

S = Nk_B\left[\ln\left(\frac{V}{N}\left(\frac{4\pi m E}{3Nh^2}\right)^{3/2}\right) + \frac{5}{2}\right]

Sackur-Tetrode Equation

Entropy counts accessible microstates in units of h³ per degree of freedom; the 5/2 is the 3/2 kinetic + 1 from volume/particle indistinguishability.

Statistical Mechanics

\Omega = -k_B T \ln \Xi, \quad \Xi = \sum_{N,i} e^{-\beta(E_{N,i} - \mu N)}

Grand Canonical Potential (Landau Free Energy)

Ω is the free energy cost of a system that can bleed particles; minimise it to find the equilibrium particle number at given T and μ.

Statistical Mechanics

\left\langle q_i \frac{\partial H}{\partial q_i} \right\rangle = k_B T

Equipartition Theorem

Classical thermal fluctuations distribute energy democratically: every quadratic term in H gets the same share k_BT/2.

Non-equilibrium Statistical Mechanics

S_{FF}(\omega) = 2k_B T\, \text{Re}[\chi(\omega)]

Fluctuation-Dissipation Theorem

A system that dissipates energy (resistance) must also fluctuate spontaneously (noise) at the same rate — you can't have one without the other at finite temperature.

Statistical Mechanics

\langle E_{\text{kin}} \rangle = -\frac{1}{2}\langle E_{\text{pot}} \rangle

Virial Theorem (Statistical Mechanics)

In a bound system, twice the kinetic energy always equals the negative of the potential energy — energy is partitioned by the power law of the force.

Kinetic Theory

f(v) = 4\pi n\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\!\left(-\frac{mv^2}{2k_B T}\right)

Maxwell Speed Distribution (3D)

The v² factor (surface area of velocity sphere) competes with the Boltzmann suppression e^{−mv²/2k_BT} to create a peaked distribution; the peak shifts right as T rises.

Laws of Thermodynamics

\Delta S_{\text{universe}} = \Delta S_{\text{sys}} + \Delta S_{\text{surr}} \geq 0

Second Law of Thermodynamics

Isolated systems drift toward disorder; entropy only ever goes up.

Laws of Thermodynamics

(A \sim C) \wedge (B \sim C) \implies (A \sim B)

Zeroth Law of Thermodynamics

If two things each match a thermometer, they match each other.

Laws of Thermodynamics

\lim_{T \to 0} S(T) = S_0 = k_B \ln g_0

Third Law of Thermodynamics

Cool toward 0 K and entropy freezes to a constant you can never fully remove.

Thermodynamic Processes

P V^{\gamma} = \text{const}, \quad \gamma = \frac{C_p}{C_v}

Adiabatic Process

No heat in or out, so work done on the gas becomes its internal energy.

Equations of State

\left(P + \frac{a n^2}{V^2}\right)(V - n b) = n R T

Van der Waals Equation

Real molecules take up space (b) and attract each other (a), bending the ideal gas law.

Heat Engines & Cycles

\text{COP}_{\text{ref}} = \frac{Q_c}{W} = \frac{Q_c}{Q_h - Q_c} \leq \frac{T_c}{T_h - T_c}

How a Refrigerator Moves Heat Uphill

Spend work to pump heat from cold to hot; the colder the inside, the more work each joule costs.

Real Gases & Phase Changes

\mu_{\text{JT}} = \left(\frac{\partial T}{\partial P}\right)_H = \frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_P - V\right]

Joule-Thomson Effect

Throttle a real gas at constant enthalpy and it cools — below the inversion temperature.

Phase Transitions

\frac{dT_m}{dP} = \frac{T\,\Delta V_{\text{melt}}}{L_f} \approx -7.4\times10^{-3}\ \text{K/atm}

Why Ice Skates Glide

A thin film of liquid water, not pressure alone, lets the blade slide.

Statistical Mechanics

W_{\text{erase}} \geq k_B T \ln 2 \ \text{per bit}

Maxwell's Demon and Information

Sorting molecules looks free, but erasing the demon's memory pays the full entropy bill.

Statistical Mechanics

\frac{n_{i+1} n_e}{n_i} = \frac{2 g_{i+1}}{g_i}\left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-\chi/k_B T}

Saha Ionization Equation

Hotter, thinner gas is more ionized; the Boltzmann factor e^(−χ/kT) sets the balance.

Heat Engines & Cycles

\eta_{\text{Otto}} = 1 - \frac{1}{r^{\gamma - 1}}, \quad r = \frac{V_1}{V_2}

Otto Cycle Efficiency

Efficiency depends only on how much you compress the fuel-air mix before ignition.

Statistical Mechanics

\Delta S_{\text{mix}} = -N k_B \sum_i x_i \ln x_i \;\to\; 0 \text{ for identical species}

Gibbs Paradox

Mixing distinct gases adds entropy; mixing identical ones adds none.

Waves & Optics

refraction

n_1 \sin\theta_1 = n_2 \sin\theta_2

Snell's Law

Light bends toward the normal when entering a denser medium.

reflection

\theta_i = \theta_r

Law of Reflection

Light bounces off a mirror at the same angle it arrives.

wave mechanics

v = f\lambda

Wave Speed Equation

Wave speed equals how many wavelengths pass a point each second.

geometric optics

\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Thin Lens Equation

Reciprocals of object and image distances always add up to the lens power.

interference

d \sin\theta = m\lambda

Young's Double Slit Interference

Two overlapping wave sources create bright and dark bands.

diffraction

a \sin\theta = m\lambda

Single Slit Diffraction

A narrow slit spreads light into a pattern of bright and dark bands.

polarization

\tan\theta_B = \frac{n_2}{n_1}

Brewster's Angle

At one special angle, reflected light becomes perfectly polarized.

wave mechanics

f' = f \frac{v + v_o}{v - v_s}

Doppler Effect

Moving toward a wave source compresses waves; moving away stretches them.

polarization

I = I_0 \cos^2\theta

Malus's Law

Only the component of the electric field aligned with the polarizer axis gets through; the rest is absorbed.

diffraction

\theta_{min} = 1.22 \frac{\lambda}{D}

Rayleigh Criterion

Two point sources are just resolvable when the central maximum of one falls on the first dark ring of the other.

diffraction

d \sin\theta = m \lambda

Diffraction Grating Equation

Light constructively interferes whenever the path difference between adjacent slits equals an integer number of wavelengths.

geometric optics

\frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)

Lensmaker's Equation

Focal length depends on how strongly light bends at each curved surface, governed by the index step and surface curvature.

geometrical optics

\frac{1}{f} = \frac{1}{v} + \frac{1}{u}

Mirror Equation

Object distance, image distance, and focal length lock together — change one and the others must rearrange.

electromagnetic waves

n = \frac{c}{v}

Refractive Index Definition

The refractive index tells you the factor by which light slows down inside a material.

electromagnetic waves

I = \frac{1}{2} c \epsilon_0 E_0^2

Intensity of an Electromagnetic Wave

Intensity scales with the square of the electric field amplitude — doubling the field quadruples the power flow.

quantum optics

E = h f = \frac{h c}{\lambda}

Photon Energy (Planck-Einstein Relation)

Higher frequency means each light particle carries more punch — color literally equals energy.

wave mechanics

\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}

Wave Equation

The curvature of the wave in space drives its acceleration in time — sharper bends snap back faster.

wave mechanics

f_n = \frac{n v}{2 L}

Standing Wave Frequencies

Only wavelengths that fit a whole number of half-wave humps between the two fixed ends survive — those are the notes.

refraction

\theta_c = \arcsin\left(\frac{n_2}{n_1}\right)

Critical Angle (Total Internal Reflection)

Past a certain angle, light heading from dense to rare can't refract out — it gets reflected back perfectly.

superposition

f_{beat} = |f_1 - f_2|

Beat Frequency

Two waves of nearly equal frequency drift in and out of phase. When they line up they add (loud); when opposed they cancel (silent). The loudness pulses at the difference frequency.

geometric optics

m = -\frac{d_i}{d_o}

Lateral Magnification

Magnification compares image height to object height. A negative sign means the image is inverted; its magnitude tells you how many times larger or smaller the image is.

interference

2 n t = \left(m + \tfrac{1}{2}\right)\lambda

Thin-Film Interference

Light reflecting off the top and bottom surfaces of a film travels different path lengths. A half-wave phase flip at the top surface makes the half-integer condition give bright reflection — and different colors satisfy it at different thicknesses.

diffraction

n\lambda = 2 d \sin\theta

Bragg's Law

Atomic planes act like a stack of partial mirrors. Reflections from successive planes add up only when their extra path length is a whole number of wavelengths — that condition pins the angle for each wavelength.

geometric optics

NA = n \sin\theta_{max}

Numerical Aperture

Numerical aperture is the sine of the widest cone of light an optical system can gather or emit, scaled by the medium's index. Bigger NA means more light collected and finer resolving power.

wave propagation

v_g = \frac{d\omega}{dk}

Group Velocity

A localized wave packet is a sum of many frequencies. The crests move at the phase velocity, but the packet's envelope — where the energy and information ride — moves at the group velocity, the slope of the dispersion curve.

interferometry

\Delta N = \frac{2\,\Delta d}{\lambda}

Michelson Interferometer Fringe Shift

Light in one arm makes a round trip, so moving the mirror by Delta d changes the path by 2*Delta d. Each whole wavelength of extra path slides the fringe pattern by exactly one fringe — turning displacement into a count.

Quantum

wave particle duality

\lambda = \frac{h}{p}

de Broglie Wavelength

Momentum and wavelength are inversely related through Planck's constant — big things have unmeasurably tiny wavelengths.

quantization

E = hf

Photon Energy (Planck Relation)

Light energy is quantized: each photon carries a fixed packet of energy set by its frequency.

quantization

K_{max} = hf - \phi

Photoelectric Effect

A photon gives all its energy to one electron; the work function is the minimum escape cost.

foundations

\Delta x \, \Delta p \geq \frac{\hbar}{2}

Heisenberg Uncertainty Principle

Position and momentum are conjugate — pinning one down spreads the other.

atomic physics

E_n = -\frac{13.6 \text{ eV}}{n^2}

Bohr Hydrogen Energy Levels

Bound electrons can only sit on a quantized energy ladder; jumping down emits a photon.

foundations

i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

Time-Dependent Schrödinger Equation

The Hamiltonian generates time evolution of the wavefunction in complex Hilbert space.

bound states

E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

Particle in a 1D Box

Standing waves must fit inside the box; only integer half-wavelengths are allowed.

relativistic qm

(i\gamma^\mu \partial_\mu - mc/\hbar)\psi = 0

Dirac Equation

A first-order relativistic wave equation whose solutions naturally carry spin and antiparticles.

wave mechanics

\hat{H}\psi = E\psi

Time-Independent Schrödinger Equation

A wavefunction is allowed only if applying the Hamiltonian gives back the same wavefunction scaled by a number — that number is the energy.

wave mechanics

E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)

Quantum Harmonic Oscillator

Energy comes in equal steps of hbar*omega, with a built-in floor of hbar*omega/2 — the zero-point motion required by Heisenberg.

atomic spectra

\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Light emerges when an electron drops between two energy rungs — the wavelength is set by the difference of two inverse squares.

many body

\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\mathbf{r}_1)

Pauli Exclusion Principle

Swap two identical fermions and the wavefunction flips sign — so the wavefunction vanishes if they share the same state.

spin

\hat{S}_i = \tfrac{\hbar}{2}\sigma_i

Spin-1/2 Operators

Three 2×2 matrices encode every spin-1/2 measurement — they don't commute, which is why spin in x and y can't be known simultaneously.

wave mechanics

T \approx \exp\!\left(-\frac{2}{\hbar}\int_a^b\sqrt{2m\bigl(V(x)-E\bigr)}\,dx\right)

Quantum Tunneling Probability

The wavefunction decays exponentially inside a barrier — make the barrier thinner or shorter and a measurable tail emerges on the far side.

spin

F_z = \mu_z\,\frac{\partial B_z}{\partial z}

Stern–Gerlach Deflection

A magnetic dipole in a field gradient feels a force along the gradient; quantum spin offers only two values of mu_z, so the beam splits in two.

open systems

\hat{\rho} = \sum_i p_i\,|\psi_i\rangle\langle\psi_i|

Density Matrix

Replace one wavefunction with a weighted bookkeeper of many — diagonal entries are probabilities, off-diagonal entries are coherences that decoherence destroys.

foundations

P(x)\,dx = |\psi(x)|^2\,dx

Born Rule

The squared magnitude of the wavefunction is the probability map for measurement outcomes.

operator algebra

[\hat{x}, \hat{p}] = i\hbar

Canonical Commutation Relation

Order matters: measuring x then p differs from p then x by exactly iℏ — the seed of uncertainty.

angular momentum

|\mathbf{L}| = \sqrt{l(l+1)}\,\hbar, \qquad L_z = m\hbar

Quantization of Angular Momentum

Angular momentum comes in rungs of ℏ: its length is √(l(l+1))ℏ and its z-shadow is mℏ.

spin dynamics

\omega_L = \gamma B

Larmor Precession

A spin in a magnetic field precesses like a gyroscope, at a rate exactly proportional to the field.

dynamics

\frac{d\langle \hat{x}\rangle}{dt} = \frac{\langle \hat{p}\rangle}{m}, \qquad \frac{d\langle \hat{p}\rangle}{dt} = -\left\langle \frac{\partial V}{\partial x}\right\rangle

Ehrenfest Theorem

Quantum averages obey Newton-like equations — the wavepacket's center moves classically.

two level systems

P_e(t) = \frac{\Omega^2}{\Omega^2 + \Delta^2}\,\sin^2\!\left(\tfrac{1}{2}\sqrt{\Omega^2+\Delta^2}\;t\right)

Rabi Oscillations

A driven two-level system cycles between ground and excited state; detuning caps the swing.

entanglement

S = E(a,b) - E(a,b') + E(a',b) + E(a',b'), \qquad |S| \le 2

Bell–CHSH Inequality

Local hidden variables cap a four-correlation sum at 2; entangled particles push it to 2√2.

particle physics

\mathcal{L}_{\text{int}} = -e\,\bar{\psi}\gamma^{\mu}\psi A_{\mu}

Standard Model Interactions

Every particle reaction in nature is built from a handful of vertices — an electron emitting a photon, a quark emitting a W boson, a gluon splitting. Glue these elementary moves together and you get beta decay, annihilation, Compton scattering, pair production. The bookkeeping rules are absolute: electric charge, lepton number, and baryon number in must equal out. If a reaction conserves them all, somewhere in the universe it happens.

Relativity

special relativity

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Lorentz Factor

Quantifies how much time, length, and mass distort as velocity approaches c.

special relativity

\Delta t = \gamma \Delta t_0

Time Dilation

A moving clock ticks slower than a stationary one, as seen by an outside observer.

special relativity

L = L_0 / \gamma

Length Contraction

Objects in motion appear shorter along their direction of travel.

special relativity

p = \gamma m v

Relativistic Momentum

Momentum diverges as velocity approaches c, preventing massive objects from reaching light speed.

special relativity

E^2 = (pc)^2 + (mc^2)^2

Energy–Momentum Relation

The full relativistic relation linking total energy, momentum, and rest mass.

special relativity

u = \frac{u' + v}{1 + u'v/c^2}

Relativistic Velocity Addition

Velocities don't simply add at high speeds; the result never exceeds c.

special relativity

f = f_0 \sqrt{\frac{1 - \beta}{1 + \beta}}

Relativistic Doppler Effect

Receding sources redshift, approaching sources blueshift — with a relativistic gamma correction.

general relativity

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

Schwarzschild Radius

The radius at which escape velocity equals c — the event horizon of a non-rotating black hole.

special relativity

E = mc^2

Mass-Energy Equivalence

Mass and energy are interchangeable currencies; any object at rest carries an enormous reservoir of energy because c^2 is huge.

special relativity

K = (\gamma - 1) m c^2

Relativistic Kinetic Energy

Kinetic energy is the extra energy a body gains by moving — it diverges as v approaches c, not 1/2 mv^2.

special relativity

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

Lorentz Transformation

Space and time mix between observers in relative motion — the geometry of spacetime is a hyperbolic rotation, not a Galilean shear.

general relativity

\frac{d\tau}{dt} = \sqrt{1 - \frac{2GM}{rc^2}}

Gravitational Time Dilation

Gravity slows time. The deeper you are in a gravitational potential, the slower your clock runs compared to a distant observer.

special relativity

s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2

Spacetime Interval

Spacetime has its own Pythagorean theorem — but with one minus sign. The interval is the only true 'distance' all observers agree on.

special relativity

d\tau = dt\sqrt{1 - v^2/c^2} = \frac{dt}{\gamma}

Proper Time

Proper time is what you'd read on a wristwatch you carry with you — the invariant 'age' of any worldline.

general relativity

\frac{\Delta\lambda}{\lambda} = \frac{GM}{rc^2}

Gravitational Redshift

Photons climbing out of a gravity well lose energy — their wavelength stretches and clocks at the bottom appear to tick slower.

black hole thermodynamics

T_H = \frac{\hbar c^3}{8\pi G M k_B}

Hawking Temperature

Quantum effects at a black hole's event horizon make it radiate like a blackbody — the temperature is inversely proportional to its mass.

special relativity

\tau_{\text{traveler}} = \tau_{\text{Earth}}\sqrt{1 - \frac{v^2}{c^2}}

The Twin Paradox

Send one twin on a fast round trip and bring her home: she returns younger than the twin who stayed. There's no paradox — the situations aren't symmetric. The traveling twin must turn around, and that acceleration breaks the symmetry, so it is unambiguously she who logs less proper time along her bent path through spacetime.

special relativity

\cos\theta' = \frac{\cos\theta + \beta}{1 + \beta\cos\theta}

Relativistic Aberration of Light

Run fast enough and the sky rearranges itself: light that arrived from your sides crowds into the direction you're heading, like rain on a windshield slanting forward as you accelerate. At near-light speed almost the entire sky compresses into a bright spot dead ahead — the relativistic headlight effect.

special relativity

I_{\text{obs}} = \delta^{\,3+\alpha}\,I_{\text{emit}},\qquad \delta = \frac{1}{\gamma(1 - \beta\cos\theta)}

Relativistic Beaming (Doppler Boosting)

A source moving toward you doesn't just blue-shift — it gets dramatically brighter, because aberration funnels its photons forward, time dilation packs more of them per second, and the Doppler shift lifts each photon's energy. These three effects multiply into a steep δ⁴ dependence, so a jet pointed at you can outshine an identical one pointed away by factors of thousands.

special relativity

E^2 = (pc)^2 + (mc^2)^2

Energy–Momentum Four-Vector

Energy and momentum aren't separate bookkeeping — they're the time and space components of one four-vector, just as duration and length are facets of spacetime. Different observers disagree on E and on p, but they all agree on the length of that four-vector, and that invariant length is the particle's rest mass. The mass is the part of energy-momentum nobody can boost away.

general relativity

h = \frac{2G}{c^4 d}\,\ddot{Q}

Gravitational-Wave Strain

Accelerating masses ripple spacetime itself, and those ripples stretch and squeeze every length they pass through by a fractional amount h — the strain. The catch is the factor c⁴ in the denominator: it makes h staggeringly tiny. Two merging black holes a billion light-years away wobble LIGO's 4 km arms by less than a thousandth the width of a proton, which is exactly what it detected.

general relativity

\Omega_{\text{LT}} = \frac{2GJ}{c^2 r^3}

Frame Dragging (Lense–Thirring Effect)

A spinning mass doesn't just curve spacetime — it drags it around, winding the very fabric of space into a slow vortex like a spoon stirring honey. A gyroscope held nearby is gently twisted by the rotation even though no force touches it; near a spinning black hole the dragging becomes so violent that, inside the ergosphere, standing still is physically impossible.

general relativity

\theta = \frac{4GM}{c^{2}b}

General Relativity: Light Bending & Curved Spacetime

Mass curves spacetime, and light follows the straightest possible path through that curved geometry — so starlight grazing the Sun bends by 1.75 arcseconds, exactly twice what Newton's gravity-on-light would give. The same curvature, turned up, gives gravitational lensing, Einstein rings, ripples in spacetime itself (gravitational waves), and at the extreme, black holes from which no path leads out.

Modern Physics

special relativity

E = mc^2

Mass-Energy Equivalence

Mass is a highly concentrated form of energy; the speed of light squared is the conversion factor.

special relativity

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

Lorentz Factor

As speed approaches c, time stretches and lengths contract by the factor γ.

quantum mechanics

K_{max} = h\nu - \phi

Photoelectric Effect

Light comes in quanta of energy hν; only photons above the work-function threshold can free electrons.

quantum mechanics

\lambda = \frac{h}{p}

De Broglie Wavelength

Every particle has a wavelength inversely proportional to its momentum — heavy/fast things have wavelengths so small they're unobservable.

quantum mechanics

\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

Heisenberg Uncertainty Principle

Position and momentum cannot both be sharply defined; nature enforces a fundamental fuzziness at small scales.

quantum mechanics

-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi

Schrödinger Equation (Time-Independent)

The total energy operator acting on the wavefunction returns the energy times the same wavefunction — an eigenvalue problem for reality.

atomic physics

E_n = -\frac{13.6\,\text{eV}}{n^2}

Bohr Energy Levels (Hydrogen)

Electrons in hydrogen are stuck on a ladder of negative energies; the gaps determine the colors of light atoms emit.

nuclear physics

N(t) = N_0 e^{-\lambda t}

Radioactive Decay Law

Each nucleus has a fixed probability per unit time of decaying, producing a smooth exponential decline in the population.

quantum mechanics

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Compton Scattering

Photons carry momentum p = h/λ. When one collides with a free electron, conservation of energy + momentum forces the photon to give up energy — its wavelength grows by an amount that depends only on the scattering angle.

atomic physics

a_0 = \frac{4\pi\varepsilon_0 \hbar^2}{m_e e^2}

Bohr Radius

A balance between two demands: Coulomb attraction wants the electron as close to the proton as possible, but the uncertainty principle penalizes localization (smaller box → bigger momentum → bigger kinetic energy). The minimum-energy compromise sits at a₀.

atomic physics

\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Each integer n labels an allowed electron energy level (the staircase). A photon's wavelength encodes the energy difference between two steps. Two integers → all hydrogen spectral lines.

quantum mechanics

B(\lambda, T) = \frac{2hc^2}{\lambda^5}\cdot\frac{1}{e^{hc/(\lambda k_B T)} - 1}

Planck Radiation Law

Light energy is quantized in packets of size hν. At low frequency, packets are cheap → many emitted (Rayleigh-Jeans). At high frequency, each packet costs more than kT → exponentially suppressed. The peak balance gives the body's color.

quantum mechanics

\lambda_{\max} T = b

Wien's Displacement Law

Hotter → more energetic photons → shorter peak wavelength. The exact peak comes from differentiating Planck's law and solving a transcendental equation; the answer is a universal product b.

quantum mechanics

j = \sigma T^4

Stefan-Boltzmann Law

Integrating the Planck spectrum over all wavelengths gives the total emitted power. Two factors of T from the peak shift (Wien) and two more from the bandwidth growth combine to a T⁴ scaling.

particle physics

E_{\gamma}^{\min} = 2 m_e c^2

Pair Production Threshold

Mass-energy equivalence says creating two particles of mass m_e demands at least 2m_e·c² of energy. But a lone photon can't do it — momentum conservation forbids it. A nearby nucleus (or another photon) absorbs the recoil and unlocks the process.

special relativity

\Delta t = \gamma \Delta t_0 = \frac{\Delta t_0}{\sqrt{1 - v^2/c^2}}

Time Dilation

The speed of light is the same for all observers. To keep that fixed when one observer moves relative to another, time itself must stretch — moving clocks run slow.

special relativity

L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - v^2/c^2}

Length Contraction

Just as moving clocks dilate, moving rulers contract. Both follow from c being invariant: lengths along motion direction shrink by 1/γ. Lengths perpendicular to motion are unchanged.

Nuclear & Particle

relativistic energy

E = mc^2

Mass–Energy Equivalence

Mass is a highly concentrated form of energy; converting even a tiny mass releases enormous energy.

radioactivity

N(t) = N_0 e^{-\lambda t}

Radioactive Decay Law

Undecayed nuclei fall exponentially; each has a fixed decay probability per unit time.

radioactivity

T_{1/2} = \frac{\ln 2}{\lambda}

Half-Life Relation

Half-life is a fixed isotope property—independent of the sample size.

nuclear structure

B = \Delta m \, c^2

Nuclear Binding Energy

The missing mass when nucleons bind into a nucleus appears as the energy holding them together.

nuclear reactions

Q = (m_i - m_f) c^2

Q-Value of Nuclear Reaction

Lighter products mean the missing mass exits as kinetic energy and radiation.

relativistic energy

E^2 = (pc)^2 + (mc^2)^2

Relativistic Energy–Momentum Relation

Energy and momentum combine to form a Lorentz invariant — the rest mass of the particle.

particle physics

\lambda = \frac{h}{p}

de Broglie Wavelength (Relativistic)

A particle’s wavelength shrinks with momentum—more momentum probes smaller scales.

radioactivity

\log_{10} T_{1/2} = a \frac{Z}{\sqrt{Q}} + b

Geiger–Nuttall Law

Alpha half-lives vary exponentially with decay energy via quantum tunneling.

nuclear structure

B(A,Z) = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_A \frac{(A-2Z)^2}{A} + \delta(A,Z)

Semi-Empirical Mass Formula

Five competing terms—volume, surface, Coulomb, asymmetry, pairing—model the nuclear binding energy.

particle physics

\frac{d\sigma}{d\Omega} = \frac{1}{2} \alpha^2 r_c^2 \left(\frac{\omega'}{\omega}\right)^2 \left[\frac{\omega}{\omega'} + \frac{\omega'}{\omega} - \sin^2\theta\right]

Klein–Nishina Cross Section

QED correction to Thomson scattering: photon cross-section shrinks at high energy.

photon interactions

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Compton Scattering Wavelength Shift

A photon hitting an electron transfers momentum, so it leaves with a longer wavelength — light behaves like a particle.

atomic structure

a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m_e e^2}

Bohr Radius

The natural size of an atom emerges from balancing electron kinetic energy with Coulomb attraction to the nucleus.

atomic structure

\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Hydrogen emits sharp, discrete colors because its electron only inhabits quantized energy levels — the spectrum is the atom's barcode.

photon interactions

K_{max} = h f - \phi

Photoelectric Effect Equation

Light comes in packets of energy hf; only packets above the work-function threshold can free an electron — no matter how intense the beam.

radioactivity

A = \lambda N

Radioactive Activity-Decay Constant Relation

Activity is just the number of unstable atoms times how likely each is to decay per second.

scattering

\frac{d\sigma}{d\Omega} = \left(\frac{Z_1 Z_2 e^2}{16\pi\epsilon_0 E}\right)^2 \frac{1}{\sin^4(\theta/2)}

Rutherford Scattering Cross Section

Rare large-angle scattering proves the atom has a tiny, dense, positively charged core — the nucleus.

nuclear force

V(r) = -g^2 \frac{e^{-m r c / \hbar}}{r}

Yukawa Potential

A short-ranged attractive force whose range is set by the mass of the exchanged particle — heavier mediators = shorter range.

transition rates

\Gamma_{i \to f} = \frac{2\pi}{\hbar} |M_{fi}|^2 \rho(E_f)

Fermi's Golden Rule

The transition rate between quantum states is proportional to the squared coupling times the number of final states available.

photon interactions

E_{\gamma}^{\min} = 2 m_e c^2

Pair Production Threshold Energy

A photon must carry at least the combined rest energy of the electron and positron it creates — 1.022 MeV.

radioactive decay

Q_\alpha = (m_P - m_D - m_\alpha)c^2

Alpha Decay Q-Value

The kinetic energy carried away by the alpha particle (and the recoiling daughter) equals the rest-mass difference between parent and decay products, expressed via E = mc².

weak interaction

\frac{dN}{dE} = \frac{G_F^2 |M|^2}{2\pi^3 \hbar^7 c^5} F(Z,E) p E (Q - E)^2

Fermi Beta Decay Spectrum

Three-body decay (nucleus + electron + neutrino) shares the released energy Q, so the electron's spectrum is continuous; the (Q-E)² factor is the neutrino phase space.

scattering

\lambda = \frac{1}{n\sigma}

Mean Free Path

If the medium has n targets per unit volume each presenting cross-sectional area σ, the average distance between interactions is 1/(nσ).

nuclear structure

\bar{B} = \frac{B(A,Z)}{A}

Binding Energy per Nucleon

Dividing total binding energy by mass number gives the per-nucleon glue strength; the curve peaks near A=56 (Fe/Ni), explaining why fusion liberates energy below A=56 and fission above.

radioactive decay

\tau = \frac{1}{\lambda}

Mean Lifetime of Radioactive Nuclei

Mean lifetime is the expectation value of the exponential decay distribution; equivalently, the time after which N drops to N₀/e.

nuclear structure

R = R_0 A^{1/3}

Nuclear Radius Formula

Nuclei have nearly constant density, so volume scales linearly with A and radius scales as A^(1/3) — like balls of incompressible nuclear fluid.

scattering

\sigma_T = \frac{8\pi}{3}\left(\frac{e^2}{4\pi\varepsilon_0 m_e c^2}\right)^2

Thomson Scattering Cross Section

An EM wave shakes a free electron, which re-radiates a dipole pattern; the total scattering rate corresponds to an effective area σ_T set by the classical electron radius r_e.

radiation

v > \frac{c}{n} \;\Leftrightarrow\; \cos\theta_C = \frac{1}{n\beta}

Cherenkov Radiation Threshold

When a charged particle moves faster than the phase velocity of light in the medium (c/n), it sets off a coherent optical shockwave at angle θ_C — the blue glow in reactor pools.

particle physics

p = uud, \quad n = udd, \quad {}^{A}_{Z}X: \; A = Z + N

Building Atoms from the Standard Model

Everything you've ever touched is three particles: up quarks, down quarks, and electrons. Two ups and a down make a proton (+1); one up and two downs make a neutron (0). The proton count Z picks the element, the neutron count N picks the isotope, and the electron count sets the charge. Stray too far from the stable Z–N balance and the weak force flips a quark — beta decay — to restore it.

fission

N_{n} = N_{0}\,k^{n}, \qquad k = \nu \, p

Nuclear Chain Reaction

One neutron splits a uranium-235 nucleus, releasing ~200 MeV and ν ≈ 2.4 fresh neutrons. If, on average, more than one of those goes on to cause another fission (k > 1), the population explodes geometrically — 2, 4, 8, … 2⁸⁰ in microseconds. A reactor is the art of pinning k at exactly 1.000; a bomb is k ≈ 2 with nothing in the way.