214 formulas and 23 constants — the same ones the simulations compute with. Every formula has a calculator that checks your units, and every constant says where its value comes from.
Showing 214 of 214
Relative error
Measurement & unitsThe uncertainty in a measurement as a fraction of the measurement itself. A millimetre is a large error on a pencil and a negligible one on a running track.
Percentage error
Measurement & unitsThe relative error written as a percentage — the same number, times one hundred.
Error in a product of powers (worst case)
Measurement & unitsFor a result Q = a^p b^q, the relative errors of the inputs add, each multiplied by the power it is raised to. A squared quantity counts twice — measure it most carefully.
Error in a product of powers (independent errors)
Measurement & unitsWhen the input errors are independent they partly cancel, so their weighted relative errors add in quadrature rather than directly. The largest term dominates — that is the measurement worth improving.
Velocity under constant acceleration
MechanicsConstant acceleration adds the same velocity every second, so final velocity is initial velocity plus acceleration times time.
Displacement under constant acceleration
MechanicsDisplacement is what the initial velocity alone would cover, plus the extra the acceleration adds, which grows with the square of time.
Velocity–displacement relation
MechanicsThe constant-acceleration relation with time eliminated — the one to reach for when the question never mentions how long anything took.
Range of a projectile on level ground
MechanicsHow far a projectile lands from where it was launched, when it lands at the height it started from. The range peaks at 45°, where sin 2θ is 1.
Newton's second law
MechanicsThe net force on a body equals its mass times its acceleration. The same force accelerates a heavier body less.
Weight
MechanicsThe gravitational force on a mass near Earth's surface. Weight is a force in newtons; mass is not.
Friction
MechanicsSliding friction is proportional to how hard the surfaces are pressed together, and independent of how much area is in contact.
Centripetal force
MechanicsThe inward net force needed to keep a mass moving in a circle. It is not an extra force — some real force (tension, friction, gravity) has to supply it.
Linear momentum
MechanicsMomentum is mass times velocity — the quantity a closed system conserves through every collision.
Impulse
MechanicsA force acting for a time changes momentum by force times time. Stretching the time — an airbag, a bent knee — lowers the force for the same change.
Work done by a constant force
MechanicsOnly the part of the force along the displacement does work. A force at right angles to the motion does none at all.
Kinetic energy
MechanicsThe energy of motion. Doubling the speed quadruples it, which is why stopping distance grows so fast with speed.
Gravitational potential energy near the surface
MechanicsThe energy stored by lifting a mass through a height h against uniform gravity. Only differences in U mean anything; where h = 0 is a choice.
Power
MechanicsPower is the rate of doing work — joules per second, which is watts.
Torque
Rotation & gravitationHow strongly a force turns something about a pivot. Only the part of the force at right angles to the lever arm counts, so pushing a door near its hinge or straight towards the hinge barely turns it.
Newton's second law for rotation
Rotation & gravitationThe net torque on a body equals its moment of inertia times its angular acceleration — rotation's version of F = ma, with I playing the part of mass.
Moment of inertia of a point mass
Rotation & gravitationA mass resists being spun in proportion to the square of its distance from the axis. Moving the same mass twice as far out makes it four times harder to turn.
Moment of inertia of a standard shape
Rotation & gravitationEvery standard shape has I equal to a pure number times M R². The number says how far out the mass sits: 1 for a hoop, ½ for a solid disc, ⅖ for a solid sphere.
Parallel-axis theorem
Rotation & gravitationSpinning a body about any axis costs its moment of inertia about a parallel axis through its centre of mass, plus M d² for the distance between the two axes.
Rotational kinetic energy
Rotation & gravitationThe energy a body has from spinning — the rotational twin of ½mv², with the moment of inertia in place of mass and angular velocity in place of speed.
Speed after rolling down a slope
Rotation & gravitationA body that rolls must spend some of its height on spinning, so it arrives slower than one that slides. Mass and radius cancel; only the shape factor k decides the race.
Angular momentum
Rotation & gravitationAngular momentum is moment of inertia times angular velocity — the quantity a spinning body keeps when no outside torque acts on it.
Conservation of angular momentum
Rotation & gravitationWith no outside torque, a body that pulls its mass inwards spins faster by exactly the factor its moment of inertia fell — the skater drawing in their arms.
Centre of mass of two bodies
Rotation & gravitationThe centre of mass is the mass-weighted average position. It always lies between the two bodies, nearer the heavier one.
Newton's law of universal gravitation
Rotation & gravitationEvery pair of masses attracts with a force proportional to both masses and falling off with the square of the distance between their centres. G is so small that only planet-sized masses make the force noticeable.
Gravitational field strength
Rotation & gravitationThe gravitational force per kilogram at a distance r from a mass M. At Earth's surface it comes out at about 9.8 N/kg — the same number as the acceleration of a falling body.
Speed in a circular orbit
Rotation & gravitationThe speed at which gravity supplies exactly the centripetal force a circle needs. It does not depend on the satellite's own mass, and it falls the farther out the orbit is.
Escape velocity
Rotation & gravitationThe launch speed at which kinetic energy just pays off the whole gravitational well, so the body never falls back. It is √2 times the circular-orbit speed at the same radius.
Kepler's third law
Rotation & gravitationThe square of an orbit's period is proportional to the cube of its size. Every planet around the Sun shares the same constant of proportionality, which is how the Sun's mass is weighed.
Pressure
FluidsPressure is force spread over area. The same push through a sharper point gives a far larger pressure, which is why a knife cuts and a snowshoe does not sink.
Pressure at a depth
FluidsBelow an open surface, the pressure is the atmosphere pressing on top plus the weight of the liquid column above. Every 10 m of water adds about one more atmosphere.
Pascal's principle (hydraulic press)
FluidsPressure applied to an enclosed liquid reaches every part of it undiminished, so a small force on a small piston becomes a large force on a large one. The work is not multiplied: the large piston moves less.
Archimedes' principle
FluidsA body in a fluid is pushed up by the weight of the fluid it displaces. It floats when that upthrust can match its own weight before it is fully under.
Volume flow rate
FluidsThe volume of fluid passing a cross-section each second: the area of the pipe times the speed of the flow.
Continuity equation
FluidsLiquid cannot pile up inside a pipe, so the same volume passes every section each second. Where the pipe narrows, the flow must speed up.
Bernoulli's equation
FluidsAlong a streamline, pressure, kinetic energy per volume and potential energy per volume add to the same total. Where the fluid speeds up, its pressure drops.
Poiseuille's law
FluidsViscous flow through a narrow pipe grows with the fourth power of its radius. Halve the radius and only a sixteenth as much flows for the same pressure difference.
Stokes' law
FluidsThe viscous drag on a small sphere moving slowly through a fluid grows in step with its speed, its radius and the fluid's viscosity.
Terminal velocity of a falling sphere
FluidsA sphere sinking through a viscous fluid stops speeding up when Stokes drag and buoyancy together balance its weight. Bigger spheres fall much faster: the speed grows with the square of the radius.
Surface tension on a film
FluidsSurface tension is a force per unit length of edge. A soap film has two surfaces, front and back, so it pulls on a wire of length L with twice T L.
Excess pressure inside a drop
FluidsA curved liquid surface squeezes what is inside it, so a drop's inside pressure exceeds the outside by 2T/r. The smaller the drop, the bigger the squeeze.
Excess pressure inside a soap bubble
FluidsA soap bubble in air has two surfaces, inside and outside, so its excess pressure is double a drop's. This is why the smaller of two joined bubbles empties into the larger.
Capillary rise (Jurin's law)
FluidsSurface tension pulls liquid up a narrow tube until the column's weight balances the pull at the rim. Halve the tube's radius and the liquid climbs twice as high.
Stress
Properties of solidsStress is the force a material carries per unit of cross-sectional area. A thick rope and a thin thread under the same load are at very different stresses.
Strain
Properties of solidsStrain is the stretch as a fraction of the original length. It has no unit: a 2 m wire stretched by 2 mm and a 1 m wire stretched by 1 mm are equally strained.
Young's modulus
Properties of solidsWithin the elastic limit, stress is proportional to strain, and the ratio is a property of the substance alone — not of the wire's length or thickness.
Stretching a loaded wire
Properties of solidsYoung's modulus written in the quantities a wire experiment measures. A longer wire stretches more, a thicker one less, and Y stays the same.
Spring constant of a wire
Properties of solidsA wire behaves as a spring whose stiffness depends on the piece, not just the substance: halve the length and k doubles, double the radius and k quadruples, while Y does not move.
Shear modulus
Properties of solidsShear stress (a force sliding one face past the opposite one, per unit area) divided by the shear strain (the small angle the block leans through) — a material's resistance to having its shape changed.
Bulk modulus
Properties of solidsHow hard a material is to squeeze into a smaller volume: the extra pressure divided by the fractional change in volume. The minus sign is there because more pressure means less volume.
Poisson's ratio
Properties of solidsA bar stretched along its length gets thinner across it. Poisson's ratio is how much thinner, as a fraction of how much longer.
Shear modulus from Young's modulus
Properties of solidsFor a material that behaves the same in every direction, the shear modulus is fixed by Young's modulus and Poisson's ratio — it is a consequence, not a third independent fact.
Bulk modulus from Young's modulus
Properties of solidsThe bulk modulus follows from the same two numbers. As Poisson's ratio approaches ½ it runs away to infinity: a material that keeps its volume when stretched cannot be compressed either.
Energy stored in a stretched wire
Properties of solidsThe work done stretching a wire is stored as elastic energy. It is half of force times extension, because the force grows from zero as the wire stretches.
Elastic energy per unit volume
Properties of solidsThe elastic energy stored in each cubic metre of a strained material — the area under the straight part of its stress–strain graph.
Deflection of a cantilever
Properties of solidsHow far the free end of a beam fixed at one end drops under a load at its tip. Length enters cubed, and the beam's shape enters through I, its second moment of area.
Hooke's law
OscillationsA spring pulls back towards its rest position with a force proportional to how far it is displaced. The minus sign is the 'back': force and displacement always point opposite ways.
Energy stored in a spring
OscillationsThe work done compressing or stretching a spring is stored in it. Double the displacement and four times the energy is stored.
Angular frequency and period
OscillationsOne full oscillation sweeps 2π radians of phase, so the angular frequency is 2π divided by the time one oscillation takes.
Angular frequency and frequency
OscillationsFrequency counts oscillations per second; angular frequency counts radians of phase per second. There are 2π radians in every oscillation.
Displacement in simple harmonic motion
OscillationsAn object in SHM traces a cosine in time: it swings between +A and −A, repeating every 2π/ω seconds, with φ setting where in the cycle it starts.
Acceleration in simple harmonic motion
OscillationsThe defining property of SHM: acceleration is proportional to displacement and always points back towards the centre.
Maximum speed in simple harmonic motion
OscillationsThe oscillator moves fastest as it passes through the centre, where its speed is the amplitude times the angular frequency.
Angular frequency of a mass on a spring
OscillationsA stiffer spring oscillates faster and a heavier mass slower. The amplitude does not appear at all.
Period of a mass on a spring
OscillationsThe time for one full oscillation of a mass on a spring. Quadruple the mass and the period doubles; how far it is pulled makes no difference.
Period of a simple pendulum
OscillationsA pendulum's period depends only on its length and gravity — not on the mass of the bob, and not on the swing, as long as the swing is small.
Period of a physical pendulum
OscillationsAny rigid body swinging about a pivot. The simple pendulum is the special case of a point mass, where I = m L² and d = L.
Total energy of a spring oscillator
OscillationsEnergy sloshes between kinetic and spring potential every cycle, but the total stays fixed at the spring energy at full stretch.
Critical damping coefficient
OscillationsThe damping at which a displaced oscillator returns to rest fastest without overshooting. Less damping than this and it oscillates as it dies away; more and it creeps back slowly.
Resonance peak of a driven oscillator
OscillationsA driven oscillator's amplitude peaks a little below its natural frequency, and the more it is damped the further below. Past ζ = 1/√2 there is no peak at all.
The wave equation
WavesA wave moves one wavelength in every period, so its speed is frequency times wavelength. The medium sets the speed; the source sets the frequency; the wavelength is whatever fits.
Electromagnetic waves in vacuum
WavesEvery electromagnetic wave — radio, light, X-ray — travels at the same speed in vacuum, so frequency alone fixes the wavelength. Higher frequency, shorter wavelength.
Angular wave number
WavesHow many radians of phase a wave turns through per metre. It does for space what angular frequency does for time.
Wave speed on a stretched string
WavesA tighter string pulls a displaced piece back harder, and a heavier one responds more slowly, so the speed grows with tension and falls with mass per unit length.
Harmonics of a string fixed at both ends
WavesA string fixed at both ends can only hold standing waves that fit a whole number of half-wavelengths, so it sounds a fundamental and whole-number multiples of it. An air column open at both ends follows the same rule.
Wavelength of a string's harmonic
WavesThe nth harmonic fits n half-wavelengths into the string's length. The fundamental's wavelength is twice the string.
Harmonics of a pipe closed at one end
WavesA pipe closed at one end has a node there and an antinode at the open end, so it fits odd quarter-wavelengths only. It sounds an octave lower than an open pipe of the same length, and has no even harmonics.
Speed of sound in a gas
WavesSound in a gas travels faster when the gas is hotter or its molecules are lighter, and does not depend on pressure. Air at 20 °C carries it at about 343 m/s.
Beat frequency
WavesTwo tones of nearly the same frequency drift in and out of step, and the loudness swells as many times a second as the frequencies differ by. The beats slow and vanish as the two are tuned together.
Doppler effect for sound
WavesA source moving towards you crowds its wavefronts together, and you moving towards it meet them more often; either way the pitch you hear rises. Moving apart lowers it.
Phase difference from path difference
WavesEvery extra wavelength of path puts a wave one full cycle behind. A whole number of wavelengths gives constructive interference; an odd number of half-wavelengths gives destructive.
Sound intensity level
WavesLoudness is measured on a logarithmic scale: every tenfold increase in intensity adds 10 decibels. The reference I_0 = 10⁻¹² W/m² is the quietest sound a typical ear can hear.
Law of reflection
OpticsLight leaves a mirror at the same angle it arrived at. Both angles are measured from the normal, not from the mirror's surface.
Refractive index
OpticsA material's refractive index is how many times slower light travels in it than in vacuum. Glass at about 1.5 carries light at two-thirds of c.
Snell's law
OpticsLight crossing into a slower medium bends towards the normal, and into a faster one bends away. The product n sin θ is the same on both sides.
Critical angle
OpticsLight inside a denser medium, striking the boundary more steeply than the critical angle, cannot get out and is totally reflected. This is what keeps light inside an optical fibre.
Apparent depth
OpticsLooking straight down into water, the bottom appears raised: a pool looks about three-quarters as deep as it is.
Thin lens and mirror equation
OpticsWhere an image forms, given where the object is and the focal length. The same equation serves a thin lens and a spherical mirror; only where the image physically sits differs.
Magnification from image and object distances
OpticsHow much larger the image is than the object, and which way up. A negative magnification is an inverted image; a size above 1 is a magnified one.
Magnification from image and object heights
OpticsThe image's height is the object's height times the magnification. A negative height is an image below the axis — upside down.
Focal length of a spherical mirror
OpticsA spherical mirror focuses parallel light halfway between its surface and its centre of curvature.
Power of a lens
OpticsA lens's power is the reciprocal of its focal length in metres, in dioptres — the number on a spectacle prescription. Positive powers converge, negative diverge.
Lensmaker's equation
OpticsA lens's focal length follows from what it is made of and how its two faces are curved. More strongly curved faces, or a higher refractive index, give a shorter focal length.
Acceptance angle of an optical fibre
OpticsLight entering the end of a fibre within this half-angle is trapped by total internal reflection all the way along; light outside it leaks into the cladding. The right-hand side is the fibre's numerical aperture.
Refractive index from a prism's minimum deviation
OpticsTurn a prism until the light is bent least; at that point the ray crosses symmetrically, and the prism's angle and the minimum deviation together give its refractive index. This is the classic way to measure n.
Simple magnifier, image at the near point
OpticsA converging lens held close to the eye lets you bring an object nearer than you could otherwise focus. With the image at the near point the magnification is largest.
Simple magnifier, image at infinity
OpticsWith the object at the lens's focus the image is at infinity and the eye can view it relaxed, at one less than the near-point magnification.
Compound microscope magnification
OpticsThe objective makes a magnified real image and the eyepiece magnifies that again, so the two magnifications multiply. Short focal lengths and a long tube give high power.
Refracting telescope magnification
OpticsA long-focus objective and a short-focus eyepiece: the ratio of their focal lengths is how much larger a distant object looks.
Wavelength in a medium
OpticsCrossing into glass, light keeps its frequency — and so its colour — but its wavelength shrinks by the refractive index, because it travels slower.
Fringe width in Young's double slit
OpticsBright fringes on the screen are evenly spaced, further apart for longer wavelengths, a more distant screen, or slits closer together. Measure the spacing and you have measured the wavelength.
Single-slit diffraction minima
OpticsLight through one slit spreads, with dark bands where the slit's top half and bottom half cancel in pairs. A narrower slit spreads the pattern wider.
Malus's law
OpticsA polariser passes only the part of the light's field along its axis, so the transmitted intensity falls as the square of the cosine. At 45° half the light gets through; at 90° none does.
Brewster's law
OpticsAt one angle of incidence, light reflected from glass or water is completely polarised; at that angle the reflected and refracted beams are exactly 90° apart. Polarising sunglasses exploit it.
Coulomb's law
ElectrostaticsTwo point charges push or pull on each other with a force proportional to each charge and to one over the square of their separation. Double the distance and the force falls to a quarter.
Electric field
ElectrostaticsThe electric field at a point is the force a small positive test charge would feel there, per coulomb of test charge. It belongs to the point, not to the test charge.
Field of a point charge
ElectrostaticsThe field around a single charge points straight away from it (or towards it, if negative) and weakens with the square of the distance.
Potential of a point charge
ElectrostaticsThe electric potential near a charge is the work per coulomb needed to bring a positive charge in from very far away. It falls off as one over the distance, not one over its square, and carries the sign of the charge.
Potential energy of two charges
ElectrostaticsThe energy stored in a pair of charges because of where they sit relative to each other. It is positive for like charges, which would fly apart if released, and negative for unlike charges, which are bound.
Work to move a charge through a potential difference
ElectrostaticsMoving a charge across a potential difference takes, or releases, energy equal to the charge times the voltage. One electron through one volt is one electronvolt.
Field between parallel plates
ElectrostaticsBetween two charged parallel plates the field is the same everywhere, and equals the voltage across them divided by the gap. That is why field strength can be quoted in volts per metre.
Gauss's law
ElectrostaticsThe total electric flux out of any closed surface depends only on the charge inside it — not on the surface's shape or size, and not on any charge outside.
Field of a large charged sheet
ElectrostaticsGauss's law applied to a flat sheet of charge: the field is the same at every distance, because the field lines never spread out.
Capacitance
ElectrostaticsA capacitor's plates hold equal and opposite charge in proportion to the voltage across them. The constant of proportionality is the capacitance, in farads.
Parallel-plate capacitor
ElectrostaticsBigger plates and a narrower gap both store more charge at the same voltage. Filling the gap with an insulator multiplies the capacitance by its relative permittivity.
Energy stored in a capacitor
ElectrostaticsCharging a capacitor stores energy in the field between its plates. The half is there because the first charge moves through almost no voltage and the last through all of it.
Electric current
CircuitsCurrent is the rate at which charge flows past a point: one ampere is one coulomb every second.
Current and drift velocity
CircuitsThe current in a wire is the number of free electrons per cubic metre, times the charge each carries, times the wire's cross-section, times how fast they drift. In copper that drift is well under a millimetre per second.
Ohm's law
CircuitsFor a resistor at constant temperature the current is proportional to the voltage across it. The resistance is the ratio: volts per amp, which is ohms.
Resistance of a wire
CircuitsA longer wire resists more and a thicker one less. The material's own contribution is its resistivity, which does not depend on the wire's shape.
Electrical power
CircuitsThe power delivered to a component is the current through it times the voltage across it — coulombs per second times joules per coulomb.
Power dissipated in a resistance
CircuitsHeat produced in a resistor grows with the square of the current. Halving the current in a transmission line cuts its losses to a quarter.
Power from voltage and resistance
CircuitsThe same power written without the current. At a fixed supply voltage a smaller resistance draws more power, which is why a heater element has a low resistance.
Two resistors in series
CircuitsResistors one after another carry the same current, and their resistances simply add. The total is always more than the largest one.
Two resistors in parallel
CircuitsResistors side by side share the same voltage, and their reciprocals add. The total is always less than the smallest one — another path only makes it easier for current to flow.
Terminal voltage of a cell
CircuitsA real cell loses some of its EMF across its own internal resistance whenever current flows, so the voltage at its terminals falls as the load draws more current.
RC time constant
CircuitsResistance times capacitance is a time: how long a capacitor takes to charge to about 63 % of the supply, or to discharge to about 37 % of where it started.
Capacitor discharging through a resistor
CircuitsA capacitor emptying through a resistor loses the same fraction of its voltage in every equal stretch of time, so the voltage falls exponentially and never quite reaches zero.
Magnetic force on a moving charge
MagnetismA magnetic field pushes on a charge only while it moves, and only on the part of its motion across the field. The force is always at right angles to the velocity, so it turns the charge without speeding it up.
Force on a current-carrying wire
MagnetismA wire carrying current across a magnetic field is pushed sideways — the force that turns every electric motor. A wire lying along the field feels nothing.
Field of a long straight wire
MagnetismThe field around a straight current circles the wire and weakens as one over the distance — not one over its square, because a wire is a whole line of sources.
Force between parallel wires
MagnetismEach wire sits in the other's field, so two parallel currents pull together when they flow the same way and push apart when they flow opposite ways.
Field at the centre of a current loop
MagnetismEvery piece of a circular loop sends its field through the centre in the same direction, so the contributions add up to a field set by the current and the loop's radius.
Field inside a solenoid
MagnetismInside a long coil the field is uniform and depends only on how tightly it is wound (turns per metre) and the current — not on how wide the coil is.
Radius of a charge's circular path
MagnetismA charge moving at right angles to a uniform field travels in a circle. Heavier or faster particles swing wider, which is how a mass spectrometer sorts ions by mass.
Cyclotron frequency
MagnetismHow many times a second a charge circles in a magnetic field. The speed has cancelled out: a faster particle takes a bigger circle in exactly the same time, which is what lets a cyclotron run at one fixed frequency.
Magnetic flux
MagnetismFlux measures how much field passes through an area. It is largest when the area faces the field squarely and zero when the field skims along its surface.
Faraday's law of induction
MagnetismA changing magnetic flux through a coil induces an EMF in it, larger for more turns and for a faster change. Lenz's law gives the direction: the induced current opposes the change that made it.
Motional EMF
MagnetismA rod moving across a magnetic field has an EMF between its ends, because the field pushes the charges inside it along the rod. It is Faraday's law for a circuit whose area is growing.
Self-inductance of a solenoid
MagnetismA coil's inductance grows with the square of its turns: doubling the turns doubles both the field each amp makes and the number of turns that field links.
Energy stored in an inductor
MagnetismBuilding up a current in a coil stores energy in its magnetic field — the energy that sparks across a switch when the current is suddenly cut.
Peak EMF of a generator
Alternating currentA coil spinning in a magnetic field makes a sinusoidal EMF whose peak grows with the turns, the field, the coil's area and how fast it spins.
Angular frequency
Alternating currentOne full cycle is 2π radians of phase, so the angular frequency is 2π times the ordinary frequency in hertz.
RMS voltage of a sine wave
Alternating currentThe RMS voltage is the steady DC voltage that would heat a resistor equally. For a sine wave it is the peak divided by √2 — a 230 V mains supply peaks at about 325 V.
RMS current of a sine wave
Alternating currentThe same √2 relates peak and RMS current. Meters read RMS, and RMS values are what make P = IV give the average power in a resistor.
Inductive reactance
Alternating currentAn inductor opposes changing current, so its opposition to AC rises with frequency. At DC it is just a wire.
Capacitive reactance
Alternating currentA capacitor passes current only while its charge is changing, so its opposition to AC falls as the frequency rises. At DC it blocks completely.
Impedance of a series LCR circuit
Alternating currentResistance and net reactance are a quarter cycle out of step, so they combine like the sides of a right-angled triangle rather than simply adding.
Phase angle of a series LCR circuit
Alternating currentHow far the current lags the voltage. Positive means the inductor wins and the current lags; negative means the capacitor wins and the current leads; zero is resonance.
Ohm's law for AC
Alternating currentIn an AC circuit the impedance takes the place of resistance: RMS voltage equals RMS current times impedance.
Average power in an AC circuit
Alternating currentOnly the part of the current in step with the voltage delivers energy on average. cos φ is the power factor: 1 for a pure resistor, 0 for a pure inductor or capacitor.
Resonant frequency of an LC circuit
Alternating currentAt one frequency the inductor's and capacitor's reactances are equal and cancel, the impedance drops to just the resistance, and the current peaks. That is how a radio tunes to one station.
Quality factor of a series LCR circuit
Alternating currentHow sharp the resonance is. A high Q picks out a narrow band of frequencies, and at resonance the voltage across the inductor or capacitor is Q times the supply.
Ideal transformer
Alternating currentA transformer multiplies the voltage by its turns ratio. Being nearly lossless, it divides the current by the same ratio — it trades voltage for current at constant power.
Heat and temperature change
ThermodynamicsThe heat needed to warm a body is its mass times its specific heat capacity times the temperature rise. Water's large c is why the sea warms more slowly than the sand.
Latent heat
ThermodynamicsMelting or boiling takes energy without any change of temperature. The heat needed is the mass times the latent heat of that change.
Ideal gas law
ThermodynamicsPressure times volume is proportional to the amount of gas and its absolute temperature. Boyle's, Charles's and Gay-Lussac's laws are each this one with two quantities held fixed.
Ideal gas law, counting molecules
ThermodynamicsThe same law written per molecule instead of per mole. Boltzmann's constant is the gas constant shared out over Avogadro's number of molecules.
Mean kinetic energy of a gas molecule
ThermodynamicsTemperature is a measure of the average kinetic energy of the molecules. It does not depend on what the gas is or how much of it there is.
Root-mean-square speed
ThermodynamicsA typical molecular speed. At the same temperature lighter molecules move faster, which is why hydrogen and helium leak out of the atmosphere.
Linear thermal expansion
ThermodynamicsA rod grows in proportion to its length and to how much it is warmed. The gaps left in rail tracks and bridges are sized with this.
Heat conduction (Fourier's law)
ThermodynamicsHeat flows through a slab at a rate proportional to its area and the temperature difference across it, and inversely to its thickness. Doubling the insulation halves the loss.
Stefan–Boltzmann law
ThermodynamicsEvery surface radiates, at a rate that grows as the fourth power of its absolute temperature. Double the temperature and it radiates sixteen times as much.
First law of thermodynamics
ThermodynamicsEnergy is conserved: the heat put into a gas either raises its internal energy or comes back out as work the gas does on its surroundings.
Efficiency of a heat engine
ThermodynamicsAn engine's efficiency is the work it delivers divided by the heat it takes in from the hot reservoir. The rest is dumped as waste heat.
Carnot efficiency
ThermodynamicsNo engine working between two temperatures can beat this. It depends only on the ratio of the reservoir temperatures, and reaching 100% would need a cold reservoir at absolute zero.
Coefficient of performance of an ideal refrigerator
ThermodynamicsHow many joules of heat an ideal refrigerator pulls out of the cold side for every joule of work. The smaller the temperature gap, the better it does.
Lorentz factor
RelativityThe factor by which moving clocks run slow and moving rulers shrink. It is almost exactly 1 at everyday speeds and grows without limit as v approaches c.
Time dilation
RelativityA clock moving past you ticks slowly. The interval you measure is longer than the proper time the moving clock itself records.
Length contraction
RelativityA moving object is shorter along its direction of motion than its rest length. Lengths across the motion are unchanged.
Relativistic velocity addition
RelativitySpeeds do not simply add. Two speeds below c always combine to something below c, and anything combined with c gives c.
Mass–energy equivalence
RelativityMass is a form of energy. A body at rest holds energy equal to its mass times c², about 9 × 10¹⁶ joules for every kilogram.
Total relativistic energy
RelativityA moving body's total energy is its rest energy multiplied by the Lorentz factor, so it climbs without limit as the speed approaches c.
Relativistic kinetic energy
RelativityKinetic energy is the total energy minus the rest energy. At low speeds it reduces to ½mv²; near c it grows much faster.
Relativistic momentum
RelativityMomentum is γmv rather than mv. It grows without limit as v approaches c, which is why no push can bring a massive body up to the speed of light.
Energy–momentum relation
RelativityTotal energy, momentum and rest mass are tied together in one relation. For a massless particle such as a photon it reduces to E = pc.
Schwarzschild radius
RelativitySqueeze a mass inside this radius and not even light can escape: it becomes a black hole. The Sun's is about 3 km, the Earth's about 9 mm.
Energy of a photon
Quantum physicsLight comes in packets whose energy is Planck's constant times the frequency. Doubling the frequency doubles the energy of every photon, however bright or dim the light.
Photon energy from wavelength
Quantum physicsThe same photon energy written with the wavelength. Shorter wavelengths carry more energy per photon, which is why ultraviolet burns and radio does not.
Einstein's photoelectric equation
Quantum physicsOne photon frees one electron. The photon's energy pays the work function to get the electron out of the metal, and whatever is left over is the fastest electron's kinetic energy.
Stopping potential
Quantum physicsThe reverse voltage that just stops the fastest photoelectrons. It is the photoelectric equation read with a voltmeter, and plotting it against frequency gives a straight line of slope h/e.
Threshold frequency
Quantum physicsThe lowest frequency that can free an electron from a metal: a photon at this frequency has exactly the work function and nothing to spare.
Wien's displacement law
Quantum physicsA hotter body's glow peaks at a shorter wavelength. The Sun at about 5800 K peaks in visible light; a person at 310 K peaks in the infrared.
de Broglie wavelength
Quantum physicsEvery moving particle has a wavelength, Planck's constant over its momentum. For anything heavier than a molecule it is far too small to notice.
Wavelength of an accelerated electron
Quantum physicsAn electron accelerated from rest through a voltage V has this de Broglie wavelength. At 54 V it is about the spacing of atoms in a crystal, which is how electron diffraction was first seen.
Position–momentum uncertainty
Quantum physicsThe more tightly a particle's position is pinned down, the more spread its momentum must have. This is the smallest spread allowed; any real measurement does no better.
Energy–time uncertainty
Quantum physicsA state that lasts only a short time cannot have a sharply defined energy. It is why short-lived excited atoms emit spectral lines with a natural width.
Radius of a Bohr orbit
Quantum physicsIn Bohr's model the electron can only orbit at certain radii, growing as n². For hydrogen's ground state (n = 1, Z = 1) this is the Bohr radius, about 0.053 nm.
Energy levels of a Bohr atom
Quantum physicsThe electron's allowed energies are negative, because it is bound, and crowd together as n grows. For hydrogen the ground state is −13.6 eV, and a jump between levels emits one photon carrying the difference.
Particle in a box
Quantum physicsA particle trapped between two walls can only have certain energies, rising as n². The smaller the box, the larger the gaps between them — which is why confinement makes energy quantised.
Radius of a nucleus
Nuclear physicsA nucleus's volume grows in proportion to the number of nucleons in it, so its radius grows as the cube root. Every nucleus therefore has about the same density.
Mass defect
Nuclear physicsA nucleus weighs less than the protons and neutrons it is made of. The missing mass is the mass defect, and it is the binding energy divided by c².
Nuclear binding energy
Nuclear physicsThe energy needed to pull a nucleus apart into separate protons and neutrons is its mass defect times c². Divided by the mass number it gives the binding-energy curve, which peaks near iron.
Q value of a nuclear reaction
Nuclear physicsAdd up the masses going into a reaction, subtract the masses coming out, and multiply by c². A positive Q is energy released — about 200 MeV for one uranium fission, 17.6 MeV for one D–T fusion.
Coulomb barrier
Nuclear physicsTwo nuclei repel until they are close enough for the strong force to take over. This is the energy needed to get them that close — about 1.4 MeV for two deuterons, far more than a star's heat supplies, which is why fusion relies on tunnelling.
Radioactive decay law
Nuclear physicsEach nucleus has the same chance of decaying every second, so the number left falls exponentially. No sample ever quite finishes decaying.
Half-life and decay constant
Nuclear physicsThe half-life is the time for half of any sample to decay. It is fixed by the decay constant and does not depend on how much material there is.
Decay counted in half-lives
Nuclear physicsThe decay law counted in half-lives: after one half-life half remains, after two a quarter, after ten about a thousandth. Radiocarbon dating reads t from the fraction left.
Activity of a sample
Nuclear physicsThe activity — decays per second, in becquerels — is the decay constant times the number of nuclei present. A short half-life means a fierce but brief activity.
Thermal voltage
SemiconductorsThe voltage equivalent of thermal energy, about 25.9 mV at room temperature. It sets the scale of every exponential in a semiconductor device.
Intrinsic carrier density
SemiconductorsHow many electrons heat alone lifts across the band gap in a pure crystal. The gap sits inside an exponential, so a slightly wider gap means vastly fewer carriers — silicon has about 10¹⁶ per cubic metre, an insulator almost none.
Mass-action law
SemiconductorsIn equilibrium the product of electron and hole densities is fixed by the material and temperature. Dope in more electrons and the holes fall in proportion.
Conductivity of a semiconductor
SemiconductorsElectrons and holes both carry current, each contributing its density times its mobility. Doping raises one density enormously, which is how a few impurity atoms per million make silicon conduct.
Built-in potential of a p-n junction
SemiconductorsThe barrier a p-n junction builds for itself as carriers diffuse across it. The doping sits inside a logarithm, so even a thousandfold change moves it only a little — which is why silicon junctions all sit near 0.7 V.
Diode equation (Shockley)
SemiconductorsThe current through a p-n junction. Forward, it grows tenfold for every 60 mV or so; reverse, it settles at the tiny saturation current. One expression covers both directions.
Transistor current gain
SemiconductorsIn its active region a bipolar transistor passes a collector current β times its base current — typically a hundred or more. A small current controls a large one.
Transistor currents
SemiconductorsEverything that flows into a transistor comes out of the emitter: the emitter current is the base and collector currents together.
LED colour from band gap
SemiconductorsAn LED emits one photon each time an electron drops across the band gap, so the gap sets the colour. Gallium arsenide's 1.42 eV glows in the infrared; a 2.6 eV gap glows blue.
Average output of a half-wave rectifier
SemiconductorsA half-wave rectifier passes only the positive half of each cycle, so its average output is the peak voltage divided by π — about 32% of the peak. A full-wave bridge doubles it.
Ripple on a smoothed rectifier
SemiconductorsA smoothing capacitor sags between recharges by this much. A bigger capacitor, a lighter load or more frequent recharging all shrink the ripple.