Reference Library

The fundamental constants of nature, gathered here for reference — each has already appeared in its own module.

Converting between unit systems means multiplying by a conversion factor equal to one — for example, multiplying by (1 m / 3.281 ft) to convert feet to metres. Chaining several such factors converts through several units at once, and any unit left over that shouldn't be there is an immediate signal that a conversion factor was inverted or omitted.

SI prefixes attach a power-of-ten multiplier to any unit, keeping numbers a manageable size across vastly different scales.

Every physics problem implicitly assumes a model: point particle versus extended body, ideal gas versus real gas, non-relativistic versus relativistic. Stating that choice explicitly — and knowing where it breaks down — prevents applying a formula outside the regime where it holds.

Exploiting a problem's symmetry — spherical, cylindrical, translational, time-reversal — before writing any equation often eliminates most of the algebra. A field problem with spherical symmetry needs Gauss's law and almost no integration; the same field computed by brute-force integration takes far more effort for the same answer.

A correct result should behave sensibly in extreme or special cases: does it reduce to a known simpler formula when a parameter goes to zero or infinity? Does it match an expected symmetric special case? A formula that fails an obvious limit is wrong, however correct the algebra leading to it looked.

Checking that a final answer has the expected dimensions catches the majority of algebra slips in under a minute, well before comparing to a numerical answer.

Coulomb's law and Newton's law of gravitation are both inverse-square laws between two point sources — only the source (charge vs mass) and the fact that gravity is always attractive differ. The electric field and the gravitational field are both defined as 'force per unit source', identically in form. A moving charge and a moving mass both trace out motion governed by the same constant-acceleration equations once the force producing that motion is known.

The exponential decay of a discharging capacitor, radioactive decay, and the extinction of light through an absorbing medium are all solutions of the same differential equation, dy/dx = −ky — recognizing the form immediately reveals the shape of the solution. Simple harmonic motion — the differential equation behind an oscillating spring, a pendulum, and an oscillating circuit's charge — is one equation, d²y/dx² = −ω²y, applied with three different meanings of ω. The standing-wave condition on a string and the allowed energy levels of a particle in a box arise from the identical boundary-value problem: fitting whole or half wavelengths into a fixed length.

The mechanical-electrical analogy (mass ↔ inductance, spring stiffness ↔ inverse capacitance, damping ↔ resistance) is a powerful aid to intuition, but not a perfect correspondence — electric charge has no negative-mass analogue, and the analogy says nothing about magnetic effects, which have no mechanical counterpart at all. An analogy is a tool for the algebra to fall out the same way, not a claim that the underlying physics is identical.

Every equation's own symbol table, opened from its popup, is the authoritative reference for that formula's notation. A handful of symbols are deliberately reused across modules with a consistent meaning: r for a radial distance, θ for an angle from a reference direction, ω for angular frequency.

Sign conventions are a recurring source of error, not because the underlying physics is hard but because equally valid choices differ from one context to the next — current direction, mirror and lens distances, and work in the first law of thermodynamics all vary by convention. When a result comes out with an unexpected sign, check the convention in use before assuming an error in the physics.

An arrow over a symbol denotes a vector, carrying both magnitude and direction; a hat denotes a unit vector, of magnitude one, giving direction alone. A vector's magnitude is written without the arrow, or between vertical bars — |A⃗| and A mean the same thing. This reference follows that convention throughout.