Connection Map
Several equations across this reference are the same mathematical statement wearing a different label.
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.