Electric power and Joule heating

Pushing current through a resistor takes work, and that work becomes heat — Joule heating, the same effect that makes a toaster's element glow and a phone charger warm up. The rate at which that work is done is power, P = IV: current times voltage, exactly as mechanical power is force times velocity.

Because a resistor obeys Ohm's law, V = IR, current and voltage are not free to vary independently in a resistor — fix one and the material fixes the other. That lets P = IV be rewritten by substitution: replacing V with IR gives P = I²R; replacing I with V/R gives P = V²/R. These are not three different quantities that happen to coincide. They are one quantity, algebra allowing you to compute it from whichever pair of values you happen to know.

The exponents matter in practice. Because power depends on the square of current, doubling the current through a fixed resistor quadruples the heat generated, not doubles it — which is why a wire carrying too much current overheats catastrophically rather than gradually, and why fuse ratings are taken seriously. And because power also depends on the square of voltage, raising a heater's voltage has an outsized effect on how hot it gets, while raising its resistance at fixed voltage actually reduces power (P = V²/R), the opposite of what "more resistance, more heating" suggests — the correct intuition depends on which of current or voltage is being held fixed.