Power Delivered to a Resistor

A resistor gets hot because the charge moving through it is losing energy to collisions inside the material. Each charge drops through a potential difference V as it crosses the resistor, which means it loses energy qV. If a current I flows, then charge is arriving at rate I, so energy is being dumped at rate P = IV. That is power in watts: joules of electrical energy converted to heat every second.

To get a formula that uses only one circuit quantity, bring in Ohm's law, V = IR. Substitute V = IR into P = IV and you get P = I^2R. Substitute I = V/R instead and you get P = V²/R. All three forms say the same physical thing — energy lost per second — just packaged around different known quantities. Use I^2R when you know the current, V²/R when you know the voltage across the resistor, and IV when you have both directly.

This sits right after Ohm's law and right before Kirchhoff's rules, because every loop and junction calculation eventually asks how much power each resistor burns. It also connects forward to batteries and internal resistance: a battery's own resistance wastes power the same way, which is why real batteries warm up and why terminal voltage sags under heavy current.