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Electric power is the rate at which electrical energy is converted to other forms: P = IV, measured in watts (W = J/s). For a resistor, this becomes P = I²R = V²/R using Ohm's law. Power dissipated as heat in a resistor (Joule heating) is always positive regardless of current direction. Over time, energy E = Pt is consumed; utility companies bill in kilowatt-hours (kWh), where 1 kWh = 3.6 MJ.
Derive the three forms of the power formula from P = IV and V = IR. Apply to real-world problems: light bulb wattage, household circuits, and transmission line losses to understand why high-voltage transmission is efficient.
From your study of work and energy, you know that power is the rate of doing work: P = dW/dt, measured in watts. In a circuit, charges are the carriers of energy. When a charge q moves through a potential difference V, the work done on it is W = qV. Dividing both sides by time, the rate at which energy is delivered is P = (q/t) × V = IV, where I = q/t is the current. This gives the fundamental formula P = IV: power equals current times voltage. Every watt delivered to a circuit element represents one joule of energy per second flowing into it.
For a resistor obeying Ohm's law (V = IR), you can substitute to get two equivalent forms. Replacing V with IR gives P = I × (IR) = I²R; replacing I with V/R gives P = (V/R) × V = V²/R. All three formulas — P = IV, P = I²R, P = V²/R — are equivalent for a resistor, but they differ in which variable is held fixed. In a series circuit, the same current flows through all elements, so P = I²R is most useful; resistance increases power dissipation. In a parallel circuit, all elements share the same voltage, so P = V²/R is most useful; resistance decreases power dissipation. Choosing the wrong formula when the context implies the other is a common error.
The energy converted in a resistor appears as heat — this is Joule heating. It is irreversible: unlike energy stored in a capacitor or compressed spring, the heat dissipated cannot be converted back to electrical energy by the resistor. The total energy consumed over time t is E = Pt. Electric utility companies measure this in kilowatt-hours: a device drawing 1 kW for one hour consumes 1 kWh = 3.6 MJ. A 100 W light bulb running for 10 hours uses 1 kWh.
Understanding power also explains long-distance electrical transmission. Moving a given amount of power P at high voltage V requires only a small current I = P/V. Since transmission line losses scale as I²R (the Joule heating formula with fixed line resistance R), halving the current cuts losses by a factor of four. This is why power plants step voltage up to hundreds of thousands of volts for transmission, then step it back down for residential use. The physics is P = I²R: reduce I to reduce wasted heat in the lines.