Power and Energy

Resistor Heating

the same power two ways

For an ohmic resistor, voltage times current and current squared times resistance give the same power.

Example

For an ohmic resistor, voltage times current and current squared times resistance give the same power. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A resistor spends power as heat

The resistor is where the circuit's electrical energy is spent. For an ohmic resistor, power can also be written as current squared times resistance.

P=I2RP = I^{2} R
A resistor turns electrical power into heatA single loop with a battery, an ammeter, and a resistor carrying current; the resistor is the load where energy is spent.12 VA3 A4 ohm3 Aheat

Square the current

The current is 3 amperes, so current squared gives 9 before the resistance factor.

I2=3⋅3=9I^{2} = 3\cdot 3 = 9

Current makes heating grow faster than linearly

Hold the resistor at 4 ohm. Increasing current raises the heat power by the square of the current, so the watts spread out quickly.

IRP1 A4 ohm4 W2 A4 ohm16 W3 A4 ohm36 W\begin{array}{c|c|c}I & R & P \\1\ \text{A} & 4\ \text{ohm} & 4\ \text{W} \\2\ \text{A} & 4\ \text{ohm} & 16\ \text{W} \\3\ \text{A} & 4\ \text{ohm} & 36\ \text{W}\end{array}
A resistor turns electrical power into heatA single loop with a battery, an ammeter, and a resistor carrying current; the resistor is the load where energy is spent.12 VA3 A4 ohm3 Aheat

At fixed current, more resistance makes more heat

Now hold the current at 3 amperes. The current-squared part is fixed, so a larger resistance spends more watts as heat.

IRP3 A2 ohm18 W3 A4 ohm36 W3 A6 ohm54 W\begin{array}{c|c|c}I & R & P \\3\ \text{A} & 2\ \text{ohm} & 18\ \text{W} \\3\ \text{A} & 4\ \text{ohm} & 36\ \text{W} \\3\ \text{A} & 6\ \text{ohm} & 54\ \text{W}\end{array}
A resistor turns electrical power into heatA single loop with a battery, an ammeter, and a resistor carrying current; the resistor is the load where energy is spent.12 VA3 A4 ohm3 Aheat

Use current squared times resistance

Multiplying by the 4 ohm resistor gives 36 watts, the same power found from voltage times current.

P=I2R=9⋅4 ohm=36 WP = I^{2}R = 9\cdot 4\ \text{ohm} = \hl{36}\ \text{W}
A resistor turns electrical power into heatA single loop with a battery, an ammeter, and a resistor carrying current; the resistor is the load where energy is spent.12 VA3 A4 ohm3 Aheat
electricity The same resistor power can be found from V times I or from I squared times R.