Result

Power

P = I Δ V = I^{2} R = \frac{( Δ V ) ^{2}}{R}

Statement

The rate at which energy is delivered to a circuit element.

Symbols and units

$P$	power delivered to the element	W = J/s
$I$	current	A
$Δ V$	voltage across it	V
$R$	resistance, for the last two forms	Ω

When it applies

Any element. The I²R and (ΔV)²/R forms assume a resistor.

Don't use when

The I²R and (ΔV)²/R forms assume a resistor specifically — they don't apply directly to a non-ohmic element where R isn't constant.

Derivation

Each coulomb passing through drops by ΔV volts, losing energy:

d U = Δ V d Q

Differentiate with respect to time:

P = \frac{d U}{d t} = Δ V \frac{d Q}{d t} = I Δ V

For a resistor, substitute ΔV = IR:

P = I (I R) = I^{2} R

Or substitute I = ΔV/R instead:

P = \frac{( Δ V ) ^{2}}{R}

Worked examples

2 kW heater on 230 V

I = \frac{2000}{230} \approx 8.7 A, R = \frac{23 0 ^{2}}{2000} \approx 26 Ω

Why transmission runs at high voltage

1 MW over a line of 5 Ω at 10 kV:

I = 100 A, P_{loss} = I^{2} R = 50 kW

The same power at 400 kV:

I = 2.5 A, P_{loss} = 31 W

A factor of 1600 in loss for the same delivered power.

Watch out for

All three are one equation — pick the form whose quantities are held fixed. In series the largest R dissipates most; in parallel the smallest R does.

In practice

Household outlets are wired in parallel at one voltage. A fuse or breaker caps the total current so the wiring cannot overheat; earthing gives fault current a safe path, and an RCD trips the instant live and neutral currents differ.

Connects to

Geometry and material Terminal voltage Combining resistors