The Physics
Opus in profectus

Electric Power

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James Joule (1818–1889) England, verified Ohm's Law and determined that the heat delivered by a conductor is directly proportional to its resistance and to the square of the current through it.

We see, therefore, that when a current of voltaic electricity is propagated along a metallic conductor, the heat evolved in a given time is proportional to the resistance of the conductor multiplied by the square of the electric intensity.

James Prescott Joule, 1841

Electric power from the definition of power. Multiply by one and commute the variables in the denominator. Look at what this gives us.

P =  W  =  W   q  =  W   q  = VI
t t q q t

Power in terms of current. Remove the voltage by substituing from Ohm's law.

P = VI = (IR)I = I2R

Power in terms of voltage. Remove the current using Ohm's law.

P = VI = V  V  =  V2

In summary…

P = VI = I2R = V2

Consumer affairs…

A portion of a residential consumer electric bill. Utility companies sell electric energy by the kilowatt hour; a unit that simplifies calculating operating costs for electrical devices. The energy consumed during this particular billing cycle was quite small (when compared to similar consumers), but the rate charged by this utility was approximately double the US average in the year 2000.

Conventional (copper-based) cables are capable of transmitting power (40 to 600 MW) at high voltages (40 to 345 kV)

A similar bill from 2015.

Line loss

Ploss = I2loadRline = 
Pload 2

and loss fraction.

loss fraction =  Ploss  =  PloadRline
Pload V2line