## Power in watts

P = i * e or watts = amps * volts

e = electromotive force or what we call voltage today.

## Thermal Power

aka: Copper Losses, Joule’s First Law, Ohmic Heating, Joule heating:

P = i^2 * r

## Why Higher Voltage is Cheaper

high voltage lines are cheaper to construct. if you accept that every gauge of copper wire/conductor has a finite(limited) amount of heat it can handle/disipate untill it glows or drips and then that heat generated is the product of resistance and current squared it you are most of the way there. Apply the first law and if you double the voltage you half the amperage which results in 1/4th the heat generated.

Eample:

we need to get 2000w to the barn for a heater.

we can chose 500v or 1000v,

by the first formula we can see that

so what about resistance? wire of any gauge has a resistance per unit of distance, multiply by length to get resistance of line. aka 2ohms per meter and we want a 6m run. that length of line is 12ohms. For the sake of simplicity/ilistration lets leave length out. we have a fat (pick a number) piece of wire that runs the distance we need and has a total resistance of 1ohm. it can take 10w of heat before it melts.

## 500v Heat Gen

heat(w) = 4^2amps * 1 ohm

16w = heat

## 1000v Heat Gen

Heat(w) =2^2 amps * 1 ohm

4w heat

Clearly if we wanted to stick to 500v we need to go buy a fatter wire (smaller gauge)

copper costs per unit of weight and obviously thicker ways more. also wire is ususally sold by cost per gauge per distance but im trying to keep it simple here.

Real world Drawbacks to higher voltage lines:

Higher risk of electricution.

Higher elevation/towers if bare wire is hung thicker insulator otherwise.

higher tower = more metal or wood but thats usually cheaper than the conductor per unit of length/weight.

why Europe is 220v can usually be summed up as copper costs.

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