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Resistivity and Power Dissipation

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Voltz

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Okay, another thing that's eluded me is how to work out the power dissipated by a wire when you know the resistivity/length/area or resistance etc...

Basically, if I wanted to build a heating element and I knew the specifications of a wire how do I work out how much power will be dissipated with a given voltage, I have no problem working out the resistance of a given length of wire R = (ρ x l)/a

But after working this out when I plug it into an ohm's law equation V²/R = W then it looks like the higher the resistance the lower the power dissipation so if I want more power dissipation I should have a very short low resistance wire? Or have I plugged the equation wrong or even used the wrong equation completely
 
When the voltage is constant, the lower the resistance the higher the power dissipation.
P = V²/R

When the current is constant, it's the other way round, the higher the resistance, the higher the power dissipation.
P = I²R

The above is only true for perfect constant current and voltage sources.

Then it isn't a perfect voltage source, the power dissipated in the resistor will increase, as the resistance is reduced up until it's equal to the power supply's internal resistance; this is known as the point of maximum power transfer. When the load resistance is lower than the power supply's internal resistance, the power dissipated in the load will fall and more power will be dissipated inside the power supply. In reality, we don't want maximum power transfer because the efficiency will only be 50%, as half of the power is being lost in the power supply. Most real power supplies will either overheat or shut down, before the load resistance is reduced to the point of maximum power transfer.

Maximum power theorem - Wikipedia, the free encyclopedia
 
So what's the e.m.f for a PP3 Battery then? If its my power supply then how much do you think I can reliably dissipate before its internal voltage becomes a limiting factor (just an example)
 
A PP3 alkaline battery (most know it as a common 9V) will have a terminal voltage very close to 9V when new. If you draw 10mA for a while, its voltage will fall to about 8.4V. As it goes dead (flat) it will begin falling more rapidly. If you draw 100mA it will fall more quickly and most brands will be dead between 1 and 2 hours.

P3 batteries are also available in NiCd, NiMH, and Lithium. Their actual voltage depends on their construction. (Some NiCd and NiMH have 6 cells = 7.2V, others have 7 cells and are 8.4V).

Do not use the information as a specification, because I didn't describe any particular battery. Manufacturers have data sheets that give detail about each product. Duracell and Eveready might be good places to look.
 
Eveready is obsolete. Now they are called "Energizer". Their website has a datasheet for every battery they sell.
Here is a graph of voltage vs current and time for a 9V alkaline battery:
 

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