Disclaimer: this is a dangerous project.
1.This is very HOT up to 300 degrees minimum so be careful.
If you get hurt during this project it is not my fault.
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Signing UpStep 1: Materials
Battery (at least half an amp other wise it wont work)
Refills for refillable pencil also known as carbon graphite
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voltage X resistance(in ohms)= amps
Ohm's law is U = I*R with U = Voltage, I = Current and R = Resistance.
Or, if you divide both sides by R and flip the around: I = U / R
Since the power for DC is P = U*I, you will find that P = U*U/R = U²/R
(double voltage gives four times the power!)
Since graphite has a pretty low resistance the current could be really high. For a valid calculation you would have the source resistance of the power supply (the battery here) into account.
So the resulting current will be:
I = U / (Rgraphite + Rbattery)
The inner resistance of NiCd or NiMH batteries (better accumulators) is very low. Bigger packs can drive ten or more Amps easily. Togehter with the low resistance of the graphite this results in the nice light show. (And is the reason why some of the laptop batteries exploded in the last years). So don't try that with alkaline or zinc-coal batteries - it's just a waste of money. The inner resistance of those is too high and will limit the current and the fun.
Boy, these accumulators would have made some of my first experiments in electronics so much more fun... ;-)
But, with a 'normal' power source (that is a voltage source) you have a constant voltage (U) and you current will follow according to I = U / R. Normal batteries, accumulators or (especially) regulated power supplies act as constant voltage sources. That is within reasonable limits, things change when the source is shorted or heavily drained (like here). That's when the inner resistance and other effects (chemical etc.) come into play and make the voltage not be a constant factor any more.
If, on the other hand, you have a constant current source (not easy to be found, but can be done easily with an electronic circuit) the resulting voltage on a resistor will follow U = I * R and you will get the power with the formula you gave.
It just depends on the type of your source. Constant voltage vs. constant current. For easy designing, accumulators are treated as constant voltage type (until near empty). In real life... well, life is complicated, isn't it?