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If I take an electrolytic capacitor, remove the aluminium casing, unroll the foil/ electrolyte-soaked paper, if charged, will it still work? And what if I took of the cathode foil and wrapped the anode foil with the paper around a metal rod -- will it work if charged? If not, why?

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Dodo, I'm not really sure why you would want to do this.

I do not think it will work when unrolled. This is also dangerous and not advisable. IF you will go through with it anyway, wear gloves and do it in a well ventilated area. If you remove the case and unroll it, it is just a flat piece of foil. The foil doesn't make the capacitor as much as it is the foil AND the geometry. If the anode foil and cathode foil were still ordered properly after removal, it would still work obviously, but I do not think it would work to the manufactories specifications.

If the rod were in direct contact with the layer you put on the rod, yes it would work. To what end you would want that for, I can only guess. Cheers.

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if this answers your question, please hit the check next to the left of this post. If not, leave it alone or ask more questions in the comments. This is the way of SE. –  Dylan Sabulsky Dec 7 '12 at 5:30
What is the use of the aluminum casing anyway? –  dodo Dec 7 '12 at 6:15
It is mostly for isolation purposes. Also it keeps the liquid in pretty well. If it is electrolytic, it also has a cross cut into the top. It the capacitor is overloaded, it will bulge out before blowing spectacularly. The aluminum casing is strong enough to usually hold from blowing in a case of overload. –  Dylan Sabulsky Dec 7 '12 at 6:17

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