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Tony Stark designed and developed an arc reactor in a cave. Did he already have a design of an arc reactor in mind?
The underlying arc reactor technology existed before the events of the film, you can see the large-scale version within the Stark building of the first film and it plays a part in the final battle. Stane describes it as a 'sop to the hippies' or something along those lines, with an implication that it is too expensive to be of practical use.
Stark's genius (in this case) was in somehow miniaturising the technology, which seems to have made it more practical. Since his company had already made a building-sized arc reactor, and he is the driving intellectual force behind his company, it seems very likely that he was already aware of the underlying principles of the thing and that he made some kind of creative leap while in the cave with a box of scraps that allowed him to improve the process.
Not likely. He managed to recreate a technology without blueprints, schematics, or a clean room, made completely of spare parts and miniaturized a technology to one ten thousandths, of it actual size AND make it produce the energy output of a small nuclear reactor. Oh yes, did I mention he did it in under a week? See: Phlebotinum
You cannot take this stuff seriously. This is one of those comic things where a willing suspension of belief is an absolute requirement.
It is completely impossible for him to have created his personal arc reactor under those conditions. The idea of Tony Stark, as a character, is that he is genius personified. In the Marvel Universe, he is one of the most inventive, creative and adaptive humans alive.
The arc reactor had existed for at least 30 years, as Obidiah Stane reminds Tony, as a publicity stunt, nothing more. It was not energy effective, implying it cost as much in energy to run it as it produced.
The very first Iron Man armor. Grey, dull, and ran on battery power. It's initial power? It deflected bullets.
And such as it is when something is adapted for movies, the adaptation always tries to exceed the original. Given its immense size, power requirements and up to this point inability to be a break-even proposition on energy, it would have been nothing more than a pretty science project in Stark's book.
So in this one, Tony will, somehow, create an energy source, faster, stronger, better, cleaner and ten thousand times smaller, than anything ever seen before from the stone knives and bearskins lying around a terrorists camp in some Third World Nation (filled with his weapons) and in addition to the miracle of the Iron Man Armor, he will also create the world's first pocket-sized tokamak fusion reactor.
This is a real tokomak fusion reactor. It is an immense thing, far larger than even the arc reactor (show above)
@user18957: Tony's genius was to minituarise the arc reactor running at Stark Industries. The 2nd installment of series shows that the original (and successful) arc reactor was prepared by his dad, Howard stark, in collaboration with the Russian physicist Anton. hope this provides a satisfactory answer to your query. :)
