Electronic – Appropriate shapes of an Electromagnet

Assuming this is a DC application, wind as many turns as you have room for.

The reason being, magnetization is the product of the number of turns and the current through them. If you double the number of turns, however, while keeping the wire gauge constant, you double the resistance and thereby get half the current from your fixed voltage source; the turns*current product comes out being the same. What's different, though, if you double the number of turns, is that the power that the coil will draw will be half.

So if you keep adding turns, the magnetization doesn't change, but the current drawn goes down, and consequently the power required to run the coil decrease, too.

(If you're intending to use AC though, even 50Hz or 60Hz, the above argument doesn't apply since it disregards inductance issues.)

First, a battery is basically a voltage source. The current rating is how much current the battery can deliver when the load demands it. It is not the current the battery always puts out. So, just because the 5 kV battery can deliver up to 500 mA, it doesn't mean that the coil will draw that much current. That is up to the coil. In steady state, the coil would have to have a resistance of 5kV / 500mA = 10 KOhms to draw the 500mA. More, and it will simply draw less current. Less, and the battery's current capacity is exceeded and its output voltage will probably droop, it could vanish into a puff of greasy black smoke, or whatever.

The magnetic field strength generated by any electromagnet is directly proportional to the current thru it. Let's say your coil had a DC resistance of 10 KOhms and therefore drew 500 mA from the 5 kV battery. The current it would draw from the 500 mV battery would be 500mV / 10kOhms = 50 uA. That means the magnetic field will be 10000 times weaker. The fact that the battery could have delivered 5 kA is irrelevant since the coil isn't trying to draw that much current.

Earlier I said that the magnetic field of a electromagnet is proportional to its current. That is true for any one electromagnet. Different electromagnets can be constructed with different proportionality constants. For example, it's quite possible to make one electromagnet that draws 1A at 10V and another one that draws 2A at 5V that have the same magnetic field strength. The first one would use a longer but thinner wire, but both could be wrapped around the same core.