Can I start by saying that in all probability nobody will actually work out calculations for you, (maybe you are not expecting to, apologies in that case, but you don't really need these, as extrapolation from existing objects and effects will suffice, and save all of us a lot of time :).
I'm writing a story in which a large (city-sized) swarm of nanoparticles are being held together and directed by a massive electromagnetic field being generated by a central "core" hidden within the swarm. In order for this amoeba-like swarm to move its arms/tendrils/psuedopods at a distance of 50km (roughly large enough to engulf most of the major Tokyo wards), how strong would the electromagnetic field controlling the swarm's movements need to be at the center?
Scale it up from the largest electromagnetic device you can find in use, by searching the Internet. You may have to make the jump from the power output and energy requirements of the LHC, which are great enough that it has to be switched off in winter or the citizens of the surrounding area will be very annoyed at the reduced supply of electricity.
The only real equation you need to seriously allow for here is the inverse power law. Hey, Isaac Asimov did all his own research, without the Internet. ....its no problem to you.
The swarm itself is comprised of superconducting nanoparticles suspended in solution similar to ferrofluids, and they likely incorporate large amounts of carbon, silicon, and metals commonly found in building materials (I'm leaving their exact composition deliberately vague to avoid dating the science too much). Each individual particle is between 1-100 nm in width.
Once you have a figure from above, assume each swarm member needs a certain amount of energy and use that to act as a check on your first figure by multiplying the individual energy by the swarm number. Assume each one needs the energy of a bacterium up to a bee, depending on their capabilities.
What kind of effect would a field this size and strength have on the humans, plants and animals inside? Would they die or receive severe cellular damage? What about buildings and electronics? How many Tesla would this field be producing?
If it's magnetism, it's not going to work as a direct drama hook. You have minimal effects from strong magnetic fields, an example is an mri scanner, which is 2 to 4 T and completely harmless, so you won't scare too many people with that. The swarm through, yeah that would worry me more than an MRI.
Find out the LHC facts and data and work from there, the very best of luck with it.
