An arc lantern uses a summoning gate to place a small patch of air in electrical contact with the ionosphere. A second summoning gate of lesser range places another patch of air in electrical contact with ground. On most habitable worlds, these two electrical contacts have a voltage difference of tens of millions of volts. This voltage difference causes an electric arc, which then gives off light.
The trick to making an arc lantern work is controlling this electric arc. If one simply allowed electricity to flow unimpeded between the two contact points, the device would be both very unsafe and very inefficient. The voltage available to it is much greater than what is necessary to break down air over the length of any reasonable sized device, so the excess voltage would cause a voltage drop on the way into the summoning gates, lighting up the sky but not the area it is supposed to illuminate. The voltage would also be much greater than what is needed to break down air between the arc lantern and its user, causing its user to get a nasty electric shock.
Instead, the electric arc is forced onto a circuitous path, winding many times around the device before actually completing its journey between the contact points. This gives it a much higher resistance, and consequently draws much less current.
Arc lanterns have the advantage that their energy source is technically free. However, unlike magma lanterns and related devices, their energy source is not unlimited. A sufficient number of arc lanterns on a given planet can deplete the ionosphere-ground potential difference faster than it can be replenished by cloud-to-sky lightning strikes. Fortunately, the cost of element control magic sufficient to handle such high voltages makes them marginally more expensive than magma lanterns, so their use is sufficiently limited to avoid such problems.
With slight variation, an arc lantern can also be used to create a magic-based arc welder.