The following is extremely rough and hand-wavy, but not totally unreasonable. My distant memory from doing chapter problems from Jackson are that magnetic fields drop off with the cube of the distance (dipole fields at reasonable distances), so 40 feet would have fields something like 1/40^3 compared to fields at 1 foot, so a factor of around 1/640. The voltage of the electrical power at that exit sign is something like 120V, while the cell phone battery is around 3v or 6v, so you might think that at the same distances, the exit sign has fields that are something like 20 or 30 times as strong, so given the distances involved, one might think that the Exit sign would have a smaller effect. Think again! The fields are not a function of the voltage, but rather a function of the current. Incandescent bulbs at 100W have about 1 Amp of current running through them, while cell phones transmit around 1-3W, giving currents in the 1 to 2A range for 3-6V (which seem absurdly high for hand-held electronics). So at similar distances, they seem to be giving similar fields, but at the 40 foot distance, the closer source would be much more important.
But of course, none of this makes any sense, since a disconnected battery has zero current flowing, and thus creates zero magnetic field beyond any intrinsic magnetism of its component parts. Magnetic fields from incandescent bulbs powered by 120V AC wiring are difficult to measure at 40 foot distances since they are so small, but they are still way bigger than zero.