Ten Amps Five Centimetres Away Rivals the Earth's Magnetic Field

Falling with distance, not with distance squared

Most fields from a point source thin out as the square of the distance, because the same amount spreads over a sphere. A long straight wire is not a point: the field spreads over a cylinder instead, whose surface grows in proportion to the radius, so the field falls only in proportion. Ten times the distance is a tenth of the field rather than a hundredth, which is why the influence of a current-carrying cable reaches further than intuition allows and why separating signal wiring from power wiring by a few centimetres helps less than expected. Twisting the pair helps far more, because the two opposite currents then cancel at any distance.

The solenoid's answer contains no distance at all

Inside a long solenoid the field is the magnetic constant times the turns per metre times the current, and where you stand inside it makes no difference — the field is uniform across the whole interior and drops to nearly nothing outside. That uniformity is the reason solenoids are the standard way to make a controlled field for an experiment, and the reason the same geometry appears in relays, contactors and the coils that move a loudspeaker cone. It also means the only levers are current and winding density: doubling the turns per metre doubles the field exactly, with no diminishing return until the wire runs out of room or the core saturates.

Frequently asked questions

Is a field of this size anything to worry about?

It is the scale of field a person already lives in. The Earth produces a comparable one everywhere on the surface, and domestic wiring produces brief spikes of the same order whenever an appliance switches. The figures worth attention are the exposure limits set by national and international bodies for occupational and public exposure, which are far above these numbers and are the only relevant reference. What a field of this size does reliably affect is instruments: a magnetometer, a compass, a hall sensor or a cathode-ray display will all notice a cable carrying ten amps, and that is an engineering constraint rather than a health one.

Why does the formula not mention the wire's thickness?

Because outside the conductor it does not matter. The field at a given distance depends only on how much current is enclosed, so a thin wire and a thick busbar carrying the same ten amps produce the same field at five centimetres. The thickness matters inside the metal, where the field rises from zero at the centre to its maximum at the surface, and it matters practically because a thin wire carrying ten amps will not survive long. Outside, the whole conductor behaves as though the current ran along its axis, which is what makes the formula so short.