Dwight's Equation for Anode Resistance to Earth

H. B. Dwight worked at MIT on the resistance of grounding electrodes and published this result in 1936. It was written for electrical earthing — the resistance between a driven rod and remote earth — and it applies unchanged to a cathodic protection anode, because the physics is identical: current spreading from a small buried conductor into a large uniform medium.

The structure of the equation says something the number alone does not. Nearly all of the resistance lives in the first few centimetres of soil around the anode, where the current is crowded into a small cross-section; once it has spread out, the remaining path to remote earth contributes almost nothing however long it is. That is why the anode's own dimensions matter so much and the distance to the structure matters so little, and it is also why two anodes placed close together interfere with each other — their high-resistance zones overlap, and the pair delivers less than twice one of them.

The diameter is the backfill column's, not the anode's. Carbonaceous backfill — coke breeze around an impressed-current anode, a bentonite-gypsum mix around a galvanic one — is a conductor, so the effective electrode is the whole augered hole. An anode 50 mm across in a 200 mm hole behaves electrically as a 200 mm anode, and that resistance reduction is most of what backfill is for. The rest of what it is for is keeping moisture against the anode and giving the corrosion products somewhere to go.

Length beats diameter, and it is not close. The length sits outside the logarithm and the diameter inside it, so doubling the length roughly halves the resistance while doubling the diameter buys a few per cent. That single asymmetry explains why groundbeds go deep rather than wide, and why deep-well anode beds — strings of anodes in a borehole tens of metres down — are the standard solution where surface soil is dry or rocky. Deep beds have a second advantage: they reach below the seasonally variable zone entirely.

Which brings up the term the equation treats as a constant and the field does not. Soil resistivity is not a property of a site; it is a property of a site on a day. It swings by a factor of ten between a wet spring and a dry August in the same trench, and freezing raises it dramatically — a system commissioned in July can quietly stop protecting in February, which is exactly when nobody is out measuring. Design on the driest and coldest condition the site will see, measure resistivity with a four-pin Wenner survey rather than trusting a soil-type table, and take readings at several pin spacings, because a resistive layer over a conductive one, or the reverse, is common and a single spacing cannot see it. Finally, note that this equation solves cleanly for resistivity and for diameter but not for length, which appears both inside and outside the logarithm; for a target length, vary it and read the resistance.