How does the field from a power line vary with the resistivity of the ground?
The nature of the ground that a power line passes over - how wet it is, what the mineral content is, how conducting it is - makes very little difference to the fields the power line produces. Where it makes a difference at all, it is only significant a long way away from the power line, where the field is already small.
When we calculate fields from overhead power lines, we normally assume that the ground is a good conductor for electric fields and not a conductor at all for magnetic fields. That may sound perverse but it gives the right answers! We explain this further here and what effect it has if we change those assumptions.
Magnetic fields - the summary
As soil gets wetter, or has a higher content of certain minerals, it becomes more conducting. In theory, this makes a difference to the magnetic field from the power line. But in practice, the difference is too small to be significant (or even detectable) until you are a long way away from the power line - hundreds of metres - and if you are that far away, the field is tiny anyway. Even then, it is only exceptionally conducting soils (perhaps because they are exceptionally wet) that will make any discernable difference.
Close to the power line, where the fields are high, the electrical properties of the ground make no practical difference at all.
Magnetic fields - more detail
The ground - any ground - is always slightly conducting because of the moisture content. The greater the moisture content, and the greater the concentration of certain minerals, the more conducting it becomes.
That means that the magnetic fields from an overhead power line can induce currents in the ground.
The currents that are induced are actually dispersed throughout the ground. But we can model their effect as if they are concentrated just at one depth. Those currents are called "image currents" and we can work out their depth if we know the resistivity of the ground. The less conducting - the more resistive - the ground, the deeper the image currents are.
For typical ground resistivities, the depth of the image currents is many hundreds of metres. That means that, when you are under the line or close to it, these image currents are too deep in the ground to have any detectable effect on the magnetic field. Only when you get several hundred metres away from the power line yourself do these image currents start making any noticeable difference. (And when you're that far away from the power line, other factors come into play anyway, such as any unbalance between the currents in the conductors.)
You can see this from this graph. It shows the field calculated, including the image currents, for four different ground resistivities - infinite (i.e. the ground not conducting at all), 100, 10 and 1 Ω m.
Most soils are in the range 10-1000 Ω m or even higher. You can see for the graph that even for the less resistive - more conductive - end of this range, this makes only a small difference to the field out to 300 m or so, when the field is already below 10 nT or 0.01 μT. For the more resistive - less conductive - soils, you can't really see a difference even at 500 m. We didn't even calculate for 1000 Ω m, even though that's still a typical soil value, because the line would have been indistinguishable from the infinite resitivity line.
So that is why it is perfectly valid to assume, as we usually do, that the ground is not conducting at all for magnetic-field calculations.
Note on the calculations: this graph comes from our detailed test of compared calculated and measured fields, so is specific to one set of conditions, but the principles apply to all power lines. There are different ways of modelling the image currents but they all give broadly similar answers.
We've talkled here about overhead power lines, where they produce image currents below the ground. With underground cables, which are already below the ground, the physics is a bit different, but the result is the same - the effect of soil conductivity is, for practical purposes, negligible.
Magnetic rocks and soils
What we've been talking about so far is the effect of the ground resistivity (or, viewed the other way round, conductivity) on the magnetic field. That's principally determined by the moisture content.
But rocks and soils can also contain magnetic material - iron, or magnetite, etc. In principle this can affect the magnetic field too. It is mainly these magnetic rocks and soils that produce what are called "magnetic anomalies" - local variations of the earth's magnetic field from place to place. Magnetic anomalies are measured (see e.g. the British Geological Survey) and they rarely amount to more than 1% - usually even less. So it's a safe bet that the effect of magnetic rocks or soils on the magnetic field from a power line is likewise only a fraction of a percent.
A bit more detail: the larger anomalies happen where the rocks have a permanent magnetisation, or remnant magnetisation, enduring from when they were laid down. There are rare places where this is big enough to interfere with a compass - the Cuillin on the Isle of Skye are notorious for this, for example. But permanent magnetisation is a DC phenomenon, and has no relevance for the magnetic fields from AC power lines. What matters for AC power lines and the AC magnetic fields they produce is the "magnetic permeability" of the rocks - the property by which magnetic materials passively distort the field lines passing nearer them. That is only a fraction of the magnetic anomaly, which is already usually less than a percent of the field - hence the confidence that magnetic rocks have a negligible effect on fields from power lines or cables.
In summary: as far as electric fields are concerned, all soils or other ground materials are already conducting, and exactly how conducting they are makes no practical difference.
In more detail:
For electric fields, it is the other way round to magnetic fields. The voltages on the power line conductors induce a set of image charges (rather than the image currents you get for magnetic fields) in the ground, but these are essentially a mirror image - they are at the same depth below ground as the real conductors are above ground. When we calculate electric fields, we always include these image charges.
There is no soil or rock that has a large enough resistivity for this model to break down, and so for practical purposes, the ground resistivity makes no difference - we can assume the ground is a perfect conductor and calculate fields on that basis.
Even more detail: how good is this approximation?
There's a quantity called the "relaxation time" - it's a measure of how quickly the charges induced in the ground can adjust to the changing field, and it's simply the resistivity times the permittivity. For a pretty wet soil, that comes out at a nanosecond or so, and for the driest, least conducting soils or rocks imaginable, it's still only a microsecond or so. That has to be compared to the period of oscillation of a 50 Hz AC electric field - 20 milliseconds. The relaxation time is so much less than the period of oscillation that the approximation is valid for the whole range of soil and rock conductivities found in practice.
This is, incidentally, a way of explaining why electric field line meet the ground at right angles:
You can imagine the mirror image of this continuing below the ground with the ground as a symmetry line - it can only be symmetrical if the field lines cross the ground at right angles.
Resistivity and conductivity
We can describe the electrical properties of the ground (or anything else for that matter) in two ways: how good a resistor it is (resistivity, usually denoted by the Greek letter ρ) or how good a conductor it is (conductivity, usually denoted by the Greek letter σ). ρ=1/σ and σ=1/ρ.
Resistivity is measured in "ohm metres" (Ω m) and conductivity in siemens per metre (S/m).