Question 2 of 10: Potential Fields — Monopole and Dipole Sources

Question 2: Potential Fields — Monopole and Dipole Sources (Choose 6 of 10 – equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Why "potential field" methods

Gravity and magnetic fields are both conservative vector fields that can be written as the gradient of a scalar function — the gravitational potential U or the magnetic scalar potential Φ — so that g = ∇U and B = -μ₀∇Φ. Because the field is derived from a single scalar potential that satisfies Laplace's equation (∇²U = 0) everywhere outside the source, both fields obey superposition (the total field of many sources is the simple sum of their individual fields), have no closed loops of force outside the source, and can be manipulated with the same mathematical toolkit (Green's functions, harmonic continuation, Fourier-domain filtering) — hence "potential field methods" as a shared discipline.

Monopolar and dipolar sources

A monopole is a single, isolated point source of field with no compensating opposite pole nearby. The gravity field is inherently monopolar: mass has only one sign, so a buried point mass (or, to a good approximation, a compact ore body or cave many diameters from the observer) radiates a purely radial field with equipotential surfaces that are concentric spheres. Magnetism has no true isolated pole (magnetic monopoles have never been observed), so the natural magnetic analogue is a dipole: two equal and opposite poles a finite distance apart, of which a bar magnet, a current loop, and — to first order — the Earth's own main field are the standard examples.

[Figure not reproduced: Left: an isolated (monopole) source radiates a purely radial field, solid lines pointing away from the source, with spherical equipotential surfaces shown dotted. Right: a dipole (two opposite poles a finite distance apart) has curved field lines running from the positive to the negative pole; equip. See the official exam paper.]

Field strength and direction versus distance

For the monopole (a point mass m, or in the magnetic analogy an isolated pole of strength p), the field strength falls off as the inverse square of distance and points radially, directly toward (gravity) or away from (a positive magnetic pole) the source:

$$g(r) = \dfrac{Gm}{r^{2}} \qquad \text{(directed radially, along } \hat{r} \text{)}$$

where G is the universal gravitational constant. The potential itself falls off as 1/r, one power slower than the field, since the field is the potential's gradient.

For the dipole (moment m_d = p·ℓ for two poles ±p separated by ℓ), the field falls off one power faster, as the inverse cube of distance, and its direction depends on the angle θ between the observation point and the dipole axis, not just on distance:

$$B(r,\theta) = \dfrac{\mu_0 m_d}{4\pi r^{3}}\sqrt{1+3\cos^{2}\theta}$$

with the field twice as strong and axially directed on the dipole's own axis (θ = 0) as it is broadside at the equatorial plane (θ = 90°), where it is directed opposite to the axial moment. The steeper 1/r³ decay (versus 1/r² for a monopole) is why magnetic anomalies from compact bodies die away with distance/depth noticeably faster than gravity anomalies of the same size — a practically important difference when choosing survey parameters for a given target depth.