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7.9 Earnshaw’s Theorem and Electromechanical Stability

It is not well known that electric and magnetic forces in mechanical systems can produce static instability,
otherwise known as elastic buckling or divergence. This is a consequence of the inverse square nature of
many electric and magnetic forces. It is well known that the electric and magnetic field potential Φ
2
satisfies Laplace’s equation, ∇ Φ = 0 . There is a basic theorem in potential theory about the impossi-
bility of a relative maximum or minimum value of a potential Φ(r) for solutions of Laplace’s equation
except at a boundary. It was stated in a theorem by Earnshaw (1829) that it is impossible for a static set
of charges, magnetic and electric dipoles, and steady currents to be in a stable state of equilibrium without
mechanical or other feedback or dynamic forces (see, for example, Moon, 1984, 1994).
One example of Earnshaw’s theorem is the instability of a magnetic dipole (e.g., a permanent magnet)
near a ferromagnetic surface (Fig. 7.17). Levitated bearings based on ferromagnetic forces, for example,
require feedback control. Earnshaw’s theorem also implies that if there is one degree of freedom with
stable restoring forces, there must be another degree of freedom that is unstable. Thus the equilibrium
positions for a pure electric or magnetic system of charges and dipoles must be saddle points. The
implication for the force potentials is that the matrix of second derivatives is not positive definite. For
example, suppose there are three generalized position coordinates {s u } for a set of electric charges. Then
if the generalized forces are proportional to the gradient of the potential, ∇Φ , then the generalized
electric stiffness matrix K ij , given by

∂ Φ
2
K ij = -------------
∂s i ∂s j
will not be positive definite. This means that at least one of the eigenvalues will have negative stiffness.
Another example of electric buckling is a beam in an electric field with charge induced by an electric
field on two nearby stationary plates as in Fig. 7.15. The induced charge on the beam will be attracted
to either of the two plates, but is resisted by the elastic stiffness of the beam. As the voltage is increased, the
combined electric and elastic stiffnesses will decrease until the beam buckles to one or the other of the
two sides. Before buckling, however, the natural frequency of the charged beam will decrease (Fig. 7.16).
This property has been observed experimentally in a MEMS device. A similar magneto elastic buckling
is observed for a thin ferromagnetic elastic beam in a static magnetic field (see Moon, 1984). Both
electroelastic and magnetoelastic buckling are derived from the same principle of Earnshaw’s theorem.
There are dramatic exceptions to Earnshaw’s stability theorem. One of course is the levitation of 50-ton
vehicles with magnetic fields, known as MagLev, or the suspension of gas pipeline rotors using feedback
controlled magnetic bearings (see Moon, 1994). Here either the device uses feedback forces, i.e., the fields

FIGURE 7.17 Magnetic force on a magnetic dipole magnet near a ferromagnetic half space with image dipole
shown.

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