Six identical point charges are fixed at the vertices of a regular octahedron of…
PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Conductors, capacitance and boundary-value electrostatics
The Boundary-Value Problem, Ch The field as a differential equation: Poisson, Laplace and the well-posed problem
Problem
Six identical point charges +Q are fixed at the vertices of a regular octahedron of "radius" b: at (±b,0,0), (0,±b,0) and (0,0,±b). A positive test charge q is placed at the origin, in the otherwise empty space between the six fixed charges. An engineer reasons: "Displace q a small distance along the x-axis. The nearer charge (at (b,0,0) or (−b,0,0), whichever is closer) repels it more strongly than the farther one, so the net force pushes q back towards the origin: restoring. By the symmetry of the arrangement the same restoring argument applies along y and along z. Since the equilibrium is restoring along all three coordinate axes, it is stable."
((a)) Using Earnshaw's theorem, decide whether the engineer's conclusion can be correct, and find the three curvatures ∂2V/∂x2, ∂2V/∂y2, ∂2V/∂z2 at the origin.
((b)) Expand the potential of the six charges to fourth order in a small displacement δ from the origin, first along the x axis and then along the body diagonal towards (1,1,1). Give the change in V for δ=0.1b in each case, in units of Q/4πε0b, and say what the engineer's argument got right, what it left out, and in which directions q escapes.