Recent discussions raised by the query from Justin Parkes
Can equipotential surfaces cross? If so, then in what direction
does the electric field point at the line of intersection? triggered
memories (from 1945 !!) of diagrams in The Mathematical Theory of
Electricity and Magnetism by Sir James Jeans (Cambridge University
Press). On digging out the ancient text I find that Jeans shows
diagrams of equipotentials for three cases:
The simplest case is of two equal charges where the midpoint C is a
point of equilibrium and the equipotential which passes through C
intersects itself at the point C. (The equipotential surface if akin
to the surface of an amoeba on the point of splitting). Jeans makes
the point that the conditions for a point of equilibrium (the partial
derivatives of V with respect to x, y and z are all zero) require that
the equipotential (V = constant) through that point should have a
double tangent plane or a tangent cone at that point.
The next case is for charges +4e and e where the equilibrium point is
on-axis beyond e. Again the same considerations apply.
The third case is three equal charges e on an equilateral triangle of
side a. Here the equipotential surface (V= 3.04e/a), which would be a
joy to a topologist, intersects itself at the three points of
equilibrium. This case provides useful exercise in 3-D thinking.
Jeans provides two diagrams, in the plane of the charges and
perpendicular to it which will aid this process. There is, of course
always the fall-back method of envisaging the lines of force pattern
and then mentally overlaying the orthogonal trajectories !!!.
Jeans is famous for his many notable contributions to Physics, but
even the great can make erroneous predictions. It is reported that in
1910 when discussing possible reforms of the mathematics curriculum at
Princeton University Jeans argued that they may as well cut out group
theory for it would never be of any use in physics. ( This anecdote
is quoted in Mathematical Apocrypha by Steven G Kranz published by
the Mathematical Association of America --- a great read).
School of Physics and Materials Engineering
Melbourne , Australia