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*From*: bill rachinger <bill.rachinger@SCI.MONASH.EDU.AU>*Date*: Mon, 23 Feb 2004 15:42:01 +1100

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).

Bill Rachinger

School of Physics and Materials Engineering

Monash University

Melbourne , Australia

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