Answer to the Question 01/02
EQUIPOTENTIAL SURFACES
The question was:
Let the equation
F(x,y,z)=c
represent a set of nonintersecting surfaces. (Each c corresponds
to a different surface.) What is the condition that this is a set of
equipotential surfaces? In other words, can we define such potential
V(c)
that its Laplacian vanishes?
(7/2002) The problem has been solved (26/6/2002)
by Thomas Garel
from Service de Physique Theorique,
CE-Saclay,
Gif-sur-Yvette (Francei) (e-mail
garel@spht.saclay.cea.fr);
the solution below follows (to large extent) his derivation.
The solution:
Derivatives of V with respect to space coordinates, can be taken
by first taking the derivative of V with respect to c
and then the derivative of c with respect to, say, x. Then
grad2V=V''(c)(gradc)2+V'(c)grad2c=0
where grad2 denotes Laplacian, and prime denotes derivative with respect
to c. This leads to the condition
[grad2c]/[(grad c)2=-V''(c)/V'(c)=G(c)
i.e. the ratio on the l.h.s. of the equation must be a function of only c.
This is the true and the only requirement of the set of surfaces determined by F.
The arbitrary function G(c) determines the potential V:
V=A {\int} exp[-{\int}G(c) dc] dc + B
where {\int} denotes the integral sign.
The solution and its applications can be found in the book of W. R. Smythe
Static and Dynamic Electricity (McGraw-Hill, 1950).
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