An introduction to the mathematical theory of attraction ...

Force acting on Element of Surface of charged Conductor. 55
element; and as the electricity cannot leave the element, the
same force tends to move the element itself relatively to the
rest of the surface, and produces stresses if this motion be
prevented by the cohesion of the material of the conductor.
The electric force just outside the element dS is 47rr, and
inside the surface, in the substance of the conductor, is zero.
To determine the force J which would act on an element
of surface dS if it contained a unit of electric mass, imagine
a surface Si described in the substance of the conductor,
inside and close to S. Since, by Article 30, the resultant
force at each point of this surface is zero, J NdS, = 0;
and therefore, by Art. 26, since there is no mass on 81, the
total mass inside it must be zero. Now imagine another
surface 82 coinciding with 81 except at the element dS where
it coincides with S; then f NdS, = 47r1I 27rM'; but by what
precedes M = 0, and M2' = adS; also, N is zero everywhere
on Sz except in the element dS, where it is the same as J; we
have therefore JdS = 27ra d8, that is, J = 27ro. The force J
is the force per unit of electric mass acting in the element.
To get the mechanical force FdS tending to move the element,
or producing stress in the material of the conductor, we must
multiply J by the electric mass in the element, that is, by
adS. HIence we obtain
R2
FdS = 2oa2ds = -dS,
87r
where R is the resultant force just outside the element; whence
F, the mechanical force acting on the unit of area of the
surface of the conductor, is given by the equation
TF =v (8)
The value of J can be obtained in another way by considering the normal force immediately inside and immediately
outside dS. The total normal force immediately outside dS
is made up of two parts, one due to the distribution on dS
itself, which may be denoted by Ni, and one due to the
electricity on the rest of the conductor, which may be denoted
by N2. Inside dS the corresponding normal forces may be