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APPLICATION OF THE PRECEDING RESULTS TO THE THEORY OF ELECTRICITY.
(8.) The first application we shall make of the foregoing principles, will be to the theory of the Leyden phial. For this, we will call the inner surface of the phial A, and suppose it to be of any form whatever, plane or curved, then, B being its outer surface, and 8 the thickness of the glass measured along a normal to A; 8 will be a very small quantity, which, for greater generality, we will suppose to vary in any way, in passing from one point of the surface A to another. If now the inner coating of the phial be put in communication with a conductor C, charged with any quantity of electricity, and the outer one be also made to communicate with another conducting body C, containing any other quantity of electricity, it is evident, in consequence of the communications here established, that the total potential function, arising from the whole system,, will be constant throughout the interior of the inner metallic coating, and of the body C. We shall here represent this constant quantity by
fi. Moreover, the same potential function within the substance of the outer coating, and in the interior of the conductor 0', will be equal to another constant quantity
Then designating by V, the value of this function, for the whole of the Bpace exterior to the conducting bodies of the system, and consequently for that within the substance of the glass itself; we shall have (art. 4)
F=£ and 7 = 0.
One horizontal line over any quantity indicating that it belongs to the inner surface A, and two showing that it belongs to the outer one B.
At any point of the surface A, suppose a normal to it to be drawn, and let this be the axes of w: then to', to", being two other rectangular axes, which are necessarily in the plane tangent to A at this point; V may be considered as a function of to, w and to", and we shall have by Tatlok's theorem, since to' = 0 and w" => 0 at the axis of w along which 0 is measured,
where, on account of the smallness of 8, the series converges very rapidly. By writing in the above, for V and F their values just given, we obtain
P P dw 1 dw* 1.2 + <Kc
In the same way, if w be a normal to B, directed towards A, and $, be the thickness of the glass measured along this normal, we shall have
P-P =— -7' + -=-. ~ + &c .
But, if we neglect quantities of the order 0, compared with those retained, the following equation will evidently hold good,
n being any whole positive number, the factor (— 1)* being introduced because w and to are measured in opposite directions. Now by article 4
- dT - dT
— Airp = -=, and — iirp = -=;
p and p being the densities of the electric fluid at the surfaces A and B respectively. Permitting ourselves, in what follows, to neglect quantities of the order 0* compared with those retained, it is clear that we may write 0 for 0,, and hence by substitution
where V and p are quantities of the order ~; /S' and /8 being the order 0* or nnity. The only thing which now remains to
be determined, is the value of -=- for any point on the sur
Throughout the substance of the glass, the potential function V will satisfy the equation 0 = 8 V, and therefore at a point on the surface of A, where of necessity w, w and to" are each equal to zero, we have
A <PV d%V d'V « =
the horizontal mark over w, w and w" being, for simplicity, omitted. Then since w = 0,
and as V is constant and equal to # at the surface A, there hence arises
B being the radius of curvature at the surface A, in the plane (to, to'). Substituting these values in the expression immediately preceding, we get
d*V I dV__-47rp
In precisely the same way we obtain, by writing S for the radius of curvature in the plane (w, w"),
d*V _ - iirp
both rays being accounted positive op the side where w, i.e. Id, is negative. These values substituted in 0 = BV, there results
d*V for the required value of jT, and thus the sum of the two
equations into which it enters, yields
and the difference of the same equations gives
which values are correct to quantities of the order 0*p, or, which is the same thing, to quantities of the order 0; these having been neglected in the latter part of the preceding analysis, as unworthy of notice.
Suppose da is an element of the surface A, the corresponding element of B, cut off by normals to A, will be
da 11 + 0 (-5 + £-, J [, and therefore the quantity of fluid on this last element will be pda jl + 0 (-g + Jt)\ ;. substituting for p its value before found, /} = — p\l — ^(p+r'jm an<^ neglecting 6*p, we obtain
the same quantity as on the element da of the first surface. If, therefore, we conceive any portion of the surface A, bounded by a closed carve, and a corresponding portion of the surface B, which would be cut of! by a normal to A, passing completely round this curve; the Bum of the two quantities of electric fluid, on these corresponding portions, will be equal to zero; and consequently, in an electrical jar any how charged, the total quantity of electricity in the jar may be found, by calculating the quantity, on the two exterior surfaces of the metallic coatings farthest from the glass, as the portions of electricity, on the two surfaces adjacent to the glass, exactly neutralise each other. This result will appear singular, when we consider the immense quantity of fluid collected on these last surfaces, and moreover, it would not be difficult to verify it by experiment
As a particular example of the use of this general theory: suppose a spherical conductor whose radius a, to communicate with the inside of an electrical jar, by means of a long slender wire, the outside being in communication with the common reservoir; and let the whole be charged: then P representing the density of the electricity on the surface of the conductor, which will be very nearly constant, the value of the potential function within the sphere, and, in consequence of the communication established, at the inner coating A also, will be iiraP very nearly, since we may, without sensible error, neglect the action of the wire and jar itself in calculating it. Hence
/3 = iiraP and £' = 0,
and the equations (8), by neglecting quantities of the order 0, give
We thus obtain, by the most simple calculation, the values of