or, substituting the values of h1 and h2,
1 T^2
X = -- --------- (cos[alpha]1 + cos [alpha]2)^2
2 [rho]ga^2
- T {1 - 1/2(sin [alpha]1 + sin[alpha]2)
- (1/12)(cos[alpha]1 + cos[alpha]2)(cot[alpha]1 + cot[alpha]2)},
the remaining terms being negligible when a is small. The force,
therefore, with which the two plates are drawn together consists first
of a positive part, or in other words an attraction, varying inversely
as the square of the distance, and second, of a negative part of
repulsion independent of the distance. Hence in all cases except that in
which the angles [alpha]1 and [alpha]2 are supplementary to each other,
the force is attractive when [alpha] is small enough, but when
cos[alpha]1 and cos[alpha]2 are of different signs, as when the liquid
is raised by one plate, and depressed by the other, the first term may
be so small that the repulsion indicated by the second term comes into
play. The fact that a pair of plates which repel one another at a
certain distance may attract one another at a smaller distance was
deduced by Laplace from theory, and verified by the observations of the
abbe Hauy.
_A Drop between Two Plates._--If a small quantity of a liquid which
wets glass be introduced between two glass plates slightly inclined to
each other, it will run towards that part where the glass plates are
nearest together. When the liquid is in equilibrium it forms a thin
film, the outer edge of which is all of the same thickness. If d is the
distance between the plates at the edge of the film and [PI] the
atmospheric pressure, the pressure of the liquid in the film is [Pi] -
(2T cos[alpha])/d, and if A is the area of the film between the plates
and B its circumference, the plates will be pressed together with a
force
2AT cos[alpha]
------------- + BT sin[alpha],
d
and this, whether the atmosphere exerts any pressure or not. The force
thus produced by the introduction of a drop of water between two plates
is enormous, and is often sufficient to press certain parts of the
plates together so powerfully as to bruise them or break them. When two
blocks of ice are placed loosely together so that the superfluous water
which melts from them may drain away, the remaining water draws the
blocks together with a force sufficient to cause the blocks to adhere by
the process called _Regelation_.
[An effect of an opposite character may be observed when the fluid is
mercury in place of water. When two pieces of flat glass are pressed
together under mercury with moderate force they cohere, the mercury
leaving the narrow crevasses, even although the alternative is a vacuum.
The course of events is more easily followed if one of the pieces of
glass constitutes the bottom, or a side, of the vessel containing the
mercury.]
In many experiments bodies are floated on the surface of water in order
that they may be free to move under the action of slight horizontal
forces. Thus Sir Isaac Newton placed a magnet in a floating vessel and a
piece of iron in another in order to observe their mutual action, and
A.M. Ampere floated a voltaic battery with a coil of wire in its circuit
in order to observe the effects of the earth's magnetism on the electric
circuit. When such floating bodies come near the edge of the vessel they
are drawn up to it, and are apt to stick fast to it. There are two ways
of avoiding this inconvenience. One is to grease the float round its
water-line so that the water is depressed round it. This, however, often
produces a worse disturbing effect, because a thin film of grease
spreads over the water and increases its surface-viscosity. The other
method is to fill the vessel with water till the level of the water
stands a little higher than the rim of the vessel. The float will then
be repelled from the edge of the vessel. Such floats, however, should
always be made so that the section taken at the level of the water is as
small as possible.
[_The Size of Drops._--The relation between the diameter of a tube and
the weight of the drop which it delivers appears to have been first
investigated by Thomas Tate (_Phil. Mag._ vol. xxvii. p. 176, 1864),
whose experiments led him to the conclusion that "other things being the
same, the weight of a drop of liquid is proportional to the diameter of
the tube in which it is formed." Sufficient time must of course be
allowed for the formation of the drops; otherwise no simple results can
be expected. In Tate's experiments the period was never less than 40
seconds.
The magnitude of a drop delivered from a tube, even when the formation
up to the phase of instability is infinitely slow, cannot be calculated
a priori. The weight is sometimes equated to the product of the
capillary tension (T) and the circumference of the tube (2[pi]a), but
with little justification. Even if the tension at the circumference of
the tube acted vertically, and the whole of the liquid below this level
passed into the drop, the calculation would still be vitiated by the
assumption that the internal pressure at the level in question is
atmospheric. It would be necessary to consider the curvatures of the
fluid surface at the edge of attachment. If the surface could be treated
as a cylindrical prolongation of the tube (radius a), the pressure would
be T/a, and the resulting force acting downwards upon the drop would
amount to one-half ([pi]aT) of the direct upward pull of the tension
along the circumference. At this rate the drop would be but one-half of
that above reckoned. But the truth is that a complete solution of the
statical problem for all forms up to that at which instability sets in,
would not suffice for the present purpose. The detachment of the drop is
a _dynamical_ effect, and it is influenced by collateral circumstances.
For example, the bore of the tube is no longer a matter of indifference,
even though the attachment of the drop occurs entirely at the outer
edge. It appears that when the external diameter exceeds a certain
value, the weight of a drop of water is sensibly different in the two
extreme cases of a very small and of a very large bore.
But although a complete solution of the dynamical problem is
impracticable, much interesting information may be obtained from the
principle of dynamical similarity. The argument has already been applied
by Dupre (_Theorie mecanique de la chaleur_, Paris, 1869, p. 328), but
his presentation of it is rather obscure. We will assume that when, as
in most cases, viscosity may be neglected, the mass (M) of a drop
depends only upon the density ([sigma]), the capillary tension (T), the
acceleration of gravity (g), and the linear dimension of the tube (a).
In order to justify this assumption, the formation of the drop must be
sufficiently slow, and certain restrictions must be imposed upon the
shape of the tube. For example, in the case of water delivered from a
glass tube, which is cut off square and held vertically, a will be the
external radius; and it will be necessary to suppose that the ratio of
the internal radius to a is constant, the cases of a ratio infinitely
small, or infinitely near unity, being included. But if the fluid be
mercury, the flat end of the tube remains unwetted, and the formation of
the drop depends upon the internal diameter only.