Page:EB1911 - Volume 05.djvu/278

where $\rho$ is the density of the liquid, or if there are two fluids the excess of the density of the lower fluid over that of the upper one.

The forces acting on the portion of liquid $P_{1}P_{2}A_{2}A_{1}$ are—first, the horizontal pressures, $-{\tfrac {1}{2}}\rho gy_{1}^{2}$ and ${\tfrac {1}{2}}\rho gy_{2}^{2}$; second, the surface-tension ${\mbox{T}}$ acting at ${\mbox{P}}_{1}$ and ${\mbox{P}}_{2}$ in directions inclined $\theta _{1}$ and $\theta _{2}$ to the horizon. Resolving horizontally we find—

${\mbox{T}}(\cos \theta _{2}-\cos \theta _{1})+{\tfrac {1}{2}}g\rho (y_{2}{}^{2}-y_{1}{}^{2})=0,$

whence

$\cos \theta _{2}=\cos \theta _{1}+{\frac {g\rho y_{1}{}^{2}}{2{\mbox{T}}}}-{\frac {g\rho y_{2}{}^{2}}{2{\mbox{T}}}},$

or if we suppose ${\mbox{P}}_{1}$ fixed and ${\mbox{P}}_{2}$ variable, we may write

$\cos \theta ={\mbox{constant}}-{\tfrac {1}{2}}g\rho y^{2}/{\mbox{T}}$

This equation gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid.

Fig. 8.

Resolving vertically we find that the weight of the liquid raised above the level must be equal to ${\mbox{T}}(\sin \theta _{2}-\sin \theta _{1})$, and this is therefore equal to the area $P_{1}P_{2}A_{2}A_{1}$ multiplied by $g\rho$. The form of the capillary surface is identical with that of the “elastic curve,” or the curve formed by a uniform spring originally straight, when its ends are acted on by equal and opposite forces applied either to the ends themselves or to solid pieces attached to them. Drawings of the different forms of the curve may be found in Thomson and Tait’s Natural Philosophy, vol. i. p. 455.

We shall next consider the rise of a liquid between two plates of different materials for which the angles of contact are $\alpha _{1}$ and $\alpha _{2}$, the distance between the plates being $a$, a small quantity. Since the plates are very near one another we may use the following equation of the surface as an approximation:—

$y=h_{1}+{\mbox{A}}x+{\mbox{B}}x^{2},\,h_{2}=h_{1}+{\mbox{A}}a+{\mbox{B}}a^{2},$

whence

$\cot \alpha _{1}=-{\mbox{A}},\,\cot \alpha _{2}={\mbox{A}}+2{\mbox{B}}a$

${\mbox{T}}(\cos \alpha _{1}+\cos \alpha _{2})=\rho ga(h_{1}+{\tfrac {1}{2}}{\mbox{A}}a+{\tfrac {1}{3}}{\mbox{B}}a^{2},$

whence we obtain

$h_{1}={\frac {\mbox{T}}{\rho ga}}(\cos \alpha _{1}+\cos \alpha _{2}){\frac {a}{6}}(2\cot \alpha _{1}-\cot \alpha _{2})$

$h_{2}={\frac {\mbox{T}}{\rho ga}}(\cos \alpha _{1}+\cos \alpha _{2}){\frac {a}{6}}(2\cot \alpha _{2}-\cot \alpha _{1}).$

Let ${\mbox{X}}$ be the force which must be applied in a horizontal direction to either plate to keep it from approaching the other, then the forces acting on the first plate are ${\mbox{T}}+{\mbox{X}}$ in the negative direction, and ${\mbox{T}}\sin \alpha _{1}+{\tfrac {1}{2}}g\rho h_{1}{}^{2}$ in the positive direction. Hence

${\mbox{X}}={\tfrac {1}{2}}g\rho (h_{1}{}^{2}-{\mbox{T}}(1-\sin \alpha _{1}).$

For the second plate

${\mbox{X}}={\tfrac {1}{2}}g\rho (h_{2}{}^{2}-{\mbox{T}}(1-\sin \alpha _{2}).$

Hence

${\mbox{X}}={\tfrac {1}{4}}g\rho (h_{1}{}^{2}+h_{2}{}^{2})-{\mbox{T}}\{1-{\tfrac {1}{2}}(\sin \alpha _{1}+\sin \alpha _{2})\},$

or, substituting the values of $h_{1}$ and $h_{2}$,

${\mbox{X}}={\tfrac {1}{2}}{\frac {{\mbox{T}}^{2}}{\rho ga^{2}}}(\cos \alpha _{1}+\cos \alpha _{2})^{2}-{\mbox{T}}\{1-{\tfrac {1}{2}}(\sin \alpha _{1}+\sin \alpha _{2})-{\tfrac {1}{12}}(\cos \alpha _{1}+\cos \alpha _{2})(\cot \alpha _{1}+\cot \alpha _{2})\},$

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 abbé Haüy.

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 $\Pi -{\frac {2{\mbox{T}}\cos \alpha }{d}}$, and if ${\mbox{A}}$ is the area of the film between the plates and ${\mbox{B}}$ its circumference, the plates will be pressed together with a force

${\frac {2{\mbox{AT}}\cos \alpha }{d}}+{\mbox{BT}}\sin \alpha$

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. Ampère 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 (${\mbox{T}}$) 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 ${\mbox{T}}/a$, and the resulting force acting downwards upon the drop would amount to one-half ($\pi a{\mbox{T}}$) of the direct upward pull of the tension along the circumference. At this