Multiplying equation (2) by χ0, and subtracting it from (4),
E − Mχ0 = S ∫ε0 (χ − χ0) ρdν (5)
In this expression M and χ0 are both constant, so that the variation of the right-hand side of the equation is the same as that of the energy E, and expresses that part of the energy which depends on the area of the bounding surface of the liquid. We may call this the surface energy.
The symbol χ expresses the energy of unit of mass of the liquid at a depth ν within the bounding surface. When the liquid is in contact with a rare medium, such as its own vapour or any other gas, χ is greater than χ0, and the surface energy is positive. By the principle of the conservation of energy, any displacement of the liquid by which its energy is diminished will tend to take place of itself. Hence if the energy is the greater, the greater the area of the exposed surface, the liquid will tend to move in such a way as to diminish the area of the exposed surface, or, in other words, the exposed surface will tend to diminish if it can do so consistently with the other conditions. This tendency of the surface to contract itself is called the surface-tension of liquids.
| Fig. 1. |
Dupré has described an arrangement by which the surface-tension of a liquid film may be illustrated. A piece of sheet metal is cut out in the form AA (fig. 1). A very fine slip of metal is laid on it in the position BB, and the whole is dipped into a solution of soap, or M. Plateau’s glycerine mixture. When it is taken out the rectangle AACC if filled up by a liquid film. This film, however, tends to contract on itself, and the loose strip of metal BB will, if it is let go, be drawn up towards AA, provided it is sufficiently light and smooth.
Let T be the surface energy per unit of area; then the energy of a surface of area S will be ST. If, in the rectangle AACC, AA = a, and AC = b, its area is S = ab, and its energy Tab. Hence if F is the force by which the slip BB is pulled towards AA,
| F = | d | Tab = Ta, (6) |
| db |
or the force arising from the surface-tension acting on a length a of the strip is Ta, so that T represents the surface-tension acting transversely on every unit of length of the periphery of the liquid surface. Hence if we write
T = ∫ε0 (χ − χ0) ρ dν, (7)