Page:Elementary Text-book of Physics (Anthony, 1897).djvu/109

 $$E = T;$$ that is, the numerical value of the surface energy per unit of surface is equal to that of the tension in the surface, normal to any line in it, per unit of length of that line.

82. Equation of Capillarity.—The surface tension introduces modifications in the pressure within the liquid mass (§§ 112 seq.) depending upon the curvature of the surface. Consider any infinitesimal rectangle (Fig. 30) on the surface. Let the length of its sides be represented by $$s$$ and $$s'$$ respectively, and the radii of curvature of those sides by $$R$$ and $$R'$$. Also let $$\phi$$ and $$\phi '$$ represent the angles in circular measure subtended by the sides from their respective centres of curvature. Now, a tension $$T$$ for every unit of length acts normal to $$s$$ and tangent to the surface. The total tension across $$s$$ is then $$Ts;$$ and if this tension be resolved parallel and normal to the normal at the point $$P$$, the centre of the rectangle, we obtain for the parallel component $$Ts \sin{\frac{\phi '}{s}}$$, or, since $$\phi '$$ is a very small angle, $$Ts \frac{\phi '}{2}$$ or $$Ts \frac{s'}{2R'}$$. The opposite side gives a similar component; the side $$s'$$ and the side opposite it give each a component $$Ts' \frac{s}{2R}$$. The total force along the normal at $$P$$ is then $$Tss'\left(\frac{1}{R'} + \frac{1}{R} \right);$$ since $$ss'$$ is the area of the infinitesimal rectangle, the force or pressure normal to the surface at $$P$$ referred to unit of surface is $$T \left(\frac{1}{R'} + \frac{1}{R} \right)$$. From a theorem given by Euler we know that the sum $$\frac{1}{R'} + \frac{1}{R}$$ is constant at any point for any position of the rectangular normal plane sections; hence the expression we have obtained fully represents the pressure at $$P$$.

If the surface be convex, the radii of curvature are positive, and the pressure is directed towards the liquid; if concave, they are