is at maximum, i.e., the radius of the tangent circle is at minimum. This is
the first principal radius of curvature R 1. R 2 is the radius of the
corresponding circle in a plane through the same normal that is at a right
angle to the first one. For a sphereR 1 ¼R 2 ¼R. For a circular cylinder,R 1
is the cylinder radiusRcyland 1/R 2 ¼0; hence,pL¼g/Rcyl.
Curved surfaces can also besaddle-shaped. Figure 10.20 shows an
example. Suppose that a surfactant film is made between the two frames.
Surface tension causes the film to assume the smallest surface area possible.
In the situation depicted, this surface is saddle-shaped. Moreover, the
surface has zero curvature. As drawn for the middle cross section of the film,
the principal radii of curvature are equal, but of opposite sign, since the
tangent circles are at opposite sides of the film (which is, actually, the
definition of a saddle-shaped surface). In other words,pL¼0 because
1/R 1 þ1/(R 2 )¼0. This is true for every part of the film surface.
It may be concluded that a confined film onto which no net external
forces act will always form a surface of zero curvature. If formation of such
a surface is geometrically impossible—for instance, if in a situation as
depicted in Figure 10.20 the two circular frames were much farther apart—a
film cannot be made.
FIGURE10.20 Representation of a soap film formed between two parallel circular
frames. At a point on the film surface two tangent circles are drawn.