Page:Elementary Text-book of Physics (Anthony, 1897).djvu/221

§ 185] potential energy, by forcing the molecules farther apart against their mutual attractions and any external forces that may resist expansion. Since the internal work to be done when a solid or liquid expands varies greatly for different substances, it might be expected that the amount of expansion for a given rise of temperature would vary greatly.

In studying the expansion of solids, we distinguish linear and voluminal expansion.

The increase which occurs in the unit length of a substance for a rise of temperature from zero to 1° C. is called the coefficient of linear expansion. Experiment shows that the expansion for a rise of temperature of one degree is very nearly constant between zero and 100°.

Kepresent by $$l_{0}$$ the distance between two points in a body at zero, by $$l_{t}$$ the distance between the same points at the temperature $$t,$$ and by $$\alpha$$ the coefficient of linear expansion of the substance of which the body is composed.

The increase in the distance $$l_{0}$$ for a rise of one degree in temperature is $$\alpha l_{0},$$ for a rise of $$t$$ degrees $$\alpha tl_{0}.$$ Hence we have, after a rise in temperature of $$t$$ degrees, The binomial $$1 + \alpha t$$ is called the factor of expansion.

In the same way, if $$k$$ represent the coefficient of voluminal expansion, the volume of a body at a temperature $$t$$ will be and if $$d$$ represent density, since density is inversely as volume, we have  For a homogeneous isotropic solid, the coefficient of voluminal expansion is three times that of linear expansion; for, if the temperature of a cube, with an edge of unit length, be raised one degree, the length of its edge becomes $$1 + \alpha,$$ and its volume $${1 + 3\alpha + 3\alpha^2 + \alpha^3 .}$$. Since $$\alpha$$ is very small, its square and cube