relationship between area and volume expansivity
I am not sure I know what you mean.
Perhaps I can guess by taking a simple example, a sphere.
A sphere of radius r has a surface area of A = (4) pi r^2
how much does the volume change if I increase the radius a little?
well, I add a volume on the outside of (4) pi r^2 times the the change in radius.
If I keep doing that from r = 0 to r = R I will get (4) pi R^3 / 3
or in other words (4/3) pi R^3
( If you have had calculus you know that the integral of r^2 dr from 0 to R is R^3 / 3 )
in other words the area is proportional to the length dimension squared
and the volume is proportional to the length dimension cubed
and the increase in volume is proportional to the area times a length increase
for similar figures
If you double a length dimension
you multiply all areas by four
and volumes by eight
The relationship between area expansivity and volume expansivity is based on the concept of thermal expansion, which describes how a material changes its dimensions with temperature.
Area expansivity, also known as linear expansivity, measures the fractional change in the area of an object per unit temperature change. It is denoted by the symbol α (alpha) and has units of 1/°C or 1/K.
Volume expansivity, on the other hand, quantifies the fractional change in the volume of an object per unit temperature change. It is denoted by the symbol β (beta) and also has units of 1/°C or 1/K.
The relationship between area and volume expansivity depends on the dimensionality of the object. Let's consider two scenarios:
1. One-dimensional object: Suppose we have a straight rod or wire. In this case, the area expansivity and volume expansivity are equivalent because the object only expands or contracts in one dimension. Therefore, α = β.
2. Three-dimensional object: If we consider a three-dimensional object, such as a solid cube, the relationship between area and volume expansivity is different. The volume expansivity, β, is directly related to the area expansivity, α, through the following equation:
β = 2α
This relationship arises from the fact that the volume of a cube is proportional to the cube of its linear dimensions, while the area is proportional to the square of its linear dimensions. By differentiating these equations with respect to temperature, we can derive the relationship.
It's important to note that different materials have different thermal expansion properties, and their area and volume expansivities can vary accordingly. The relationship described above assumes that the material's thermal expansion is isotropic, meaning it expands equally in all directions.