# Thermal expansion of Sphere

How would one go about writing an expression of the expansion of the volume of a sphere of a given material? I noticed a few sources give it as

$\Delta V= 3\gamma V\Delta T$
where V is the initial volume; $\gamma$ is the expansivity coefficient and $\Delta T$ is change in temperature of sphere.

Other texts leave out the 3, but with everything else the same.

Any suggestions?

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Other texts leave out the 3, but with everything else the same.

Presumably because ...

"For exactly isotropic materials, and for small expansions, the linear thermal expansion coefficient is one third the volumetric coefficient."

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How small are we talking? I'm looking at the change in sea level by thermal expansion and so the change in V is kinda big. The section you linked to talks about using it for a cube, not a sphere. Does the shape make a difference? – kuantumbro Nov 18 '12 at 22:46
@kuantumbro No, the shape doesn't matter. The link given here explains where the factor of 3 emerges from. If you understand the mathematics behind that, you should be able to resolve the discrepancy between books that you ran into (look at the coefficient's definition, units, so on). You could even use that link's equations to give numerical metrics for "how small" of a change the approximation is valid within your desired tolerance. – Alan Rominger Nov 19 '12 at 0:57

## protected by Qmechanic♦Mar 20 '13 at 13:20

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