A hollow sphere and a solid sphere of equal radii and of same material are heated to raise their temperature by equal amounts. How will the change in their volume expansion be related?

Explanation

In case of a solid sphere, there is only thermal expansion in volume. Let V be the original volume of the solid sphere. `Delta V` be the change in volume when temperature rises by `1^(@)C` (i.e. `Delta T = 1^(@)C)`. Then coefficient of cubical expansion of solid sphere, `gamma = (Delta V)/(V Delta T)`
or `Delta V = gamma V Delta T = gamma xx V xx 1 = gamma V`
In case of a hollow sphere, apart from thermal expansion, there will be increase in volume `Delta upsilon` due to increase in pressure `Delta P` of the enclosed air on heating. then bulk modules of elasticity of air is `B = (VDeltaP)/(Delta upsilon) = or Delta upsilon = (V Delta P)/(B)`
total increase in volume of the hollow sphere is
`Delta V' = Delta V + Delta upsilon = gamma V =(V Delta P)/(B)`
Total increase in volume of the hollow sphere is
`Delta V' = Delta V + Delta upsilon = gamma V + (V Delta P)/(B)`
or `(Delta V)/(Delta V) = (gamma V+V Delta P//B)/(gamma V) = 1+(Delta P)/(gamma B)`.