The radius of a metal sphere at room temperature T is R, and the coefficient of linear expansion of the metal is `alpha`. The sphere is heated a little by a temperature `Delta T` so that its new temperature is `T+ Delta T`. The increase in the volume of the sphere is approximately

Options

A. `2pi R prop Delta T`
B. `pi R^2 alpha Delta T`
C. `4pi R^3 alpha Delta T//3`
D. `4pi R^3 alpha Delta T`

Answer

D

Explanation

Volume of a sphere of radius R at temperature T is
`V = 4/3 pi R^3`
If `Delta T` is the increase in temperature, the increase in volume is given by
`Delta V = gamma V Delta T`
` = 3alpha 4/3 pi R^3 Delta T " " [ because gamma = 3 alpha]`
` =4pi R^3 alpha Delta T`