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. `2 pi R alpha Delta T`
B. `pi R^(2) alpha Delta T`
C. `4 pi R^(3) alpha Delta T//3`
D. `4 pi R^(3) alpha Delta T`

Answer

D

Explanation

Original volume `V_(o) = (4)/(3) pi R^(3)`
Coeff of linear expansion `= alpha`
`:.` Coeff of volume expansion `= 3 alpha`
`:. (1)/(V) (dV)/(dT) = 3 alpha`
`rArr dV = 3V alpha dT ~~ 4 pi R^(3) alpha Delta T`