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 T 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

Let the radius of the sphere is R. As the temperature increases radius of the sphere increases as shown in fig.

Original volume `V_0 = (4/3) pi R^3`
Coefficient of linear expansion` = alpha`
`therefore` Coefficient of volume expansion `3 alpha`
`therefore (1)/(V) (dV)/(dT) = 3 alpha rArr dV = 3V alpha dT cong 4 pi R^(3) alpha Delta T`
= Increase in the volume.