Question

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:

• 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 1

Answer: (D)

$4\pi R^3\alpha \Delta T$

As $\gamma =\frac{\Delta V}{V\times \Delta T}$ and $\gamma =3\alpha$,

$\therefore 3\alpha =\frac{\Delta V}{(\frac{4\pi }{3}\times R^3)\Delta T}$
which gives, $\Delta V=4\pi R^3\alpha \Delta T$