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. $\pi R\alpha\Delta T$
• C. $4\pi R^{3}\,\alpha\Delta T/3$
• D. $4\pi R^{3}\,\alpha\Delta T$

Answer 1

Answer: (D)

$4\pi R^{3}\,\alpha\Delta T$

Answer 2

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