Heat and Thermal ExpansionMedium

Question

The temperature of a metallic sphere of radius $R$ is increased by a small amount $\Delta T$. If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is:

Options

A.$4\pi R^{3}\alpha\Delta T$
B.$6\pi R^{3}\alpha\Delta T$
C.$2\pi R^{3}\alpha\Delta T$
D.$3\pi R^{3}\alpha\Delta T$

Solution

Sol. [Key Concept: Thermal expansion of a solid sphere]

For the metallic sphere, $\frac{\Delta R}{R} = \alpha\Delta T$

Volume of the sphere, $V = \frac{4}{3}\pi R^{3}$

$$\ \therefore\frac{\Delta V}{V} = 3\frac{\Delta R}{R} $$Increase in volume $= \Delta V = 3\left( \frac{\Delta R}{R} \right)\left( \frac{4}{3}\pi R^{3} \right) = 4\pi R^{3}\alpha\Delta T$

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