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:

• A. \(6\pi R^3\alpha\Delta T\)
• B. \(2\pi R^3\alpha\Delta T\)
• C. \(3\pi R^3\alpha\Delta T\)
• D. \(4\pi R^3\alpha\Delta T\)

Hint:

Volume coefficient equals \(3\alpha\). For isotropic solids use \(\gamma=3\alpha\). Remember volume of sphere \(=\frac43\pi R^3\). Use approximation for small temperature changes.

Answer 1

The Correct Option is D

Concept:

• Volume coefficient of expansion is \[ \gamma=3\alpha \]

• Increase in volume is \[ \Delta V=\gamma V\Delta T \]

Step 1: Write initial volume of sphere
\[ V=\frac43\pi R^3 \]

Step 2: Apply volume expansion formula
\[ \Delta V=3\alpha V\Delta T \] \[ =3\alpha \left(\frac43\pi R^3\right)\Delta T \]

Step 3: Simplify expression
\[ \Delta V = 4\pi R^3\alpha\Delta T \]