Question

If the ratio of radii of two spheres of same material is $1:4$, then the ratio of their heat capacities will be:

• A. $1/64$
• B. $1/32$
• C. $1/2$
• D. $1/4$

Answer 1

The correct option is A $1/64$

As material of both the spheres is same, therefore their specific heat capacity $\left(s\right)$ and density will also be same.
Also,
$\text{Mass}\left(m\right)=\text{Volume}\left(V\right)\times \text{density}\left(\rho \right)$
For sphere;
Volume $\left(V\right)=\frac{4}{3}\times \pi \times r^3$
$\therefore$ Mass of 1st sphere; $m_A=V_A\times {\rho }_A$
$=\frac{4}{3}\pi r_A^3\times {\rho }_A$
and
Mass of 2nd sphere,
$m_B=\frac{4}{3}\pi r_B^3\times {\rho }_B$
$\text{Heat capacity}\left(C\right)=\text{Mass}\left(m\right)\times \text{Specific heat capacity}\left(s\right)$
$\therefore \frac{C_A}{C_B}=\frac{m_A{s}_A}{m_B{s}_B}=\frac{r_A^3{\rho }_A}{r_B^3{\rho }_B}=\frac{r_A^3}{r_B^3}\because {\rho }_A={\rho }_B\therefore \frac{C_A}{C_B}={\left(\frac{1}{4}\right)}^3\Rightarrow \frac{C_A}{C_B}=\frac{1}{64}$