Question

Two spheres made of the same substance have diameters in the ratio $1:2$. Their thermal capacities are in the ratio of:

• A. $1:8$
• B. $1:4$
• C. $1:16$
• D. $1:32$

Answer 1

The correct option is A $1:8$

Let $V_1$ be the volume of sphere $1$,
$V_2$ be the volume of sphere $2$.
$\because$ Both spheres are made of the same material, we can say that density of both materials will be same. Let it be denoted by $\rho$. Also the specific heat capacity of both spheres will be same. (denoted by $s$).

Thermal capacity $\left(C\right)$ = Mass $\left(m\right)$ $\times$ specific heat $\left(s\right)$
Also Mass $\left(m\right)$ = Volume $\left(V\right)\times$ Density $\left(\rho \right)$

$\therefore$ Ratio of thermal capacities of both spheres is given by:
$\frac{C_1}{C_2}=\frac{m_{1}s}{m_{2}s}=\frac{V_1\rho }{V_2\rho }$
$\Rightarrow \frac{C_1}{C_2}=\frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}={\left(\frac{r_1}{r_2}\right)}^3$
From the data given in the question,
$\frac{r_1}{r_2}=\frac{1}{2}$
$\therefore \frac{C_1}{C_2}={\left(\frac{1}{2}\right)}^3=1:8$
Thus, option (a) is the correct answer.