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

The correct option is 1:8.

Here is the detailed step-by-step derivation to find the ratio of their thermal capacities:

1. Understand the formula for Thermal Capacity:
The thermal capacity (or heat capacity) $C$ of a body is defined as the amount of heat required to raise its temperature by one unit. Mathematically, it is given by the product of its mass ($m$) and its specific heat capacity ($s$):
$C=m\cdot s$

2. Identify the constants:
Since both spheres are made of the same substance, they share the same density ($\rho$) and specific heat capacity ($s$).
Therefore, the specific heat capacity $s$ is a constant value for both spheres.

3. Relate Mass to Diameter:
The mass of a sphere is equal to its volume multiplied by its density:
$m=V\cdot \rho$
For a sphere of diameter $D$, the volume $V$ is given by:
$V=\frac{4}{3}\pi R^3=\frac{4}{3}\pi {\left(\frac{D}{2}\right)}^3=\frac{1}{6}\pi D^3$
Substituting this volume back into the mass formula:
$m=\frac{1}{6}\pi D^3\cdot \rho$
This shows that the mass of the sphere is directly proportional to the cube of its diameter:
$m\propto D^3$

4. Relate Thermal Capacity to Diameter:
Since thermal capacity is $C=m\cdot s$ and $s$ is constant, the thermal capacity $C$ is directly proportional to the mass, which in turn is proportional to the cube of the diameter:
$C\propto D^3$

5. Calculate the Ratio:
Let the diameters of the two spheres be $D_1$ and $D_2$, with a ratio of:
$\frac{D_1}{D_2}=\frac{1}{2}$
The ratio of their thermal capacities is:
$\frac{C_1}{C_2}={\left(\frac{D_1}{D_2}\right)}^3$
Substituting the given ratio:
$\frac{C_1}{C_2}={\left(\frac{1}{2}\right)}^3=\frac{1}{8}$
Thus, the thermal capacities are in the ratio of 1:8.