Two sphere with radii in the ratio `1 : 2` have specific heats in the ratio `x : y` and densities in the ratio `z : x`. The ratio of their thermal capacities is

Options

A. `z : 2y`
B. `zy : 8`
C. `z : 8y`
D. `xy : 2z`

Explanation

To find the ratio of the thermal capacities of two spheres with given properties, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The thermal capacity \( H \) of an object is given by the formula:
\[ H = M \cdot S \]
where \( M \) is the mass and \( S \) is the specific heat capacity.

2. **Express Mass in Terms of Density and Volume**:
- The mass \( M \) of each sphere can be expressed as:
\[ M = \rho \cdot V \]
where \( \rho \) is the density and \( V \) is the volume of the sphere.

3. **Volume of a Sphere**:
- The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
- For two spheres with radii \( R_1 \) and \( R_2 \), the volumes will be:
\[ V_1 = \frac{4}{3} \pi R_1^3 \quad \text{and} \quad V_2 = \frac{4}{3} \pi R_2^3 \]

4. **Calculate the Thermal Capacities**:
- The thermal capacities of the two spheres can be expressed as:
\[ H_1 = M_1 \cdot S_1 = \rho_1 \cdot V_1 \cdot S_1 = \rho_1 \cdot \left(\frac{4}{3} \pi R_1^3\right) \cdot S_1 \]
\[ H_2 = M_2 \cdot S_2 = \rho_2 \cdot V_2 \cdot S_2 = \rho_2 \cdot \left(\frac{4}{3} \pi R_2^3\right) \cdot S_2 \]

5. **Find the Ratio of Thermal Capacities**:
- The ratio of the thermal capacities \( \frac{H_1}{H_2} \) can be expressed as:
\[ \frac{H_1}{H_2} = \frac{\rho_1 \cdot V_1 \cdot S_1}{\rho_2 \cdot V_2 \cdot S_2} \]
- Substituting the volumes:
\[ \frac{H_1}{H_2} = \frac{\rho_1 \cdot \left(\frac{4}{3} \pi R_1^3\right) \cdot S_1}{\rho_2 \cdot \left(\frac{4}{3} \pi R_2^3\right) \cdot S_2} \]
- The \( \frac{4}{3} \pi \) cancels out:
\[ \frac{H_1}{H_2} = \frac{\rho_1 \cdot R_1^3 \cdot S_1}{\rho_2 \cdot R_2^3 \cdot S_2} \]

6. **Substituting the Ratios**:
- Given the ratios:
- Radii: \( \frac{R_1}{R_2} = \frac{1}{2} \)
- Specific heats: \( \frac{S_1}{S_2} = \frac{x}{y} \)
- Densities: \( \frac{\rho_1}{\rho_2} = \frac{z}{x} \)
- Substitute these into the ratio:
\[ \frac{H_1}{H_2} = \frac{\frac{z}{x} \cdot \left(\frac{1}{2}\right)^3 \cdot x}{\rho_2 \cdot S_2} = \frac{z}{x} \cdot \frac{1}{8} \cdot \frac{x}{y} \]
- Simplifying gives:
\[ \frac{H_1}{H_2} = \frac{z}{8y} \]

7. **Final Result**:
- The ratio of the thermal capacities of the two spheres is:
\[ \frac{H_1}{H_2} = \frac{z}{8y} \]