Two spheres `A` and `B` have diameters in the ratio `1:2`, densities in the ratio `2:1` and specific heat in the ratio `1:3`. Find the ratio of their thermal capacities.

Options

A. `1:6`
B. `1:12`
C. `1:3`
D. `1:4`

Explanation

To find the ratio of the thermal capacities of two spheres A and B, we can follow these steps:

### Step 1: Understand the formula for thermal capacity
Thermal capacity (C) is defined as the product of mass (m) and specific heat capacity (c):
\[ C = m \cdot c \]

### Step 2: Express mass in terms of volume and density
The mass of a sphere can be expressed as:
\[ m = \text{Volume} \times \text{Density} \]
The volume (V) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass of sphere A is:
\[ m_A = \frac{4}{3} \pi r_A^3 \cdot \rho_A \]
And for sphere B:
\[ m_B = \frac{4}{3} \pi r_B^3 \cdot \rho_B \]

### Step 3: Substitute the expressions for mass into the thermal capacity formula
Now, we can express the thermal capacities of spheres A and B:
\[ C_A = m_A \cdot c_A = \left(\frac{4}{3} \pi r_A^3 \cdot \rho_A\right) \cdot c_A \]
\[ C_B = m_B \cdot c_B = \left(\frac{4}{3} \pi r_B^3 \cdot \rho_B\right) \cdot c_B \]

### Step 4: Find the ratio of thermal capacities
Now, we can find the ratio of thermal capacities \( \frac{C_A}{C_B} \):
\[ \frac{C_A}{C_B} = \frac{\left(\frac{4}{3} \pi r_A^3 \cdot \rho_A\right) \cdot c_A}{\left(\frac{4}{3} \pi r_B^3 \cdot \rho_B\right) \cdot c_B} \]
The \( \frac{4}{3} \pi \) cancels out:
\[ \frac{C_A}{C_B} = \frac{r_A^3 \cdot \rho_A \cdot c_A}{r_B^3 \cdot \rho_B \cdot c_B} \]

### Step 5: Substitute the known ratios
Given the ratios:
- Diameter ratio \( \frac{d_A}{d_B} = \frac{1}{2} \) implies \( \frac{r_A}{r_B} = \frac{1}{2} \)
- Density ratio \( \frac{\rho_A}{\rho_B} = \frac{2}{1} \)
- Specific heat ratio \( \frac{c_A}{c_B} = \frac{1}{3} \)

Now substituting these ratios into the equation:
\[ \frac{C_A}{C_B} = \frac{\left(\frac{1}{2}\right)^3 \cdot 2 \cdot \frac{1}{3}}{1} \]
Calculating this gives:
\[ \frac{C_A}{C_B} = \frac{\frac{1}{8} \cdot 2 \cdot \frac{1}{3}}{1} = \frac{2}{24} = \frac{1}{12} \]

### Conclusion
Thus, the ratio of the thermal capacities of spheres A and B is:
\[ \frac{C_A}{C_B} = \frac{1}{12} \]