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

Options

A. `1:2`
B. `1:8`
C. `1:4`
D. `2:1`

Answer

To find the ratio of thermal capacities of two spheres made of the same substance with diameters in the ratio of 1:2, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between diameter and radius**:
- Let the diameter of the first sphere be \( d_1 \) and the second sphere be \( d_2 \).
- Given the ratio \( \frac{d_1}{d_2} = \frac{1}{2} \).
- Therefore, the radii \( r_1 \) and \( r_2 \) can be expressed as:
\[ r_1 = \frac{d_1}{2}, \quad r_2 = \frac{d_2}{2} = \frac{1}{2} \cdot \frac{d_2}{2} = \frac{1}{2} \cdot 1 = 1 \]
- Thus, \( \frac{r_1}{r_2} = \frac{1}{2} \).

2. **Calculate the volumes of the spheres**:
- The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
- For the first sphere:
\[ V_1 = \frac{4}{3} \pi r_1^3 \]
- For the second sphere:
\[ V_2 = \frac{4}{3} \pi r_2^3 \]

3. **Express the volumes in terms of the radius ratio**:
- Since \( r_1 = \frac{1}{2} r_2 \), we can express \( V_1 \) in terms of \( V_2 \):
\[ V_1 = \frac{4}{3} \pi \left(\frac{1}{2} r_2\right)^3 = \frac{4}{3} \pi \cdot \frac{1}{8} r_2^3 = \frac{1}{6} \cdot \frac{4}{3} \pi r_2^3 = \frac{1}{6} V_2 \]

4. **Calculate the thermal capacities**:
- The thermal capacity \( C \) of a substance is given by:
\[ C = m \cdot c \]
where \( m \) is the mass and \( c \) is the specific heat capacity.
- The mass \( m \) of the sphere can be expressed as:
\[ m = \rho V \]
where \( \rho \) is the density of the material.
- Therefore, the thermal capacities for both spheres can be written as:
\[ C_1 = \rho V_1 c, \quad C_2 = \rho V_2 c \]

5. **Find the ratio of thermal capacities**:
- The ratio of thermal capacities \( \frac{C_1}{C_2} \) becomes:
\[ \frac{C_1}{C_2} = \frac{\rho V_1 c}{\rho V_2 c} = \frac{V_1}{V_2} \]
- Substituting the volume ratio:
\[ \frac{C_1}{C_2} = \frac{\frac{1}{6} V_2}{V_2} = \frac{1}{6} \]

6. **Final Ratio**:
- Thus, the ratio of their thermal capacities is:
\[ \frac{C_1}{C_2} = \frac{1}{6} \]

### Conclusion:
The thermal capacities of the two spheres are in the ratio of \( 1:6 \).