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 solve the problem of finding 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 1: Understand the relationship between diameter and radius
The diameters of the two spheres are given in the ratio:
\[ \frac{d_1}{d_2} = \frac{1}{2} \]
Since the radius is half of the diameter, the ratio of the radii will be the same:
\[ \frac{r_1}{r_2} = \frac{1}{2} \]

### Step 2: Write the formula for thermal capacity
The thermal capacity (C) of an object is given by:
\[ C = m \cdot s \]
where \(m\) is the mass and \(s\) is the specific heat capacity.

### Step 3: Express mass in terms of volume and density
The mass \(m\) can be expressed as:
\[ m = \rho \cdot V \]
where \(\rho\) is the density and \(V\) is the volume. The volume \(V\) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the thermal capacity can be rewritten as:
\[ C = \rho \cdot \left(\frac{4}{3} \pi r^3\right) \cdot s \]

### Step 4: Calculate thermal capacities for both spheres
For the first sphere (with radius \(r_1\)):
\[ C_1 = \rho \cdot \left(\frac{4}{3} \pi r_1^3\right) \cdot s \]
For the second sphere (with radius \(r_2\)):
\[ C_2 = \rho \cdot \left(\frac{4}{3} \pi r_2^3\right) \cdot s \]

### Step 5: Find the ratio of thermal capacities
Now, we can find the ratio of the thermal capacities:
\[ \frac{C_1}{C_2} = \frac{\rho \cdot \left(\frac{4}{3} \pi r_1^3\right) \cdot s}{\rho \cdot \left(\frac{4}{3} \pi r_2^3\right) \cdot s} \]
The terms \(\rho\), \(\frac{4}{3} \pi\), and \(s\) will cancel out:
\[ \frac{C_1}{C_2} = \frac{r_1^3}{r_2^3} \]

### Step 6: Substitute the ratio of the radii
Since we have \(\frac{r_1}{r_2} = \frac{1}{2}\), we can substitute this into the equation:
\[ \frac{C_1}{C_2} = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

### Conclusion
Thus, the ratio of the thermal capacities of the two spheres is:
\[ C_1 : C_2 = 1 : 8 \]