Two sphere of copper of diameters `10 cm` and `20 cm` will have thermal capacities in the ratio

Options

A. `(1)/(8)`
B. `(1)/(2)`
C. `(1)/(4)`
D. `(1)/(6)`

Explanation

To find the ratio of thermal capacities of two copper spheres with diameters of 10 cm and 20 cm, we can follow these steps:

### Step 1: Determine the Radii of the Spheres
The diameter of the first sphere is 10 cm, so its radius (R1) is:
\[ R_1 = \frac{10 \, \text{cm}}{2} = 5 \, \text{cm} \]

The diameter of the second sphere is 20 cm, so its radius (R2) is:
\[ R_2 = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} \]

### Step 2: Write the Formula for Thermal Capacity
The thermal capacity (H) of an object is given by the formula:
\[ H = M \times S \]
where M is the mass and S is the specific heat capacity.

### Step 3: Relate Mass to Volume and Density
The mass (M) of the spheres can be expressed in terms of volume (V) and density (ρ):
\[ M = \rho \times V \]

For a sphere, the volume (V) is given by:
\[ V = \frac{4}{3} \pi R^3 \]

Thus, we can write the thermal capacity as:
\[ H = \rho \times V \times S = \rho \times \left(\frac{4}{3} \pi R^3\right) \times S \]

### Step 4: Express Thermal Capacities for Both Spheres
For the first sphere (R1 = 5 cm):
\[ H_1 = \rho \times \left(\frac{4}{3} \pi (5)^3\right) \times S \]

For the second sphere (R2 = 10 cm):
\[ H_2 = \rho \times \left(\frac{4}{3} \pi (10)^3\right) \times S \]

### Step 5: Find the Ratio of Thermal Capacities
Now, we can find the ratio of the thermal capacities:
\[ \frac{H_1}{H_2} = \frac{\rho \times \left(\frac{4}{3} \pi (5)^3\right) \times S}{\rho \times \left(\frac{4}{3} \pi (10)^3\right) \times S} \]

Since ρ and S are the same for both spheres, they cancel out:
\[ \frac{H_1}{H_2} = \frac{(5)^3}{(10)^3} = \frac{125}{1000} = \frac{1}{8} \]

### Final Answer
Thus, the ratio of the thermal capacities of the two spheres is:
\[ H_1 : H_2 = 1 : 8 \]