Two spheres of radii `R_(1)` and `R_(2)` are made of the same material and are at the same temperature. The ratio of their thermal capacities is:

Options

A. `R_(1)^(4)//R_(2)^(4)`
B. `R_(1)^(3)//R_(2)^(3)`
C. `R_(1)^(2)//R_(2)^(2)`
D. `R_(1)//R_(2)`

Explanation

To find the ratio of the thermal capacities of two spheres made of the same material and at the same temperature, we can follow these steps:

### Step 1: Understand the concept of thermal capacity
Thermal capacity (or heat capacity) is defined as the amount of heat required to change the temperature of an object by one degree Celsius (or Kelvin). It is given by the formula:
\[ C = m \cdot c \]
where \( C \) is the thermal capacity, \( m \) is the mass of the object, and \( c \) is the specific heat capacity of the material.

### Step 2: Determine the mass of each sphere
Since both spheres are made of the same material, they will have the same specific heat capacity \( c \). The mass \( m \) of a sphere can be calculated using the formula:
\[ m = \rho \cdot V \]
where \( \rho \) is the density of the material and \( V \) is the volume of the sphere. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass of each sphere can be expressed as:
- For sphere 1 (radius \( R_1 \)):
\[ m_1 = \rho \cdot \frac{4}{3} \pi R_1^3 \]
- For sphere 2 (radius \( R_2 \)):
\[ m_2 = \rho \cdot \frac{4}{3} \pi R_2^3 \]

### Step 3: Write the thermal capacities for both spheres
Using the mass calculated above, we can express the thermal capacities for both spheres:
- For sphere 1:
\[ C_1 = m_1 \cdot c = \left(\rho \cdot \frac{4}{3} \pi R_1^3\right) \cdot c \]
- For sphere 2:
\[ C_2 = m_2 \cdot c = \left(\rho \cdot \frac{4}{3} \pi R_2^3\right) \cdot c \]

### Step 4: Find the ratio of thermal capacities
Now, we can find the ratio of the thermal capacities \( C_1 \) and \( C_2 \):
\[ \frac{C_1}{C_2} = \frac{\left(\rho \cdot \frac{4}{3} \pi R_1^3\right) \cdot c}{\left(\rho \cdot \frac{4}{3} \pi R_2^3\right) \cdot c} \]
The terms \( \rho \), \( \frac{4}{3} \pi \), and \( c \) cancel out:
\[ \frac{C_1}{C_2} = \frac{R_1^3}{R_2^3} \]

### Step 5: Conclusion
Thus, the ratio of the thermal capacities of the two spheres is:
\[ \frac{C_1}{C_2} = \left(\frac{R_1}{R_2}\right)^3 \]