Q.No 4-57, the ratio of their initial rates of cooling is:

Options

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

Explanation

To solve the problem of finding the ratio of the initial rates of cooling of two solid spheres with radii \( R_1 \) and \( R_2 \), we will follow these steps:

### Step 1: Understand the Heat Loss
The heat loss from a body is given by the Stefan-Boltzmann law, which states that the power loss \( P \) is proportional to the surface area \( A \) and the fourth power of the temperature \( T \):
\[ P \propto A \cdot \sigma \cdot T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant.

### Step 2: Calculate the Surface Area
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi R^2 \]
Thus, for the two spheres:
- Surface area of sphere 1: \( A_1 = 4 \pi R_1^2 \)
- Surface area of sphere 2: \( A_2 = 4 \pi R_2^2 \)

### Step 3: Relate Heat Loss to the Surface Area
Since both spheres are made of the same material and are at the same temperature, we can express the ratio of their heat loss:
\[ \frac{P_1}{P_2} = \frac{A_1}{A_2} = \frac{4 \pi R_1^2}{4 \pi R_2^2} = \frac{R_1^2}{R_2^2} \]

### Step 4: Understand the Rate of Cooling
The rate of cooling \( \frac{dT}{dt} \) is influenced not only by the heat loss but also by the mass of the spheres. The mass \( m \) of a sphere is given by:
\[ m = \frac{4}{3} \pi R^3 \cdot \rho \]
where \( \rho \) is the density of the material.

### Step 5: Calculate the Mass
Thus, for the two spheres:
- Mass of sphere 1: \( m_1 = \frac{4}{3} \pi R_1^3 \cdot \rho \)
- Mass of sphere 2: \( m_2 = \frac{4}{3} \pi R_2^3 \cdot \rho \)

### Step 6: Find the Ratio of the Initial Rate of Cooling
The initial rate of cooling can be expressed as:
\[ \frac{dT}{dt} \propto \frac{P}{m} \]
Thus, the ratio of the rates of cooling for the two spheres is:
\[ \frac{R_{c1}}{R_{c2}} = \frac{P_1/m_1}{P_2/m_2} = \frac{P_1 \cdot m_2}{P_2 \cdot m_1} \]

Substituting the expressions for \( P \) and \( m \):
\[ \frac{R_{c1}}{R_{c2}} = \frac{\frac{R_1^2}{R_2^2} \cdot \frac{4}{3} \pi R_2^3 \cdot \rho}{\frac{R_2^2}{R_1^2} \cdot \frac{4}{3} \pi R_1^3 \cdot \rho} \]

### Step 7: Simplify the Expression
The \( \frac{4}{3} \pi \) and \( \rho \) cancel out, leading to:
\[ \frac{R_{c1}}{R_{c2}} = \frac{R_1^2 \cdot R_2^3}{R_2^2 \cdot R_1^3} = \frac{R_2}{R_1} \]

### Final Result
Thus, the ratio of the initial rates of cooling of the two spheres is:
\[ \frac{R_{c1}}{R_{c2}} = \frac{R_2}{R_1} \]