Two solid spheres A and B made of the same material have radii `r_(A)` and `r_(B)` respectively . Both the spheres are cooled from the same temperature under the conditions valid for Newton's law of cooling . The ratio of the rate of change of temperature of A and B is

Options

A. `(r_(A))/(r_(B))`
B. `(r_(B))/(r_(A))`
C. `(r_(A)^(2))/(r_(B)^(2))`
D. `(r_(B)^(2))/(r_(A)^(2))`

Answer

To solve the problem, we need to find the ratio of the rate of change of temperature of two solid spheres A and B, which are made of the same material and are cooled under Newton's law of cooling.

### Step 1: Understand Newton's Law of Cooling
Newton's law of cooling states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, it can be expressed as:
\[ -\frac{dT}{dt} = k(T - T_0) \]
where \(T\) is the temperature of the object, \(T_0\) is the ambient temperature, and \(k\) is a constant that depends on the properties of the object and the medium.

### Step 2: Define the Variables for Spheres A and B
Let:
- \( r_A \) = radius of sphere A
- \( r_B \) = radius of sphere B
- \( \rho \) = density of the material (same for both spheres)
- \( V_A \) = volume of sphere A = \(\frac{4}{3} \pi r_A^3\)
- \( V_B \) = volume of sphere B = \(\frac{4}{3} \pi r_B^3\)
- \( m_A \) = mass of sphere A = \( \rho V_A = \rho \frac{4}{3} \pi r_A^3 \)
- \( m_B \) = mass of sphere B = \( \rho V_B = \rho \frac{4}{3} \pi r_B^3 \)

### Step 3: Write the Rate of Change of Temperature for Each Sphere
For sphere A:
\[ -\frac{dT_A}{dt} = k_A (T_A - T_0) \]
For sphere B:
\[ -\frac{dT_B}{dt} = k_B (T_B - T_0) \]

### Step 4: Relate the Cooling Constants
The cooling constant \(k\) is proportional to the surface area and inversely proportional to the mass:
\[ k \propto \frac{A}{m} \]
where \(A\) is the surface area and \(m\) is the mass.

The surface area \(A\) for a sphere is given by:
\[ A = 4 \pi r^2 \]
Thus, for sphere A:
\[ k_A \propto \frac{4 \pi r_A^2}{\rho \frac{4}{3} \pi r_A^3} = \frac{3}{\rho r_A} \]
And for sphere B:
\[ k_B \propto \frac{4 \pi r_B^2}{\rho \frac{4}{3} \pi r_B^3} = \frac{3}{\rho r_B} \]

### Step 5: Find the Ratio of the Rates of Change of Temperature
Now we can write the ratio of the rates of change of temperature:
\[ \frac{-\frac{dT_A}{dt}}{-\frac{dT_B}{dt}} = \frac{k_A (T_A - T_0)}{k_B (T_B - T_0)} \]
Since both spheres are cooled from the same initial temperature and reach the same final temperature, we can simplify this to:
\[ \frac{dT_A}{dT_B} = \frac{k_B}{k_A} = \frac{r_B}{r_A} \]

### Conclusion
Thus, the ratio of the rate of change of temperature of spheres A and B is:
\[ \frac{dT_A}{dT_B} = \frac{r_B}{r_A} \]

### Final Answer
The ratio of the rate of change of temperature of A and B is \( \frac{r_B}{r_A} \).

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