There are two thin spheres A and B of the same material and same thickness. They behave like black bodies, Radius of A is double that of B and Both have same temperature T. When A and B are kept in a room of temperature `T_0 (lt T)`, the ratio of their rates of cooling is (assume negligible heat exchange between A and B).

Options

A. `2:1`
B. `1:1`
C. `4:1`
D. `8:1`

Explanation

To solve the problem step by step, we need to analyze the cooling rates of the two spheres A and B using the principles of calorimetry and Stefan's law.

### Step 1: Understand the Problem
We have two spheres, A and B, made of the same material and thickness. Sphere A has a radius that is double that of sphere B. Both spheres are at the same initial temperature \( T \) and are placed in a room at a lower temperature \( T_0 \) (where \( T_0 < T \)). We need to find the ratio of their rates of cooling.

### Step 2: Apply Stefan's Law
According to Stefan's law, the rate of heat loss (cooling) for a black body is given by:
\[ \frac{dQ}{dt} = \sigma A (T^4 - T_0^4) \]
where:
- \( \sigma \) is Stefan's constant,
- \( A \) is the surface area of the sphere,
- \( T \) is the temperature of the sphere,
- \( T_0 \) is the temperature of the surroundings.

### Step 3: Calculate the Surface Areas
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
Let the radius of sphere B be \( r \). Then, the radius of sphere A is \( 2r \).

- Surface area of sphere B:
\[ A_B = 4\pi r^2 \]

- Surface area of sphere A:
\[ A_A = 4\pi (2r)^2 = 4\pi (4r^2) = 16\pi r^2 \]

### Step 4: Find the Rate of Cooling for Each Sphere
Using Stefan's law, we can express the rates of cooling for both spheres.

- For sphere A:
\[ \frac{dQ_A}{dt} = \sigma A_A (T^4 - T_0^4) = \sigma (16\pi r^2)(T^4 - T_0^4) \]

- For sphere B:
\[ \frac{dQ_B}{dt} = \sigma A_B (T^4 - T_0^4) = \sigma (4\pi r^2)(T^4 - T_0^4) \]

### Step 5: Calculate the Ratio of Rates of Cooling
Now, we can find the ratio of the rates of cooling for spheres A and B:
\[ \frac{\frac{dQ_A}{dt}}{\frac{dQ_B}{dt}} = \frac{\sigma (16\pi r^2)(T^4 - T_0^4)}{\sigma (4\pi r^2)(T^4 - T_0^4)} \]

The \( \sigma \), \( \pi \), and \( r^2 \) terms cancel out:
\[ \frac{\frac{dQ_A}{dt}}{\frac{dQ_B}{dt}} = \frac{16}{4} = 4 \]

### Step 6: Conclusion
The ratio of the rates of cooling of spheres A and B is:
\[ \frac{dQ_A/dt}{dQ_B/dt} = 4 \]

### Final Answer
The ratio of their rates of cooling is \( 4:1 \).

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