Two spheres A and B have radius but the heat capacity of A is greater than that of B. The surfaces of both are painted black. They are heated to the same temperature and allowed to cool. Then initially

Options

A. A cools faster than B
B. both A and B cool at the same rate
C. at any common temperature, the ratio of their rates of cooling is a constant
D. B cools faster than A

Answer

To solve the problem step by step, we will analyze the cooling rates of the two spheres based on their heat capacities.

### Step 1: Understand the given information
We have two spheres, A and B, which have the same radius but different heat capacities. Sphere A has a greater heat capacity than sphere B. Both spheres are painted black and heated to the same initial temperature before being allowed to cool.

**Hint:** Recall that heat capacity is a measure of the amount of heat energy required to change the temperature of an object.

### Step 2: Define heat capacity and its implications
The heat capacity (C) of an object is defined as the amount of heat required to change its temperature by a certain amount. Since sphere A has a greater heat capacity than sphere B, we can express this as:
\[ C_A > C_B \]

**Hint:** A higher heat capacity means that sphere A can store more heat energy compared to sphere B for the same temperature change.

### Step 3: Apply Newton's Law of Cooling
According to Newton's Law of Cooling, the rate of heat loss of a body is proportional to the difference in temperature between the body and its surroundings. The rate of cooling can be expressed as:
\[ \frac{d\theta}{dt} = -k(T^4 - T_0^4) \]
where \( k \) is a constant, \( T \) is the temperature of the sphere, and \( T_0 \) is the ambient temperature.

**Hint:** Remember that the cooling rate depends on the temperature difference between the object and its surroundings.

### Step 4: Set up the equations for both spheres
For sphere A:
\[ \frac{d\theta_1}{dt} = -k_A (T^4 - T_0^4) \]

For sphere B:
\[ \frac{d\theta_2}{dt} = -k_B (T^4 - T_0^4) \]

Since both spheres are initially at the same temperature, we can analyze their cooling rates based on their heat capacities.

**Hint:** The cooling rate is influenced by the heat capacity of the object.

### Step 5: Relate the cooling rates to heat capacities
The relationship between the cooling rates and heat capacities can be expressed as:
\[ \frac{d\theta_1}{dt} \propto \frac{C_B}{C_A} \]
This implies that the cooling rate of sphere A is inversely related to its heat capacity compared to sphere B.

**Hint:** If one sphere has a greater heat capacity, it will cool down more slowly.

### Step 6: Compare the cooling rates
Since \( C_A > C_B \), we conclude that:
\[ \frac{d\theta_1}{dt} < \frac{d\theta_2}{dt} \]
This means that sphere B cools faster than sphere A.

**Hint:** The sphere with the lower heat capacity (B) will lose heat more quickly.

### Conclusion
Thus, the correct statement is that sphere B cools faster than sphere A.

**Final Answer:** Sphere B cools faster than sphere A.