Two substance of same size are made of same material but one is hollow and the other is solid. They are heated to same temperature, then

Options

A. Rate of heat radiation in solid is more
B. Rate of heat radiation in hollow is more
C. Rate of heat radiation in both are equal
D. Depends on mass of the sphere

Answer

To solve the problem, we need to analyze the rate of heat radiation for both the hollow and solid substances. Here’s a step-by-step breakdown:

### Step 1: Understand the Concept of Heat Radiation
The rate of heat radiation from a body is given by the Stefan-Boltzmann law, which states that the power (P) radiated by a black body is proportional to the fourth power of its absolute temperature (T) and its surface area (A). The formula is:
\[ P = \sigma \cdot A \cdot T^4 \]
where \(\sigma\) is the Stefan-Boltzmann constant.

### Step 2: Identify the Given Information
- Both substances are made of the same material.
- Both substances are of the same size.
- One substance is hollow, and the other is solid.
- Both substances are heated to the same temperature.

### Step 3: Analyze the Surface Area
Since both substances are of the same size and made from the same material, their surface areas will be the same. The hollow substance may have a different volume, but since the problem states they are the same size, we can assume that their outer dimensions (and hence surface areas) are identical.

### Step 4: Apply the Stefan-Boltzmann Law
Since both substances have the same surface area (A) and are heated to the same temperature (T), we can substitute these values into the Stefan-Boltzmann equation for both substances:
\[ P_{\text{hollow}} = \sigma \cdot A \cdot T^4 \]
\[ P_{\text{solid}} = \sigma \cdot A \cdot T^4 \]

### Step 5: Compare the Rates of Heat Radiation
From the equations derived in Step 4, we can see that:
\[ P_{\text{hollow}} = P_{\text{solid}} \]
This means that the rate of heat radiation for both the hollow and solid substances is equal.

### Conclusion
Thus, the rate of heat radiation for both substances is the same.

### Final Answer
The rate of heat radiation for both the hollow and solid substances is equal.

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