A metal piece is heated up to T K. The temperature of the surrounding is t K. The heat lost to the surrounding due to radiation is proportional to

Options

A. `T^(4)-t^(4)`
B. `(T-t^(4))`
C. `(T^(4)+t^(4))`
D. `(T^(4)+t^(4))`

Explanation

To solve the problem of determining how the heat lost to the surrounding due to radiation is proportional, we can follow these steps:

### Step 1: Understand the Concept of Radiation Heat Loss
The heat lost due to radiation from a surface is described by Stefan-Boltzmann Law, which states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature.

### Step 2: Apply Stefan-Boltzmann Law
According to Stefan-Boltzmann Law, the heat loss \( Q \) due to radiation can be expressed as:
\[ Q \propto A (T^4 - t^4) \]
where:
- \( A \) is the surface area of the metal piece,
- \( T \) is the absolute temperature of the metal piece,
- \( t \) is the absolute temperature of the surroundings.

### Step 3: Identify the Proportionality
From the equation \( Q \propto A (T^4 - t^4) \), we can see that the heat loss is proportional to the difference in the fourth powers of the temperatures of the metal piece and its surroundings.

### Step 4: Conclude the Proportionality
Thus, the heat lost to the surrounding due to radiation is proportional to:
\[ (T^4 - t^4) \]

### Final Answer
The heat lost to the surrounding due to radiation is proportional to \( (T^4 - t^4) \).

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