Which one of the following graphs best represents the ways in which the total power P radiated by a black body depends upon the thermodynamic temperature T of the body?

Options

A.
B.
C.
D.

Explanation

To solve the question of how the total power \( P \) radiated by a black body depends on the thermodynamic temperature \( T \), we can follow these steps:

### Step 1: Understand Stefan-Boltzmann Law
According to the Stefan-Boltzmann Law, the total power \( P \) radiated by a black body is directly proportional to the fourth power of its absolute temperature \( T \). This can be mathematically expressed as:
\[ P = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant and \( A \) is the surface area of the body.

### Step 2: Analyze the Relationship
From the equation \( P = \sigma A T^4 \), we can see that as the temperature \( T \) increases, the power \( P \) increases with the fourth power of \( T \). This indicates a non-linear relationship, specifically a polynomial relationship of degree 4.

### Step 3: Identify the Graph Type
Since the relationship is \( P \propto T^4 \), the graph of \( P \) versus \( T \) will be a curve that starts at the origin (0,0) and rises steeply as \( T \) increases. This is characteristic of a parabolic curve.

### Step 4: Select the Correct Option
Among the options provided, we need to identify which graph represents a fourth-degree polynomial relationship. The correct graph will show a steep increase in power as temperature increases, confirming that it is a parabolic curve.

### Conclusion
Based on the analysis, the graph that best represents the relationship between the total power \( P \) radiated by a black body and the thermodynamic temperature \( T \) is the one that shows a parabolic curve, which corresponds to option C.