Which of the following graphs correctly represents the relation between In (E) and In (T), where E is the amount of radiation emitted per unit time from a unit area of the body and T is the absolute temperature ?

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to establish the relationship between the amount of radiation emitted per unit time from a unit area of the body (E) and the absolute temperature (T). We will use Stefan-Boltzmann Law, which states that the power (P) emitted by a black body per unit area is proportional to the fourth power of its absolute temperature (T).

### Step-by-Step Solution:

1. **Understand the Stefan-Boltzmann Law**:
The law states that:
\[ P = \sigma A T^4 \]
where \(P\) is the power emitted, \(A\) is the area, and \(\sigma\) is the Stefan-Boltzmann constant.

2. **Define E**:
The amount of radiation emitted per unit time from a unit area (E) can be expressed as:
\[ E = \frac{P}{A} \]
Substituting the expression for \(P\):
\[ E = \frac{\sigma A T^4}{A} = \sigma T^4 \]

3. **Take the Natural Logarithm**:
To find the relationship in terms of natural logarithms, we take the natural logarithm of both sides:
\[ \ln(E) = \ln(\sigma T^4) \]

4. **Apply Logarithmic Properties**:
Using the properties of logarithms, we can expand the right side:
\[ \ln(E) = \ln(\sigma) + \ln(T^4) = \ln(\sigma) + 4\ln(T) \]

5. **Rearranging the Equation**:
We can rearrange this equation to match the form \(y = mx + c\):
\[ \ln(E) = 4\ln(T) + \ln(\sigma) \]
Here, \(y = \ln(E)\), \(x = \ln(T)\), the slope \(m = 4\), and the y-intercept \(c = \ln(\sigma)\).

6. **Analyze the y-intercept**:
Since \(\sigma\) is a constant between 0 and 1, \(\ln(\sigma) < 0\). This means that the y-intercept is negative.

7. **Conclusion about the Graph**:
The graph of \(\ln(E)\) versus \(\ln(T)\) will be a straight line with a slope of 4 and a negative y-intercept.

### Final Answer:
The correct graph will be a straight line with a positive slope (4) and a negative y-intercept.