The current curve between `log_e""(j)/(T^2)" and "l/T`

Options

A.
B.
C.
D.

Explanation

To solve the question regarding the current curve between \( \log_e\left(\frac{j}{T^2}\right) \) and \( \frac{1}{T} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
We start with the relationship given in the problem:
\[ j = a T^2 e^{-\frac{b}{T}} \]
where \( a \) and \( b \) are constants.

2. **Rearranging the Equation**:
We can express \( \frac{j}{T^2} \):
\[ \frac{j}{T^2} = a e^{-\frac{b}{T}} \]

3. **Taking the Natural Logarithm**:
Now, we take the natural logarithm (base \( e \)) of both sides:
\[ \log_e\left(\frac{j}{T^2}\right) = \log_e(a) + \log_e\left(e^{-\frac{b}{T}}\right) \]

4. **Simplifying the Logarithm**:
Using the property of logarithms \( \log_e(e^x) = x \), we can simplify the right side:
\[ \log_e\left(\frac{j}{T^2}\right) = \log_e(a) - \frac{b}{T} \]

5. **Rearranging the Equation**:
We can rearrange this equation to express it in the form of a straight line:
\[ \log_e\left(\frac{j}{T^2}\right) = -\frac{b}{T} + \log_e(a) \]

6. **Identifying the Variables**:
In this equation, we can identify:
- \( y = \log_e\left(\frac{j}{T^2}\right) \)
- \( x = \frac{1}{T} \)
- The slope \( m = -b \)
- The y-intercept \( c = \log_e(a) \)

7. **Conclusion**:
The relationship between \( \log_e\left(\frac{j}{T^2}\right) \) and \( \frac{1}{T} \) is linear, with a negative slope. Thus, the correct option that represents this relationship is:
\[ \log_e\left(\frac{j}{T^2}\right) \text{ vs } \frac{1}{T} \]