The log - log graph between the energy `E` of an electron and its de - Broglie wavelength `lambda` will be

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Explanation

To solve the problem of determining the log-log graph between the energy \( E \) of an electron and its de Broglie wavelength \( \lambda \), we can follow these steps:

### Step-by-step Solution:

1. **Understand the de Broglie wavelength formula**:
The de Broglie wavelength \( \lambda \) of a particle is given by the formula:
\[ \lambda = \frac{h}{\sqrt{2mE}} \]
where \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( E \) is the energy of the electron.

2. **Rearranging the formula**:
We can rearrange the formula to express it in terms of \( E \):
\[ \lambda = \frac{h}{\sqrt{2m}} \cdot \frac{1}{\sqrt{E}} \]

3. **Taking logarithms**:
To analyze the relationship in a log-log graph, we take the logarithm of both sides:
\[ \log \lambda = \log \left( \frac{h}{\sqrt{2m}} \right) - \frac{1}{2} \log E \]

4. **Breaking down the logarithm**:
We can separate the logarithm on the right-hand side:
\[ \log \lambda = \log h - \frac{1}{2} \log (2m) - \frac{1}{2} \log E \]

5. **Identifying the linear relationship**:
This equation can be rewritten as:
\[ \log \lambda = \left(-\frac{1}{2}\right) \log E + \left(\log h - \frac{1}{2} \log (2m)\right) \]
Here, we can identify:
- The slope of the graph is \( -\frac{1}{2} \)
- The y-intercept is \( \log h - \frac{1}{2} \log (2m) \)

6. **Graph characteristics**:
Since the slope is negative, the graph of \( \log \lambda \) versus \( \log E \) will be a straight line that slopes downwards.

### Final Result:
The log-log graph between the energy \( E \) of an electron and its de Broglie wavelength \( \lambda \) will be a straight line with a negative slope of \( -\frac{1}{2} \).