If E and `lambda` represent the energy and wavelenght respectively of an electron, then the graph between log `lambda` and log E will have

Options

A. Positive slope
B. Negative slope
C. Zero slope
D. Infinite slope

Explanation

To solve the problem of determining the nature of the graph between log λ (wavelength) and log E (energy) for an electron, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between wavelength and momentum**:
The de Broglie wavelength (λ) of an electron is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the electron.

2. **Express momentum in terms of kinetic energy**:
The momentum \( p \) can be expressed in terms of kinetic energy (KE) as:
\[ p = \sqrt{2m \cdot KE} \]
where \( m \) is the mass of the electron.

3. **Substitute the expression for momentum into the wavelength formula**:
Substituting \( p \) into the wavelength equation gives:
\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]

4. **Relate kinetic energy to energy (E)**:
Since kinetic energy (KE) can be considered as the total energy (E) for a free electron, we can rewrite the wavelength as:
\[ \lambda = \frac{h}{\sqrt{2mE}} \]

5. **Take the logarithm of both sides**:
Taking the logarithm of the wavelength, we have:
\[ \log \lambda = \log h - \frac{1}{2} \log(2mE) \]

6. **Expand the logarithm of the product**:
Using the property of logarithms, we can expand this:
\[ \log \lambda = \log h - \frac{1}{2} \log(2m) - \frac{1}{2} \log E \]

7. **Rearranging the equation**:
Rearranging gives:
\[ \log \lambda = \left(-\frac{1}{2}\right) \log E + \left(\log h - \frac{1}{2} \log(2m)\right) \]
This can be expressed in the form:
\[ y = mx + c \]
where \( y = \log \lambda \), \( x = \log E \), and the slope \( m = -\frac{1}{2} \).

8. **Determine the slope**:
Since the slope \( m \) is negative, the graph of log λ versus log E will have a negative slope.

### Conclusion:
Thus, the graph between log λ and log E will have a **negative slope**.