Which one of the following graphs in following figure represents variation of de Broglie wavelength `(lamda)` of a particle having linear

Options

A.
B.
C.
D.

Answer

To solve the question regarding the variation of de Broglie wavelength (λ) of a particle with linear momentum (p), we can follow these steps:

### Step 1: Understand the de Broglie Hypothesis
According to the de Broglie hypothesis, the relationship between the momentum (p) of a particle and its wavelength (λ) is given by the equation:
\[ p = \frac{h}{\lambda} \]
where \( h \) is the Planck constant.

### Step 2: Rearranging the Equation
We can rearrange the equation to express the wavelength in terms of momentum:
\[ \lambda = \frac{h}{p} \]
This shows that the wavelength is inversely proportional to the momentum.

### Step 3: Analyzing the Relationship
From the equation \( \lambda = \frac{h}{p} \), we can see that as the momentum \( p \) increases, the wavelength \( \lambda \) decreases. This indicates a hyperbolic relationship between \( p \) and \( \lambda \).

### Step 4: Graphical Representation
The graph of \( p \) versus \( \lambda \) will resemble the shape of the function \( y = \frac{1}{x} \), which is a hyperbola. Specifically, as \( p \) increases, \( \lambda \) approaches zero, and as \( p \) approaches zero, \( \lambda \) increases indefinitely.

### Step 5: Identifying the Correct Graph
In the provided options, we need to identify the graph that represents this hyperbolic relationship. The correct graph will show a curve that decreases as it moves to the right, indicating that as momentum increases, the wavelength decreases.

### Conclusion
Based on the analysis, the correct option that represents the variation of de Broglie wavelength (λ) with linear momentum (p) is option 4.

---