The graph that correctly represents the relation of frequency `v` of a particular characteristic `X` - ray with the atomic number `Z` of the material is

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Answer

To solve the question regarding the relationship between the frequency \( \nu \) of a particular characteristic X-ray and the atomic number \( Z \) of the material, we can follow these steps:

### Step 1: Understand Moseley's Law
Moseley's law states that the frequency \( \nu \) of the characteristic X-rays emitted by an element is related to its atomic number \( Z \) by the equation:
\[ \sqrt{\nu} = a(Z - b) \]
where \( a \) and \( b \) are constants specific to the material.

### Step 2: Rearranging the Equation
From the above equation, we can square both sides to eliminate the square root:
\[ \nu = a^2(Z - b)^2 \]
This shows that the frequency \( \nu \) is proportional to the square of the difference between the atomic number \( Z \) and the constant \( b \).

### Step 3: Identify the Graph Shape
The equation \( \nu = a^2(Z - b)^2 \) is a quadratic equation in terms of \( Z \). This means that when we plot \( \nu \) (on the y-axis) against \( Z \) (on the x-axis), we will get a parabola that opens upwards.

### Step 4: Determine Key Points
1. When \( Z = b \), \( \nu = 0 \). This is the vertex of the parabola.
2. As \( Z \) increases beyond \( b \), \( \nu \) increases because the square of a positive number is positive.
3. When \( Z = 0 \), substituting into the equation gives:
\[ \nu = a^2(0 - b)^2 = a^2b^2 \]
This indicates that the frequency is positive at \( Z = 0 \).

### Step 5: Sketch the Graph
- The vertex of the parabola is at \( (b, 0) \).
- The parabola opens upwards, indicating that as \( Z \) increases from \( b \), \( \nu \) increases.
- The graph will intersect the y-axis at \( (0, a^2b^2) \).

### Conclusion
The graph that correctly represents the relationship between the frequency \( \nu \) of a particular characteristic X-ray and the atomic number \( Z \) is a parabola that opens upwards, with its vertex at \( (b, 0) \) and intersecting the y-axis at \( (0, a^2b^2) \).