Highly energetic electrons are bombarded on a target of an element containing `30` neutrons. The ratio of radii of nucleus to that of Helium nucleus is `141//3`. The atomic number of nucleus will be.

Options

A. 25
B. 26
C. 56
D. 30

Explanation

To solve the problem step by step, we will use the relationship between the radius of a nucleus and its mass number, as well as the information given in the question.

### Step 1: Understand the relationship between the radius and mass number
The radius \( R \) of a nucleus is related to its mass number \( A \) by the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant.

### Step 2: Set up the ratio of the radii
Given that the ratio of the radius of the unknown nucleus to that of the helium nucleus is \( \frac{R_n}{R_{He}} = 14^{1/3} \), we can express this as:
\[ \frac{R_n}{R_{He}} = \frac{R_0 A^{1/3}}{R_0 (4)^{1/3}} = \frac{A^{1/3}}{4^{1/3}} \]
where \( A \) is the mass number of the unknown nucleus and \( 4 \) is the mass number of helium.

### Step 3: Equate the two expressions
From the ratio given in the problem, we have:
\[ \frac{A^{1/3}}{4^{1/3}} = 14^{1/3} \]
This implies:
\[ A^{1/3} = 14^{1/3} \times 4^{1/3} \]

### Step 4: Simplify the equation
Cubing both sides to eliminate the cube root gives:
\[ A = 14 \times 4 = 56 \]

### Step 5: Determine the atomic number
The atomic number \( Z \) can be found using the relationship:
\[ Z = A - N \]
where \( N \) is the number of neutrons. Given that there are \( 30 \) neutrons:
\[ Z = 56 - 30 = 26 \]

### Conclusion
The atomic number of the nucleus is \( 26 \).