If there are N nucleons in a nucleus of radius R, then the number of nucleons in a nucleus of radius 2R will be :

Options

A. N
B. 2N
C. 8N
D. `2^(1//3) N`

Answer

To solve the problem, we will use the relationship between the radius of a nucleus and the number of nucleons it contains. The key concept is that the volume of the nucleus is proportional to the number of nucleons.

### Step-by-Step Solution:

1. **Understanding the relationship between radius and nucleons**:
The radius \( R \) of a nucleus is related to the number of nucleons \( N \) in the nucleus by the formula:
\[ R \propto N^{1/3} \]
This means that if the radius of the nucleus changes, the number of nucleons will change according to the cube of the ratio of the radii.

2. **Setting up the problem**:
Let the radius of the first nucleus be \( R \) and the number of nucleons in it be \( N \). Therefore, we can write:
\[ R = r_0 N^{1/3} \]
where \( r_0 \) is a constant.

3. **Finding the new radius**:
Now, consider a nucleus with a radius \( 2R \). We want to find the number of nucleons \( N' \) in this nucleus. According to the same relationship:
\[ 2R = r_0 N'^{1/3} \]

4. **Relating the two equations**:
We can express the new radius in terms of the original radius:
\[ 2R = 2 (r_0 N^{1/3}) = r_0 N'^{1/3} \]

5. **Dividing the equations**:
Dividing the two equations gives:
\[ \frac{2R}{R} = \frac{N'^{1/3}}{N^{1/3}} \]
Simplifying this we get:
\[ 2 = \left(\frac{N'}{N}\right)^{1/3} \]

6. **Cubing both sides**:
To eliminate the cube root, we cube both sides:
\[ 2^3 = \frac{N'}{N} \]
This simplifies to:
\[ 8 = \frac{N'}{N} \]

7. **Finding \( N' \)**:
Rearranging gives us:
\[ N' = 8N \]

### Conclusion:
Thus, the number of nucleons in a nucleus of radius \( 2R \) will be \( 8N \).