N identical drops of mercury are charged simultaneously to 10 V. When combined to form one large drop, the potential is found to be 40 V, the value of N is:

Options

A. 4
B. 6
C. 8
D. 10

Explanation

To solve the problem, we need to find the value of \( N \) (the number of identical drops of mercury) given that they are charged to 10 V and when combined into one large drop, the potential is found to be 40 V.

### Step-by-Step Solution:

1. **Understanding the Potential of a Single Drop:**
The potential \( V \) of a charged drop is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge on the drop, and \( r \) is the radius of the drop. For \( N \) identical drops charged to 10 V, we have:
\[ \frac{kQ}{r} = 10 \quad \text{(Equation 1)} \]

2. **Combining the Drops into One Large Drop:**
When \( N \) drops are combined into one large drop, the total charge becomes \( NQ \) and the radius of the large drop is \( R \). The potential of the large drop is given by:
\[ \frac{k(NQ)}{R} = 40 \quad \text{(Equation 2)} \]

3. **Volume Conservation:**
The volume of the drops remains constant before and after combining. The volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Therefore, the volume of \( N \) small drops is:
\[ N \left(\frac{4}{3} \pi r^3\right) = \frac{4}{3} \pi R^3 \]
From this, we can derive the relationship between \( R \) and \( r \):
\[ N r^3 = R^3 \implies \frac{R}{r} = N^{1/3} \quad \text{(Equation 3)} \]

4. **Substituting Equation 3 into Equations 1 and 2:**
We can now substitute \( R \) from Equation 3 into Equation 2:
\[ \frac{k(NQ)}{N^{1/3} r} = 40 \]
Simplifying gives:
\[ \frac{kQ}{r} \cdot N^{2/3} = 40 \quad \text{(Equation 4)} \]

5. **Dividing Equation 4 by Equation 1:**
Now, we divide Equation 4 by Equation 1:
\[ \frac{N^{2/3} \cdot \frac{kQ}{r}}{\frac{kQ}{r}} = \frac{40}{10} \]
This simplifies to:
\[ N^{2/3} = 4 \]

6. **Solving for \( N \):**
To find \( N \), we raise both sides to the power of \( \frac{3}{2} \):
\[ N = 4^{3/2} = 8 \]

### Final Answer:
Thus, the value of \( N \) is \( 8 \).