1000 small water drops each of capacitance C join togethter to form one large spherical drop. The capacitance of bigge sphere is

Options

A. C
B. 10 C
C. 100 C
D. 1000 c

Answer

To solve the problem of finding the capacitance of a larger spherical drop formed by combining 1000 small water drops, we can follow these steps:

### Step 1: Understand the Capacitance of a Sphere
The capacitance \( C \) of a small spherical drop of radius \( r \) is given by the formula:
\[ C = 4 \pi \epsilon_0 r \]
where \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Volume Conservation
When 1000 small drops combine to form one larger drop, the volume of the larger drop must equal the total volume of the small drops. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the total volume of 1000 small drops is:
\[ V_{\text{small}} = 1000 \times \frac{4}{3} \pi r^3 \]
The volume of the larger drop of radius \( R \) is:
\[ V_{\text{large}} = \frac{4}{3} \pi R^3 \]
Setting these volumes equal gives us:
\[ 1000 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi R^3 \]

### Step 3: Simplifying the Volume Equation
We can cancel \( \frac{4}{3} \pi \) from both sides:
\[ 1000 r^3 = R^3 \]
Taking the cube root of both sides, we find:
\[ R = 10 r \]

### Step 4: Calculate the Capacitance of the Larger Sphere
Now, we can find the capacitance \( C' \) of the larger sphere using its radius \( R \):
\[ C' = 4 \pi \epsilon_0 R \]
Substituting \( R = 10r \):
\[ C' = 4 \pi \epsilon_0 (10r) = 10 \times (4 \pi \epsilon_0 r) \]
Since \( C = 4 \pi \epsilon_0 r \), we can substitute:
\[ C' = 10C \]

### Conclusion
The capacitance of the larger spherical drop is:
\[ C' = 10C \]

### Final Answer
Thus, the capacitance of the larger sphere is \( 10C \).

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