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

Options

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

Answer

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

### Step 1: Understand the relationship between capacitance and radius
The capacitance \( C \) of a sphere is given by the formula:
\[ C = 4 \pi \epsilon_0 R \]
where \( R \) is the radius of the sphere and \( \epsilon_0 \) is the permittivity of free space.

### Step 2: Determine the radius of the small drops
Let the radius of each small drop be \( r \). The capacitance of each small drop is given as \( C = 4 \pi \epsilon_0 r \).

### Step 3: Calculate the total volume of the small drops
The volume \( V \) of one small drop is:
\[ V = \frac{4}{3} \pi r^3 \]
For 1000 small drops, the total volume \( V_{total} \) is:
\[ V_{total} = 1000 \times \frac{4}{3} \pi r^3 = \frac{4000}{3} \pi r^3 \]

### Step 4: Find the radius of the large drop
Let the radius of the large drop be \( R \). The volume of the large drop is:
\[ V_{large} = \frac{4}{3} \pi R^3 \]
Setting the total volume of the small drops equal to the volume of the large drop gives:
\[ \frac{4000}{3} \pi r^3 = \frac{4}{3} \pi R^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides, we have:
\[ 1000 r^3 = R^3 \]
Taking the cube root of both sides:
\[ R = 10 r \]

### Step 5: Calculate the capacitance of the large drop
Now, substituting \( R = 10r \) into the capacitance formula for the large drop:
\[ C_{large} = 4 \pi \epsilon_0 R = 4 \pi \epsilon_0 (10r) = 10 \times (4 \pi \epsilon_0 r) = 10C \]

### Conclusion
Thus, the capacitance of the larger spherical drop is:
\[ C_{large} = 10C \]