1000 drops of same size are charged to a potential of 1 V each. If they coalesce to form in single drop, its potential would be

Options

A. `1V`
B. `10V`
C. `100V`
D. `1000V`

Explanation

To solve the problem, we need to find the potential of a single drop formed by the coalescence of 1000 smaller drops, each charged to a potential of 1 V. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the initial conditions
We have 1000 small drops, each with a potential \( V = 1 \, \text{V} \).

### Step 2: Calculate the charge on each small drop
Let the charge on each small drop be \( Q \). The potential \( V \) of a drop can be expressed as:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant and \( r \) is the radius of the small drop. Given that \( V = 1 \, \text{V} \), we can write:
\[ 1 = \frac{kQ}{r} \]

### Step 3: Coalescence of drops
When the 1000 drops coalesce into a single larger drop, the volume of the drops remains constant. The volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the total volume of 1000 small drops is:
\[ 1000 \times \frac{4}{3} \pi r^3 \]
The volume of the larger drop (with radius \( R \)) is:
\[ \frac{4}{3} \pi R^3 \]
Setting these equal gives:
\[ 1000 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi R^3 \]
Cancelling \( \frac{4}{3} \pi \) from both sides, we have:
\[ 1000 r^3 = R^3 \]

### Step 4: Solve for the radius of the larger drop
Taking the cube root of both sides, we find:
\[ R = 10 r \]

### Step 5: Calculate the total charge on the larger drop
The total charge \( Q_{\text{total}} \) on the larger drop is the sum of the charges from the 1000 smaller drops:
\[ Q_{\text{total}} = 1000 Q \]

### Step 6: Determine the potential of the larger drop
The potential \( V' \) of the larger drop can be expressed as:
\[ V' = \frac{k Q_{\text{total}}}{R} \]
Substituting \( Q_{\text{total}} = 1000 Q \) and \( R = 10 r \):
\[ V' = \frac{k (1000 Q)}{10 r} \]
This can be simplified to:
\[ V' = 100 \cdot \frac{k Q}{r} \]

### Step 7: Substitute the known potential
From Step 2, we know that \( \frac{k Q}{r} = 1 \, \text{V} \). Therefore:
\[ V' = 100 \cdot 1 \, \text{V} = 100 \, \text{V} \]

### Final Answer
The potential of the larger drop formed by the coalescence of 1000 smaller drops is:
\[ \boxed{100 \, \text{V}} \]