Twenty seven drops of same size are charged at 220 V each. They combine to form a bigger drop. Calculate the potential of the bigger drop.

The correct option is 1980 V.

Step-by-step Explanation:

To find the potential of the bigger drop formed by combining 27 smaller, identical charged drops, we can use the principles of conservation of volume and conservation of charge.

1. Conservation of Volume:
Let $r$ be the radius of each small drop and $R$ be the radius of the combined bigger drop.
Since the volume of the 27 small drops equals the volume of the single large drop, we write:
$\frac{4}{3}\pi R^3=27·\left(\frac{4}{3}\pi r^3\right)$
Taking the cube root on both sides:
$R={27}^{\frac{1}{3}}r=3r$

2. Conservation of Charge:
Let $q$ be the charge on each small drop. The total charge $Q$ on the bigger drop is the sum of the charges of all 27 drops:
$Q=27q$

3. Calculating the Potential:
The potential $v$ of a small drop is given by:
$v=\frac{1}{4\pi {\epsilon }_0}·\frac{q}{r}=220\text{V}$
The potential $V$ of the bigger drop is given by:
$V=\frac{1}{4\pi {\epsilon }_0}·\frac{Q}{R}$
Substitute the values of $Q$ and $R$ into this expression:
$V=\frac{1}{4\pi {\epsilon }_0}·\frac{27q}{3r}$
Simplifying the fraction:
$V=9·(\frac{1}{4\pi {\epsilon }_0}·\frac{q}{r})$
Since $\frac{1}{4\pi {\epsilon }_0}·\frac{q}{r}=v=220\text{V}$:
$V=9·220\text{V}=1980\text{V}$

Thus, the potential of the bigger drop is 1980 V.