Question

A spherical drop of liquid splits into \(1000\) identical spherical drops. If \(u_i\) is the surface energy of the original drop and \(u_f\) is the total surface energy of the resulting drops, the (ignoring evaporation). \(u_f/u_i = (10/x)\). Then value of \(x\) is

• A. \(1\)
• B. \(3\)
• C. \(7\)
• D. \(9\)

Hint:

Radius of each small drop is \(R/10\); total area grows by a factor 10.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Splitting conserves volume. If the original radius is \(R\) and each small drop has radius \(r\), then \(1000r^3 = R^3\), so \(r = \frac{R}{10}\).

Step 2: Surface energies:
\(u_i = 4\pi R^2T\).
\(u_f = 1000\times4\pi r^2T = 1000\times4\pi\frac{R^2}{100}T = 10\times4\pi R^2T\).

Step 3: Ratio:
\(\frac{u_f}{u_i} = 10 = \frac{10}{x}\), so \(x = 1\).

Final Answer:
The value of \(x\) is \(1\), option (A). \[ \boxed{1} \]