Determine the ratio of the surface energy of | vedclass

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(B) Let the radius of each small drop be $r$ and the radius of the large drop be $R$.Since the volume is conserved, the volume of $1000$ small drops e

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$1 : 10$

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(B) Let the radius of each small drop be $r$ and the radius of the large drop be $R$.Since the volume is conserved, the volume of $1000$ small drops equals the volume of $1$ large drop:$1000 \cdot (\frac{4}{3}\pi r^3) = \frac{4}{3}\pi R^3$$1000 r^3 = R^3 \implies R = 10r$.The surface energy of a drop is given by $U = T \cdot A$, where $T$ is surface tension and $A$ is the surface area.Surface energy of $1$ large drop: $U_{large} = T \cdot 4\pi R^2 = T \cdot 4\pi (10r)^2 = 100 T \cdot 4\pi r^2$.Total surface energy of $1000$ small drops: $U_{small_total} = 1000 \cdot (T \cdot 4\pi r^2) = 1000 T \cdot 4\pi r^2$.The ratio of the surface energy of $1$ large drop to $1000$ small drops is:$\frac{U_{large}}{U_{small_total}} = \frac{100 T \cdot 4\pi r^2}{1000 T \cdot 4\pi r^2} = \frac{100}{1000} = \frac{1}{10}$.Thus, the ratio is $1 : 10$.

Determine the ratio of the surface energy of $1$ large drop to $1000$ small drops, if $1000$ small drops combine to form $1$ large drop.

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