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Young-Laplace law states that the excess pressure inside a soap bubble of radius $R$ is given by $\Delta P=4 \sigma / R$, where $\sigma$ is the coefficient of surface tension of the soap. The EOTVOS number $E_0$ is a dimensionless number that is used to describe the shape of bubbles rising through a surrounding fluid. It is a combination of $g$, the acceleration due to gravity $\rho$ the density of the surrounding fluid $\sigma$ and a characteristic length scale $L$ which could be the radius of the bubble. A possible expression for $E_0$ is
$\frac{\rho g}{\sigma L^3}$
$\frac{\rho L^2}{\sigma g}$
$\frac{\rho g L^2}{\sigma}$
$\frac{g L^2}{\sigma \rho}$
Solution
(c)
As EOTVOS number $E_{\text {s }}$ is dimensionless, we check dimensions of options given to the choose correct answer.
Now, $\left[\frac{\rho g I^2}{\sigma}\right]=\frac{[\rho] \cdot[g] \cdot[L]^2}{[\sigma]}$
$=\frac{\left[ ML ^{-3}\right] \cdot\left[ LT ^{-2}\right] \cdot\left[ I ^2\right]}{\left[ MT ^{-2}\right]}$
$=\left[ M ^{0} L ^{0} T ^0\right]=\text { Dimensionless }$
