Question

The excess pressure inside a soap bubble is thrice the excess pressure inside a second soap bubble. The ratio between the volume of the first and the second bubble is :

• A. 1 : 9
• B. 1 : 3
• C. 1 : 81
• D. 1 : 27

Answer 1

The Correct Option is D

The excess pressure \( P_{\text{excess}} \) inside a soap bubble is given by the formula:

\[ P_{\text{excess}} = \frac{4T}{r}, \] where \( T \) is the surface tension and \( r \) is the radius of the bubble.

Let the radii of the two bubbles be \( r_1 \) and \( r_2 \), and their excess pressures be \( P_1 \) and \( P_2 \), respectively.

Given:

\[ P_1 = 3P_2. \]

Using the formula for excess pressure:

\[ \frac{4T}{r_1} = 3 \times \frac{4T}{r_2}. \]

Cancelling common terms:

\[ \frac{1}{r_1} = 3 \times \frac{1}{r_2} \Rightarrow r_1 = \frac{r_2}{3}. \]

Since the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), the ratio of the volumes is:

\[ \frac{V_1}{V_2} = \left( \frac{r_1}{r_2} \right)^3 = \left( \frac{1}{3} \right)^3 = \frac{1}{27}. \]

Thus, the ratio of the volumes is \( 1 : 27 \).