Two soap bubble have volumes in the ratio `8:1`. The ratio of excess pressures inside them will be

Options

A. `1:2`
B. `1:4`
C. `2:1`
D. `4:1`

Answer

To solve the problem of finding the ratio of excess pressures inside two soap bubbles with volumes in the ratio of 8:1, we can follow these steps:

### Step 1: Understand the relationship between volume and radius
The volume \( V \) of a soap bubble is given by the formula:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the bubble.

### Step 2: Set up the ratio of volumes
Given that the volumes of the two soap bubbles are in the ratio \( 8:1 \), we can denote the volumes as:
\[ V_1 = 8V \quad \text{and} \quad V_2 = V \]

### Step 3: Relate the volumes to the radii
Using the volume formula, we can express the radii in terms of the volumes:
\[ V_1 = \frac{4}{3} \pi R_1^3 \quad \Rightarrow \quad R_1^3 = \frac{3V_1}{4\pi} \]
\[ V_2 = \frac{4}{3} \pi R_2^3 \quad \Rightarrow \quad R_2^3 = \frac{3V_2}{4\pi} \]

### Step 4: Substitute the volume ratio into the radius equation
Since \( V_1 = 8V_2 \), we can write:
\[ R_1^3 = \frac{3(8V)}{4\pi} = 6V \quad \text{and} \quad R_2^3 = \frac{3V}{4\pi} = \frac{3V}{4\pi} \]

### Step 5: Find the ratio of the radii
Taking the cube root of both sides, we find:
\[ \frac{R_1^3}{R_2^3} = 8 \quad \Rightarrow \quad \frac{R_1}{R_2} = 2 \]

### Step 6: Use the formula for excess pressure
The excess pressure \( P_x \) inside a soap bubble is given by:
\[ P_x = \frac{4T}{R} \]
where \( T \) is the surface tension.

### Step 7: Set up the ratio of excess pressures
Using the formula for excess pressure, we can express the ratio of excess pressures for the two bubbles:
\[ \frac{P_{x1}}{P_{x2}} = \frac{4T/R_1}{4T/R_2} = \frac{R_2}{R_1} \]

### Step 8: Substitute the radius ratio into the pressure ratio
From our earlier calculation, we have:
\[ \frac{R_2}{R_1} = \frac{1}{2} \]
Thus, we can conclude:
\[ \frac{P_{x1}}{P_{x2}} = \frac{1}{2} \]

### Final Answer
The ratio of excess pressures inside the two soap bubbles is:
\[ \frac{P_{x1}}{P_{x2}} = 1:2 \]