The Question is from neet-physics , and it was created by the author NAGA SAI

Question

Two soap bubbles of radii $r_1$ and $r_2$ coalesce to form a bigger bubble of radius $R$. If the atmospheric pressure is $P_0$, the surface tension of the soap solution is:

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Excess pressure inside a soap bubble and conservation of volume.
	**Related Topics**: Surface Tension, Fluid Mechanics
	**Prerequisites**: Pressure difference across curved surfaces, volume of a sphere.
## SOLUTION PATHWAY
**Given Information**:
	* Radii of smaller bubbles: $r_1$ and $r_2$
	* Radius of the larger bubble: $R$
	* Atmospheric pressure: $P_0$
**Method & Steps**:
	1. **Excess pressure inside a soap bubble:**
		Reasoning: The excess pressure inside a soap bubble is given by $P_{ex} = \frac{4T}{r}$, where $T$ is the surface tension and $r$ is the radius.
		Formula/Rule: $P_{ex} = \frac{4T}{r}$
	2. **Volume conservation:**
		Working: Assuming the air inside the bubbles is conserved during coalescence, the sum of the volumes of the two smaller bubbles equals the volume of the larger bubble.
		Key Point: Volume of a sphere = $\frac{4}{3}πr^3$
	3. **Relating pressures and volumes:**
		Result:  $P_1 = P_0 + \frac{4T}{r_1}$, $P_2 = P_0 + \frac{4T}{r_2}$, and $P_R = P_0 + \frac{4T}{R}$. Also, $\frac{4}{3}πr_1^3 + \frac{4}{3}πr_2^3 = \frac{4}{3}πR^3$, which simplifies to $r_1^3 + r_2^3 = R^3$.
		Verification:  Since the air inside combines, we can write $(P_1 - P_0)V_1 + (P_2 - P_0)V_2 = (P_R - P_0)V_R$. Substituting the expressions for excess pressure and simplifying using the volume conservation equation gives $T = \frac{P_0(r_1^3 + r_2^3 - R^3)}{4(R^2 - r_1^2 - r_2^2)}$.
 ## OPTION ANALYSIS
[A] ✓: This option correctly represents the derived formula for surface tension: $T = \frac{P_0(r_1^3 + r_2^3 - R^3)}{4(R^2 - r_1^2 - r_2^2)}$
[B] :x: Incorrectly relates the powers of radii in the numerator and denominator.
[C] :x: Incorrectly relates the powers of radii and has the wrong signs in the numerator and denominator.
[D] :x: This formula doesn't consider the pressure difference and uses incorrect powers of radii.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrectly applying the volume conservation principle: Ensure that the sum of the volumes of the initial bubbles equals the final bubble's volume.
		2. Confusing the formula for excess pressure: Remember that it's $\frac{4T}{r}$ for a soap bubble (two surfaces) and $\frac{2T}{r}$ for a liquid drop (one surface).
	**Quick Tips**:
		1. Remember the analogy of adding two inflated balloons into a larger balloon to visualize volume conservation.
		2. The excess pressure is inversely proportional to the radius. Smaller bubbles have higher internal pressure.

NAGA SAI · Answer

The correct Option is "T = \frac{P_0(R^3 - r_1^3 - r_2^3)}{4(r_1^2 + r_2^2 - R^2)}"

NAGA SAI · Answer

T = \frac{P_0(r_1^3 + r_2^3 - R^3)}{4(r_1^2 + r_2^2 - R^2)}

NAGA SAI · Answer

T = \frac{P_0(R^3 - r_1^3 - r_2^3)}{4(R^2 - r_1^2 - r_2^2)}

NAGA SAI · Answer

T = \frac{P_0(r_1^3 + r_2^3 + R^3)}{4(r_1^2 + r_2^2 + R^2)}

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