On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius \(R\) and making a circular contact of radius \(r\) with the bottom of the vessel. If \(r < < R\), and the surface tension of water is \(T\), value of \(r\) just before bubbles detach is (density of water is \(\rho_{w}\))

A \(R^{2} \sqrt{\frac{\rho_{w} g}{T}}\)
B \(R^{2} \sqrt{\frac{2 \rho_{w} g}{3 T}}\)
C \(R^{2} \sqrt{\frac{3 \rho_{w} g}{T}}\)
D \(R^{2} \sqrt{\frac{\rho_{w} g}{6 T}}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(R^{2} \sqrt{\frac{2 \rho_{w} g}{3 T}}\)

Step-by-step Solution

Detailed explanation

When the bubble gets detached, Buoyant force = force due to surface tension \(\int T \times d l \sin \theta=\frac{4}{3} \pi R^{3} \rho_{w} g\) \(\Rightarrow T \times 2 \pi r \times \frac{r}{R}=\frac{4}{3} \pi R^{3} \rho_{w} g\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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