On heating water, bubbles being formed at the bottom of the...

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 $\rho_{w}$ ) [2014](A) $R^{2} \sqrt{\frac{\rho_{w} g}{3 T}}$ (B) $R^{2} \sqrt{\frac{\rho_{w} g}{6 T}}$ (C) $R^{2} \sqrt{\frac{\rho_{w} g}{T}}$ (D) $R^{2} \sqrt{\frac{3 \rho_{w} g}{T}}$

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 $\rho_{w}$ ) [2014](A) $R^{2} \sqrt{\frac{\rho_{w} g}{3 T}}$
(B) $R^{2} \sqrt{\frac{\rho_{w} g}{6 T}}$
(C) $R^{2} \sqrt{\frac{\rho_{w} g}{T}}$
(D) $R^{2} \sqrt{\frac{3 \rho_{w} g}{T}}$

Key Concepts

Surface Tension

Surface tension is the property of a liquid’s surface that allows it to contract and resist an external force. In problems like this, it creates a force along the perimeter of the contact circle between the bubble and the vessel, which acts to ‘hold’ the bubble against the surface. Although the force acts tangentially, its vertical component plays a decisive role in resisting the detachment of the bubble from the surface.

Buoyant Force

Buoyancy is the upward force experienced by a body immersed in a fluid, equal to the weight of the fluid it displaces. In the context of bubbles forming at the bottom of a heated vessel, it is this upward force – proportional to the bubble’s volume and the density of the water – that eventually overcomes the adhesion provided by surface tension, triggering the bubble’s detachment.

Laplace Pressure

Laplace pressure is the pressure difference across a curved interface caused by surface tension. For a spherical bubble the pressure inside exceeds that outside by an amount proportional to the surface tension divided by the radius. This pressure difference is closely connected to the forces exerted over the area of contact and helps determine the stability of the bubble at the moment of detachment.

Force Balance (Equilibrium)

In many capillarity problems the point of detachment is identified by a balance between two opposing forces – the upward buoyant force and the downward force due to surface tension (or adhesion). The equilibrium condition sets the scale at which the buoyancy becomes large enough to overcome the holding force provided by the surface tension along the small contact line.

Small Parameter Approximation

The approximation r << R indicates that the contact radius is very small compared to the bubble’s radius. This simplification allows one to linearize geometric relations and capture the leading–order dependence (often revealing scaling laws in terms of R, ?, g, and T) in the force–balance analysis without the complications introduced by higher–order corrections.

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