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 lt lt R`, and the surface tension of water is T, value of r just before bubbles detach is : (density of water is `rho_(w)`)

Options

A. ` R^ 2 sqrt ((2 rho_w g )/(3 T)) `
B. ` R ^ 2 sqrt( ( 3 rho _w g )/(T)) `
C. ` R^ 2 sqrt ((rho _w g )/(6 T )) `
D. `R ^2 sqrt ((rho _w g )/(3 T )) `

Answer

A

Explanation

Force acting on bubble due to surface tension `=intTd1sintheta`
`F_("surface")=(Tsintheta)intd1=T((r)/(R))(2pir)`
This force will balance the force of buoyancy.
So, `T(2pir)((r)/(R))=((4)/(3)piR^(3))rho_(w)g" "Rightarrow" "r^(2)=((2)/(3)(rho_(w)g)/(T)R^(4)RightarrowR^(2)sqrt((2rho_(w)g)/(3T))`