Answer

Verified

359.7k+ views

**Hint**For bubble to form beyond ring at a radius $b$ greater than $R$, then excess pressure must exist within the bubble. Because there is no acceleration of the molecule, then the force of the air must be equal to the force due to excess pressure within the bubble. On equating the formulas we get the solution.

**Formula used**In the solution we will be using the following formulas,

$\Rightarrow P = \dfrac{{4T}}{R}$ where $P$ is the excess pressure within the bubble, $T$ is the surface tension, and $R$ is the radius of the bubble.

Force exerted by air molecules ${F_a} = \rho A{v^2}$ where $\rho $ is the density of air, $A$ is the area on which the force is exerted on, and $v$ is the speed of the air.

**Complete step by step answer**

To solve the above equation, we note the bubble must contain an excess pressure within it to expand outwards. The excess pressure in a bubble is given by

$\Rightarrow P = \dfrac{{4T}}{R}$ where, $T$ is the surface tension and $R$ is the radius of the bubble.

This pressure pushes outward from the inside of the bubble.

According to the question, air collides with the bubble. These air molecules must exert a force on the bubble.

The force exerted by the molecules is given by

$\Rightarrow {F_a} = \rho A{v^2}$where$\rho $ is the density of air, $A$ is the area the force exerted on, and $v$ is the speed of the air.

Now, since there’s no acceleration of the bubble, the force exerted due to the excess pressure inside the bubble is equal to the force exerted by the air molecules hence on equating we get,

$\Rightarrow {F_a} = PA \Rightarrow \rho A{v^2} = \dfrac{{4AT}}{R}$ (Force due to excess pressure is $PA$. )

Making $R$ subject of the formula and cancelling the like terms, we have

$\Rightarrow R = \dfrac{{4T}}{{\rho {v^2}}}$

**Hence, the correct option is (C).**

**Note**

Alternatively, we can eliminate option B and D using a unit check. The unit of surface tension is N/m. Hence on substituting the units of the variables in the answers given for option B and D we have

$\Rightarrow \dfrac{N}{m} \div \left( {\dfrac{{kg}}{{{m^3}}} \times m{s^{ - 1}}} \right)$

$\Rightarrow \dfrac{{kgm{s^{ - 2}}}}{m} \times \left( {\dfrac{{{m^2}}}{{kg{s^{ - 1}}}}} \right) = {m^2}{s^{ - 1}}$ This is not the unit of distance.

Also from knowledge that

$\Rightarrow P = \dfrac{{4T}}{R}$ since force exerted by air is a dimensionless constant, we can safely say that option C is the solution.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Describe the poetic devices used in the poem Aunt Jennifers class 12 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE

State the laws of reflection of light

State and prove Bernoullis theorem class 11 physics CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE