Question

In air, a charged soap bubble of radius 'R' breaks into 27 small soap bubbles of equal radius 'r '. Then the ratio of mechanical force acting per unit area of big soap bubble to that of a small soap bubble is ______.

Options

• `1/81`

• `3/1`

• `1/3`

• `9/1`

Answer 1

In air, a charged soap bubble of radius 'R' breaks into 27 small soap bubbles of equal radius 'r '. Then the ratio of mechanical force acting per unit area of big soap bubble to that of a small soap bubble is `underline(1/3)`.

Explanation:

The force per unit area Is pressure Pressure inside a soap bubble of radius R is given by

P = `"4T"/"R"`

where, T = surface tension

and R = radius of the drop

Pressure inside a soap bubble, P = `(4"T")/"R"`.

If a bubble is break into 27 small soap bubbles then the volume of single bubble of radius Rand the combined volume of 27 bubbles of radius r would be constant.

27 × volume of small bubbles = volume of larger bubble

`=> 27(4/3 pi"r"^3) = 4/3 pi"R"^3`

`=> 27"r"^3 = "R"^3`

`=> "r" = "R"/3`    ....(i)

Now, the pressure inside smaller soap bubble,

`"P"_"small" = (4"T")/"r" = (12"T")/"R"`  (Using the relation)

and similarly `"P"_"large" = "4t"/"R"`

∴ Ratio of pressure of the smaller and larger soap bubble is given as,

`"P"_"large"/"P"_"Small" = "4T"/"R" xx "R"/(12"T") = 1/3`

`"P"_"large" : "P"_"Small"` = 1 : 36

Hence, the ratio of mechanical force acting per unit area of big soap bubble to that of a small bubble is 1 : 3.