Two spherical soap bubbles of radii `r_(1)` and `r_(2)` in vacuum combine under isothermal conditions. The resulting bubbles has a radius equal to

Options

A. `(r_(1)+r_(2))/2`
B. `(r_(1)r_(2))/(r_(1) + r_(2))`
C. `sqrt(r_1r_2)`
D. `sqrt(r_(1)^(2) + r_(2)^(2))`

Answer

C

Explanation

`P_(1)` and `P_(2)` are pressures inside the small bubble with radius `r_(1)` and `r_(2)` respectively
`P_(1)=(4T)/r_(1) , (4T)/r_(2)`
Pressure inside large bubble, `P=(4T)/R`
`PV=P_(1)V_(1) + P_(2)V_(2)` [`because` Isothermal]
`(4T)/R (4/3 pi R^(3)) = (4T)/r_(1) (4/3 pi r_(1)^(3)) + (4T)/r_(2) (4/3 pi r_(2)^(3))`
`therefore R=sqrt(r_(1)^(2) + r_(2)^(2)`