In air , a charged soap bubble of radius 'r' is in equilibrium having outside and inside pressures being equal . The charge on the drop is (`epsilon_(0)` = permittivity of free space , T = surface tension of soap solution )

Options

A. `4pir^(2)sqrt((2Tepsilon_(0))/r)`
B. `4pir^(2)sqrt((4Tepsilon_(0))/r)`
C. `4pir^(2)sqrt((6Tepsilon_(0))/r)`
D. `4pir^(2)sqrt((8Tepsilon_(0))/r)`

Answer

D

Explanation

This problem is based upon the surface tension and electrostatic force on a changed conductor . For a charged soap bubble . There is an outward electrostatic pressure and hence the bubble expands .
` P_("ele")= 1/2 (sigma^(2))/(Kepsilon_(0) )` where K = 1
and ` sigma ` surface charge densit ` = Q/(4pir^(2))`
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For the charged soap bubble .
Atmospheric pressure `(P_(a))` and pressure due to S.T , `((4T)/r)` act radially inwards while `P_("ele")` ats outwards
Excess pressure inside the charged bubble
` = P_("int") -P_("at") = (4T)/r - (sigma^(2))/(2epsilon_(0))`
If `P_("int") - P_("at") =0` , then excess pressure = 0
and then ` (4T)/r = (sigma^(2))/(2epsilon_(0) )= 1/2 (Q^(2))/((4pir^(2))^(2)epsilon_(0))`
` :. Q^(2) = (8Tepsilon_(0))/r (4pir^(2))^(2)`
` :. Q = 4pir^(2)sqrt((8Tepsilon_(0))/r)`