Two identical beads each have a mass m and charge q. When placed in a hemispherical bowl of radius R with frictionless, non-conducting walls, the beads move, and at equilibrium they area distance R apart (figure). Determine the charge on each bead.

Options

A. `q=R((mg)/(Ksqrt(3)))^(1//2)`
B. `q=(R(mg)/(Ksqrt(3)))^(1//2)`
C. `q=R((sqrt(3)mg)/(K))^(1//2)`
D. `q=(R(sqrt(3)mg)/(K))^(1//2)`

Answer

a

Explanation

Two bowl, exerts a normal force `N` on each bead, directed along he radius line or at `60^(@)` above the horizontal. Consider the free-body diagram of the bead on the left with electric force `F_(e)` applied.

`SigmaF_(y)= N sin 60^(@)-mg-0, implies N= mg//sin 60^(@)`
`SigmaF_(X)=-F_(e )+N cos 60^(@)=0`
`implies (Kq^(2))/(R^(2))= N cos 60^(@)= (mg)/(tan 60^(@))= (mg)/(sqrt(3))`
Thus `q=R((mg)/(K sqrt(3)))^(1//2)`