Q.

Two identical small beads each have mass m and charge q. When placed in a hemispherical bowl of radius R with frictionless, non-conductive walls, the beads move and at equilibrium the line joining the beads is horizontal and the distance between them is R (see figure). Neglect any induced charge on the hemispherical bowl. Then the charge on each bead is  (here K=14πε0

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a

q=R3mgK1/2

b

q=R3mgK1/2

c

q=RmgK31/2

d

q=RmgK31/2

answer is A.

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Detailed Solution

The bowl exerts a normal force N on each bead, directed along the radius line or at 60° above the horizontal. Consider the free-body diagram of the bead on the left with the electric force Fe applied.

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Fy=Nsin60mg=0,N=mg/sin60 Fx=Fe+Ncos60=0 Kq2R2=Ncos60=mgtan60=mg3  Thus, q=RmgK31/2

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