A small charged ball q of mass m is suspended on an insulating string of length ℓ . Another identical charged ball is slowly moved by a student towards the first one from a large distance. Eventually,

Question

A small charged ball q of mass m is suspended on an insulating string of length $ℓ$ . Another identical charged ball is slowly moved by a student towards the first one from a large distance. Eventually, the second ball is placed at the original location of the first one as shown in figure. At that moment, the first ball is elevated a small distance h above its original position and remains at rest. Then :-

TestHub · Accepted Answer

Let θ be the string's angle with the vertical. Ball 1 rises h = ℓ ( 1 − cos θ ) , and the separation is r 2 = ( ℓ sin θ ) 2 + h 2 = 2 ℓ 2 ( 1 − cos θ ) = 2 ℓ h . The joining line makes angle α = 9 0 ∘ − θ /2 with the vertical, so the repulsion F = k q 2 / r 2 acts on ball 1 at θ /2 above horizontal. Equilibrium: T sin θ = F cos ( θ /2 ) and T cos θ + F sin ( θ /2 ) = m g , giving F = 2 m g sin ( θ /2 ) = 2 m g cos α . 2 ℓ h k q 2 ​ = 2 m g cos α ⇒ cos α = 4 m g h ℓ k q 2 ​ . Hence (C).

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