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Two identical positive charges Q each are fixed at a distance of 2 a apart from each other. Another point charge q 0 with mass m is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge q 0 executes SHM. The time period of oscillation of charge q 0 will be

Options

  1. A4 π 3 ε 0 m a 3 q 0 Q
  2. Bq 0 Q 4 π 3 ε 0 m a 3
  3. C2 π 2 ε 0 m a 3 q 0 Q
  4. D8 π 3 ε 0 m a 3 q 0 Q

Correct answer

A. 4 π 3 ε 0 m a 3 q 0 Q

Step-by-step solution

Let the charged particle be displaced slightly towards right by distance x x ≪ d . Q   a t   A ,   q : F 1 = 1 4 π ε 0 Q q d + x 2 , towards right Q   at   B ,   q :   F 2 = 1 4 π ε 0 Q q d - x 2 , towards left   F → n e t = F → 1 + F → 2 , will bring it towards O . F → n e t = 1 4 π ϵ 0 1 d + x 2 - 1 d - x 2 i ^ = - Q q 4 π ϵ 0 4 x d d 2 - x 2 2 i ^ Since x ≪ d ,   x 2 is negligible. F →

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