Two identical positive charges Q each are fixed a distance 2a apart. Another point charge q₀ of mass m is placed at the midpoint between the two fixed charges. For a small displacement along the line joining the fixed charges, q₀ executes SHM. The time period of oscillation of q₀ is

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(1) √(4π³ε₀ m a³)/(q₀Q). Displace q₀ by x (x a) towards one charge. The net restoring force is F=kQq₀[1/((a-x)²)-1/((a+x)²)]. Expanding to first order, 1/((a∓ x)²)≈1/(a²)(1±(2x)/a), so F≈(4kQq₀)/(a³)x, directed back towards the midpoint. ω²=(4kQq₀)/(ma³)=(Qq₀)/(πε₀ ma³) using k=1/(4πε₀). T=(2π)/ω=2π√(πε₀ ma³)/(Qq₀)=√(4π³ε₀ ma³)/(q₀Q).

Two identical positive charges Q each are fixed a distance 2a apart. Another point charge q₀ of mass m is placed at the midpoint between the two fixed charges. For a small displacement along the line joining the fixed charges, q₀ executes SHM. The time period of oscillation of q₀ is

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