Two identical positive charges Q are fixed on the x-axis at x = -a and x = +a. A particle of mass m and charge q is placed near the origin. For q > 0 it performs small oscillations along the x-axis with period T₁. If q < 0, it performs small oscillations along the y-axis with period T₂. The ratio T₁/T₂ is:
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Question

Two identical positive charges Q are fixed on the x-axis at x = -a and x = +a. A particle of mass m and charge q is placed near the origin. For q > 0 it performs small oscillations along the x-axis with period T₁. If q < 0, it performs small oscillations along the y-axis with period T₂. The ratio T₁/T₂ is:

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G1 Physics · Accepted Answer

For a small displacement x along the x-axis, E x E_x E x ​ ≈ -4kQx/a³. Thus, ω₁² = 4kQ|q|/(ma³). For a small displacement y along the y-axis, E y E_y E y ​ ≈ 2kQy/a³. For q < 0 this gives restoring force, so ω₂² = 2kQ|q|/(ma³). Since T = 2π/ω, T₁/T₂ = ω₂/ω₁ = 1/√2.