A thin ring of radius R carries total charge Q uniformly. A particle of mass m and charge q, with qQ

Question

A thin ring of radius R carries total charge Q uniformly. A particle of mass m and charge q, with qQ<0, is constrained to move along the ring axis. It passes through the centre with speed v. Under the small-displacement approximation |x|≪R, its maximum displacement is:

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G1 Physics · Accepted Answer

Near the centre, Eₓ ≈ Qx/(4πε₀R³), so for qQ<0 the force is F≈-|qQ|x/(4πε₀R³). Thus k e f f = ∣ q Q ∣ / ( 4 π ε 0 R 3 ) . k_eff=|qQ|/(4πε₀R³). k e ​ f f = ∣ q Q ∣/ ( 4 π ε 0 ​ R 3 ) . Equating ½mv²=½ k e f f k_eff k e ​ f f A² gives A=v√(4πε₀mR³/|qQ|).

A thin ring of radius R carries total charge Q uniformly. A particle of mass m and charge q, with qQ&lt;0, is constrained to move along the ring axis. It passes through the centre with speed v. Under the small-displacement approximation |x|≪R, its maximum displacement is:

A thin ring of radius R carries total charge Q uniformly. A particle of mass m and charge q, with qQ<0, is constrained to move along the ring axis. It passes through the centre with speed v. Under the small-displacement approximation |x|≪R, its maximum displacement is: