Exams › JEE Main › Physics

A particle of mass m and charge -Q can move only along the axis of a ring of radius a. The ring carries a uniform positive linear charge density +lambda. The particle is released slightly displaced from the centre of the ring (its equilibrium point on the axis). Find the period of its small oscillations.

1. T = 4*pi*sqrt(epsilon0*m*a² / (lambda*Q))
2. T = 2*pi*sqrt(2*epsilon0*m*a² / (lambda*Q))
3. T = 2*pi*sqrt(4*epsilon0*m*a² / (lambda*Q))
4. T = 2*pi*sqrt(epsilon0*m*a² / (2*lambda*Q))

Correct answer: T = 2*pi*sqrt(2*epsilon0*m*a² / (lambda*Q))

Solution

The ring's axial field at distance x from the centre is E = q*x/(4*pi*epsilon0*(a²+x²)^(3/2)), which for x << a reduces to E = q*x/(4*pi*epsilon0*a³) with total charge q = lambda*2*pi*a. The force on the charge -Q is restoring: F = -[Q*q/(4*pi*epsilon0*a³)]*x, giving omega² = Q*q/(4*pi*epsilon0*m*a³). Substituting q = 2*pi*a*lambda yields omega² = Q*lambda/(2*epsilon0*m*a²), so T = 2*pi/omega = 2*pi*sqrt(2*epsilon0*m*a²/(lambda*Q)).

Related JEE Main Physics questions

• What is the SI unit of the permittivity of free space, ε₀?
• What is the SI unit used for electric flux?
• A constant electric field E is directed along the positive x-axis. If a charge of 0.5 C is displaced by 2 m along a line inclined at 60° to the x-axis, and the work done in this process is 10 J, what is the magnitude of the electric field?
• An electric dipole is located at the origin O along the x-axis. A point P lies 20 cm from O, and the line OP makes an angle π/3 with the x-axis. If the electric field at P is inclined at an angle θ to the x-axis, what is θ?
• A uniformly charged semicircular wire has radius a and linear charge density λ. The magnitude of the electric field at the centre of the semicircle is
• A point Q is located on the perpendicular bisector of an electric dipole whose dipole moment is p. If Q is at a distance r from the dipole and r is much greater than the dipole size, the electric field at Q varies as

⚔️ Practice JEE Main Physics free + battle 1v1 →