A circular ring of radius a with uniform charge density λ is in the xy plane with centre at origin. A particle of mass m and charge q is projected from P (0, 0, a√3) on +z-axis towards origin with initial velocity u. The minimum value of the velocity so that the particle does not return to P is λ q x ϵ 0 m \sqrt{\frac{\lambda q}{x\epsilon_0m}} x ϵ 0 m λ q .Find 'x' (neglect gravity).
Physics · Electrostatic · potential energy and electric potential
JEE Mains

Question

A circular ring of radius a with uniform charge density λ is in the xy plane with centre at origin. A particle of mass m and charge q is projected from P (0, 0, a√3) on +z-axis towards origin with initial velocity u. The minimum value of the velocity so that the particle does not return to P is λ q x ϵ 0 m \sqrt{\frac{\lambda q}{x\epsilon_0m}} x ϵ 0 m λ q .Find 'x' (neglect gravity).

Physics · Electrostatic · potential energy and electric potential

G1 Physics · Accepted Answer

Step 1: Convert the density to charge using dq = ρ dV, dq = λ dl, or dq = σ dA as appropriate. Step 2: Integrate over the stated region and then use Coulomb's law/Gauss's law for the field or flux. Step 3: Apply the given radius/position limits and simplify. Final: 2. In this specific setup, apply the calculation to: A circular ring of radius a with uniform charge density λ is in the xy plane with centre at origin. A par.