Two concentric circular arcs are fixed in space. The inner arc is a semicircle of radius R with a uniform linear charge density + . The outer arc is a quarter-circle of radius 2R with a uniform linear charge density - . A particle of mass m and positive charge +q is released from rest at the common centre of the arcs. The kinetic energy of the particle when it reaches a point very far away (infinity) is:

Step-by-step solution

The electric potential at the centre due to a circular arc of radius r subtending an angle and having linear charge density is V = 1 4 ₀ (r ) r = 4 ₀ . For the inner semicircle ( = ), the potential at the centre is: V₁ = (+ )( ) 4 ₀ = 4 ₀ For the outer quarter-circle ( = /2 ), the potential at the centre is: V₂ = (- )( /2) 4 ₀ = - 8 ₀ The net electric potential at the centre is: V_ net = V₁ + V₂ = 4 ₀ - 8 ₀ = 8 ₀ By conservation of mechanical energy, the kinetic energy of the particle at infinity equals its initial