Question

a circular ring of radius r with uniform positive charge density

Answer 1

A circular ring of radius R with uniform charge density λ per unit length is located in the YZ plane with its centre at origin. A particle of mass m and charge q is projected from the point P(R√3,0) on the positive axis towards the center of the ring with initial velocity u. Find the smallest value of u such that the particle does not return to P​.​  

Answer 2

Dear student,

Let the particle be at a distance x from the origin at a point P at any instant.The electric field at point P due to charge on the circular ring is

E=14πε02πRλx(R2+x2)32/ from P to x

Force acting on q at P is F=qE

Amount of work done in moving a particle towards O for small distance dx is

dW =Fdx

dW=14πε02πRλxqdx(R2+x2)32/The total work done is W=2πRλq4πε0∫03R√x.dx(R2+x2)32/ →eqn 1Now ∫03R√x.dx(R2+x2)32/,(R2+x2)=t2xdx=dtxdx=dt2∫dt2t32/=−1t√=[−1R2+x2√]03R√=−12RSubstituting in 1 we get,W=λq4ε0

The work done needs to be supplied by kinetic energy given to charge initially

12mv2=λq4ε0v=λq2mε0−−−−√

This is the minimum velocity the charge q must be projected to reach the orgin.

Regards.