Positive charge Q is distributed uniformly over a circular ring of radius R. A particle having a mass m and a negative charge q, is placed on its axis at a distance x from the centre. Find the force on the particle. Assuming x ltltR, find the time period of oscillation of the particle if it is released from there.

Explanation

`dq=Q/(2piR).Rd theta=Q/(2pi) .d theta `
` dE=(qdq)/(4piepsilon_0)(1/r^2 )`
` =(q.Q/(2pi)d theta )/(4piepsilon_0)1/(R^2+x^2 )`
` or dE cos theta =(qQd theta)/(8pi^2epsilon_0)x/((R^2+x^2)^(3/2) )`
Total force
` =intdFcos theta `
` =(qQ)/(8pi^2epsilon_0)x/(R^2+x^2)^(3/2) (:.int dtheta=2pi) `
` xltltR `
` F=(Qqx)/(4piepsilon_0R^3)=mw^2x `
` or m((2pi)/T)^2=(Qq)/(4piepsilon_0R^3) `
` T=[(16pie^3epsilon_0mR^3)/(Qq)]^(1/2)`.