First EM problem that I solved by myself

A non-conducting disc of radius \(a\) and uniform positive charge density \(\sigma\) is placed on ground, with its axis vertical. A particle of mass \(m\) and positive charge \(q\) is dropped along the axis of disc from a height \(H\) with zero initial velocity. The particle has charge to mass ratio \(\frac{q}{m}=\frac{4g\epsilon_0}{\sigma}\). The ratio \(\frac{H}{a}\) such that the particle barely reaches the disc can be expressed as \(\frac{n_1}{n_2}\) where \(n_1,n_2\) are coprime integers. Calculate the value of \(n_1+n_2.\)

Details and assumptions

\(\bullet\) \(\epsilon_0\) is permittivity of free space and \(g\) is acceleration due to gravity.

\(\bullet\) Particle does not radiate while accelerating.

\(\bullet\) This problem is taken from IIT 1999 question paper.

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