Electric Field Locus

A solid infinite cylinder of equation \(x^2+y^2=R^2\) has a uniform charge density \(\rho\). It also has a spherical cavity which is represented by the equation \(x^2+y^2+z^2=R^2\). The locus, in the \(xy\) plane, where the electric field is maximum is a circle of radius \(\frac{a}{b}R\), where \(a\) and \(b\) are coprime positive integers.

Find the value of \(a+b\).

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