Testing Coulomb's law

Suppose that a new theory predicts a small deviation of Coulomb's law. That is, imagine that the force of interaction between two charges \(q_{1}\) and \(q_{2}\) separated by a distance \(r\) is given by \[F(r)=k \frac{q_{1}q_{2}}{r^{2}}+ \delta{F}(r) \quad \text{with} \quad \delta F(r)= k' \frac{q_{1}q_{2}}{r^{2-\alpha}}\] where \(k'\) and \(\alpha\) are constants. Consider now an experiment in which we place a point charge \(q\) with mass \(m\) at the center O of a heavy nonconducting spherical shell of radius \(R\) with uniformly distributed charge \(Q\). It turns out that if both charges \(q\) and \(Q\) are positive and \(\alpha<0\) the charge \(q\) will oscillate about the point O. If the deviation \(\delta F(r)\) were zero, then the force of interaction between the shell and the charge would also be zero. Therefore, from dimensional analysis, we deduce that the period of small oscillations \(T\) must be proportional to \[ T_{0}=2 \pi \sqrt{\frac{m R}{|\delta{F(R)}|}},\] i.e., \(T=c T_{0}\). If \(\alpha=-1\), determine the constant of proportionality \(c\). You may assume that \(\frac{x}{R}\ll 1\).

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