# 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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