# 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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