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 q1q_{1} and q2q_{2} separated by a distance rr is given by F(r)=kq1q2r2+δF(r)withδF(r)=kq1q2r2αF(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 kk' and α\alpha are constants. Consider now an experiment in which we place a point charge qq with mass mm at the center O of a heavy nonconducting spherical shell of radius RR with uniformly distributed charge QQ. It turns out that if both charges qq and QQ are positive and α<0\alpha<0 the charge qq will oscillate about the point O. If the deviation δF(r)\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 TT must be proportional to T0=2πmRδF(R), T_{0}=2 \pi \sqrt{\frac{m R}{|\delta{F(R)}|}}, i.e., T=cT0T=c T_{0}. If α=1\alpha=-1, determine the constant of proportionality cc. You may assume that xR1\frac{x}{R}\ll 1.

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