Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
The author told us to assume that the proton charge's magnitude differs from the electron charge's by a mere 1 part in $10^9$. This means that
$$Q=|q_{\rm proton}|-|q_{\rm electron}|=\dfrac{1}{10^9}=10^{-9}\;\rm C\tag 1 $$
This difference should produce a net charge of the atoms since the number of protons and electrons, in neutral atoms, is equal.
And we need to work with two copper spheres, so we need to find the number of atoms in each sphere.
We know, from density law, that $\rho =m/V$
Hence, the mass of one sphere is
$$m=\rho V=\rho \left( \frac{4}{3}\pi r^3\right)$$
$$m= \frac{4}{3}\pi r^3 \rho$$
And we know that the number of atoms is given by
$$N=\dfrac{m}{M}$$
where $M$ is the atomic mass,
$$N=\dfrac{ \frac{4}{3}\pi r^3\rho}{M}$$
Plug the known;
$$N=\dfrac{ \frac{4}{3} \pi (1\times 10^{-3})^3(8920)}{(63.5\times 1.66\times 10^{-27})} $$
$$N_{Cu}=\bf 3.5446\times 10^{20}\;\rm atom$$
Now we know that the number of electrons or protons in one copper atom is 29.
Hence,
$$N_e=29 N_{Cu}=\bf 1.0279\times 10^{22}\;\rm electron \;or\; proton$$
This is the number of electrons, or the number of protons, in one sphere. Hence, the net charge of one sphere's electrons is then given by
$$Q_e^-=1.0279456\times 10^{22}\cdot e^- = 1.0279\times 10^{22}\times 1.6\times 10^{-19} $$
$$Q_e^-=1644.64\;\rm C$$
This is supposedly the net charge of the electrons and the protons in normal cases, but here, there is a difference between the protons and the electrons charge as the author assumed.
Thus, the net charge of one sphere is
$$Q_{net}= Q_e^- \times 10^{-9}=\bf 1644.64\times 10^{-9}\;\rm C$$
Thus the electric force between the two copper spheres must be a repulsive force since both spheres have the same net charge, and its magnitude is given by
$$F=\dfrac{kQ_{net}Q_{net}}{r^2}=\dfrac{(8.99\times 10^9)(1644.64\times 10^{-9})^2}{(1\times 10^{-2})^2}$$
$$F=\color{red}{\bf 243}\;\rm N$$
$$\color{blue}{\bf [a]}$$
Yes, this is a detectable force. It can accelerate both spheres away from each other since it is an electric repulsive force. And since, in the real world, we can notice this motion, then this assumption must be wrong. Or the difference between the electron's and proton's charges is less than $10^{-9}$.
