How did Coulomb charge his spheres to verify Coulomb's law?

[Samyukta]

We are doing an experiment to verify coulomb's law, and we are currently using a stable voltage source. However, we were assigned to do it using static electricity, and we were wondering how to establish inverse r^2 dependence if the amount of the charge on the ball isn't the same every time we do the experiment. When we knew the charge was the same every time (assuming same voltage begets same charge), we were able to cancel the q1q2 term and be left with only 1/ r^2 dependence.
This is apart from the issue that we aren't actually able to create the forces required to measure significant deflection.
Any help would be much appreciate.

 

[sk1105]

Do you really need the charge to be the same each time? If you measure the charge at different distances away from the ball, you should detect inverse r^2 dependence irrespective of what the charge is at the ball itself.

 

[vanhees71]

But that can be very tricky. By just charging a (usually metallic) sphere, you can never be sure whether this charge is kept constant during the whole time. To keep it on constant DC voltage relative to a given point, circumvents this problem, and everything is still static. So why would you like to change this setup?

That it's always the same charge in this case, follows from the electrostatic Maxwell equations. However, you might argue that this is a circular argument, because electrostatics, i.e.,
$$\vec{\nabla} \cdot \vec{E}=\rho, \quad \vec{\nabla} \times \vec{E}=0,$$
is more or less equivalent to Coulomb's Law, which is nothing else than the Green's function of the Laplace operator...

 

[litup]

You could use multiple voltage detectors at various distances that way you just read all the detectors at the same time and correlate the readings.

 

[Samyukta]

Well, how would we measure the charge? Basically the task is to perform the experiment and verify Coulomb's Law without the aid of modern voltage supplies or measuring instruments.
@litup : could you elaborate a little more?

 

[dipole]

But that can be very tricky. By just charging a (usually metallic) sphere, you can never be sure whether this charge is kept constant during the whole time. To keep it on constant DC voltage relative to a given point, circumvents this problem, and everything is still static. So why would you like to change this setup?

That it's always the same charge in this case, follows from the electrostatic Maxwell equations. However, you might argue that this is a circular argument, because electrostatics, i.e.,
$$\vec{\nabla} \cdot \vec{E}=\rho, \quad \vec{\nabla} \times \vec{E}=0,$$
is more or less equivalent to Coulomb's Law, which is nothing else than the Green's function of the Laplace operator...

You can just reset the distance and check that the force is still the same. If you started at some [itex] r_0 [/itex] at [itex]t=0[/itex] then resetting the apparatus to [itex] r_0 [/itex] at some later time [itex] t [/itex], and finding the measurements to be equal, will reassure one that the charge has remained constant.