Charging a small metal ball from a charged hollow sphere

[cianfa72]

TL;DR Summary

About the Feynman discussion about charging a small metal ball from a charged hollow sphere

I keep reading the amusing lectures from Feynman on Electrodynamics. In particular in Lecture 5-8 figure 5-10, he argues that touching a charged hollow sphere externally with a litte metal ball will cause the ball to pick up charge from the charged sphere (contrary touching the hollow sphere from the inside will not cause any charge to be pushed on the small ball since the electric field inside the hollow sphere vanishes).

I was reasoning about the reason behind it. In section 19-1 Feynman uses the principle of minimum action to derive some interesting results. In this specific case the charge volume density vanishes (charges exist only as surface charge density on the conductors), so the principle claims that the target charge distribution when the small ball is in touch with the charged hollow sphere, corresponds to the surface charge distribution that produces a potential function that minimize the total electrostatic energy in all the space surrounding the two conductors.

I believe one can get the same result considering the matrix capacitance of the system of two conductors next to each other (i.e. the charged hollow sphere and the small metal ball). Then one may imagine that the two conductors are effectively "connected" through a non-capactive wire. This way the net result is setting up a Dirichlet problem where the capacitance matrix and the total charge are given/assigned.

What do you think about, does it make sense?
Thanks.

 

[hutchphd]

I would have to work it out. Which is what you should do, and present the calculation for consideration. I see no obvious reason why it should not work out but the devil is always in the detail, and the communication of said details.

 

[cianfa72]

Which is what you should do, and present the calculation for consideration.

About the second approach, we know the total charge $Q$ (initially stored as surface density on the hollow sphere). Suppose we also know the 2x2 symmetric capacitance matrix for the "hollow sphere + near small metal ball" system. Next I believe one must solve the linear system that includes the charge/potential linear relationship via the capacitance matrix plus the two equations $Q_1 + Q_2 = Q$ and $V_1 = V_2$ (in total 4 linear equations in 4 unknown).

This way one gets as solutions the charges $Q_1$ and $Q_2$.

 

[hutchphd]

I was hoping for something that starts with "let the large sphere have radius R and the small one radius r" and proceeds mathematically to a conclusion. Anything less is hand-waving, however vigorous (and potentially correct).....but I'm not going to do the calculation ab initio. Please do the work. I will happilly check it.

 

[cianfa72]

As explained here, the capacitance matrix coefficients $C_{11}$ and $C_{22}$ are not the self-capacitances of each conductor if it were in isolation. I know the formula for the self-capacitance of a spherical conductor of radius R, however I think it doesn't help here...