Splitting of charge in a symmetry argument Suppose we have some object that has charge $q$ and we take an identical object and we bring the two objects together. One asks what the charge on either object would be after separating them. According to prof Shankar (lecture @43:47) the objects will split the charge evenly (a symmetry argument).
Question 1. Is it assumed $q = q_e(2k)$ i.e an even multiple of the elementary charge?
Question 2. Does it make sense to consider where one object has precisely $q_e$ charge? If yes, how would the charge split if brought to contact with an identical object and separated?
Question 3. Assuming protons or electrons are 'irreducible', is any charge non-integer multiple of $q_e$ attainable? E.g $\frac{1}{2}q_e, \pi q_e$?
In the same lecture, it is also mentioned that charge is quantized (multiples of $|q_e| \approx 1.6\cdot 10^{-19}\ \mbox{C}$). Hence I think Q3 is answered in the negative, however, I didn't understand which multiples of $q_e$ were meant: rational, integral?
 A: Q1. No, but to claim two charges will be equal is true upto $10^{-19}$ error (if our initial charge was about 1 Coulomb). That is, for everyday charges we can assume with extremely low error that the charge is not quantized.
Q2. Yes, a nice example would be $_1\text{H}^+$ ($q=+q_e$) vs. $_1\text{H}$ ($q=0$). After a collision, either one of the protons will retain the electron. Notice the error here is $10^0$, since our initial charge was 1 $q_e$ to begin with (I guess the second proton gets the $\text{e}^-$ due to conservation of momentum but I'm not sure.).
Q3. Well, the lowest charge among elementary particles (i.e. matter) is $1/3\cdot q_e$ (anti-down and anti-strange quarks, which are always bound in nature). Since every matter is made up from these particles, we always get a charge as an integer multiple of $q_e$ for normal matter, and an integer multiple of $1/3\cdot q_e$ if we are looking at a particle collider data.
*edit: anti-quarks are usually not bound due to matter-antimatter interactions, but their normal counterparts are.
**edit: Down and strange quarks themselves have $-1/3\cdot q_e$ charges. But for heuristic reasons I gave the positive charged antiparticles (sorry about that).
