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In the Space Exploration SE question Is there any possible reason to direct the electron gun specifically towards the ion trail behind an ion thruster? there is somewhat of a consensus that there isn't a strong compelling reason to intentionally inject the electrons directly into the ion plume.

Question: What would happen if it weren't and the electrons were emitted in some other direction? Just for example, would enough space charge be built up in the semi-infinite cone of ions behind the spacecraft to eventually impede ion emission?

The Dawn Spacecraft might be useful as a canonical example for numbers. There is also some information in the following Space SE questions and answers:

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How does an Ion Thruster work?

An ion thruster basically just ionizes (generally) Xenon atoms to produce ions by electron bombardment, in this process , a highly energetic electron is made to collide with a neutral propellant atom (e.g. Xenon), which knocks out electrons from the atom, and leaves a net positive charge on the Xenon atoms. The resultant product is a plasma.

If we had m Xenon atom and n Electrons initially,
we now have m positive Xenon ions and (m+n) electrons.
Thus we have extra m electrons (extra negative charge).

These electrons that are used to bombard the fuel atoms are generated by a process called Thermionic emission at the discharge cathode. These thus generated electrons are emitted into a discharge chamber. The discharge chamber walls has high positive potential (to attract the electrons emitted to the chamber) and is lined with strong magnets in a specific arrangement (to make it difficult for the electrons to reach the positively charged wall). The fuel atoms as well as the electrons when enter this chamber , collide and form the positive ions and extra electrons starting a chain reaction. These positive ions migrate towards a screen grid which is like a disk with tiny bores drilled in it uniformly all over the surface and is subjected to a highly positive voltage that accelerates the positively charged fuel ions towards a negatively charged electrode to very high speeds and are ejected in direction opposite to direction of motion [Newtons Third Law].

The main part:

Now finally, we have that extra m electrons on our space object that gives our object a net negative charge. This can lead to serious problem as some if not all of the ejected Xenon ions can be drawn back to our space object reducing thrust and leading to inefficiency and also in severe cases to space object erosion on its external walls.

Hence, we shoot a beam of electrons in the ion beam to neutralize the spacecraft, expelling an equal amount of electrons to make the total charge of the exhaust beam neutral.

And, neutralizing the positively charged beam in the direction of flow will also help in gaining some positive momentum in the direction of motion.

You can neutralize it by shooting those electrons right into the ion beam, but that would not add up to more momentum in the direction of motion.

Also:

Ion current extracted from an ion source (ion thruster) can be increased above
the Child Langmuir limit if the ion space charge is neutralized. 
Similarly, the limiting kinetic energy density of the plasma flow
in a Hall thruster might be exceeded if additional mechanisms of
space charge neutralization are introduced. Space charge neutralization 
with high-mass negative ions or negatively charged dust particles seems,
in principle, promising for the development of a
high current or high energy density source of positive light ions.
Source:[https://www.osti.gov/servlets/purl/836621]

Helpful links::

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  • $\begingroup$ Let us continue this discussion in chat. I've moved our conversation to a chat room so it can continue without growing too large here, and I'm deleting my comments here since we don't need two copies. $\endgroup$
    – uhoh
    Commented Jul 25, 2020 at 10:16

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