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If there was a sufficient charge density, would it be possible for that charge to have an event horizon, a radius at which a charged particle would need to be going faster than the speed of light to escape? If it is possible, what would be the radius for this charge event horizon?

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Not in the same way as a black hole - at least, not for the electric force itself.

You see, a big part of the reason that black holes have an event horizon is that they "act on everything equally": that's the famed weak equivalence principle of gravity, that gravitational "charge", or gravitational mass, is strictly and exactly proportional to inertial mass, and thus gravity has a strictly kinematic effect. Naively, i.e. in terms of Newtonian mechanics and without invoking general relativity, it makes gravity into a strictly kinematic effect: one that can be understood solely in terms of acceleration and without needing to bring in the mass of the object being acted upon. Or, to put it another way, it allows you to cancel the small "m" in

$$\frac{GMm}{r^2} = ma$$

from Newton's second law combined with Newton's law of universal gravitation. The small "m" on the left is the gravitational mass of the small test body, while that on the right is the inertial mass. Only by these two being equal (or proportional, but then we absorb that proportionality into $G$) can we just cancel them out. When this is taken to its fullest extent in General Relativity, this fact ultimately makes gravity an interaction that is not a force - where a "force" very specifically means an interaction that exchanges four-momentum (energy/momentum combined into a single "space-time momentum") between the two objects: it's an interaction that occurs because objects change the spacetime around them so that "motion in a straight line" is, in effect, redefined. In the extreme, this eventually forms the event horizons of a black hole, where that the geometry has been bent to the extent that all allowable straight-line motions end up in the black hole core where, of course, general relativity itself fails as a theory.

Now for electromagnetism. Electromagnetism is a real force: it does result in an interchange of four-momentum and so causes, even in the context of general relativity's curved space-time, genuine acceleration that can be picked up by an accelerometer. Particles interacting electromagnetically follow paths that are not straight lines even in the modified sense thereof caused by ambient gravitational effects.

Thus, in particular, one consequence is there isn't a single escape speed for an arbitrary charged object. It will depend on the mass of that object, as @Nihar Karve mentions in his comment, and it will depend on the charge.

But moreover than that, I believe it is possible to actually show that even the escape speed for a specific charged object never exceeds the speed of light in a relativistic electromagnetic context. The simplest way to see this is to consider what happens with a charge falling from infinity down an endless electric potential. As this is a real force, it gains more and more momentum - but momentum can never carry anything past the speed of light. Likewise, then, the time-reverse of this situation is the charged particle climbing against the electric potential. This will, then, necessarily happen at speeds below that of light. Hence, there are no "electromagnetic event horizons" generated solely by the electromagnetic interaction alone.

This argument doesn't work, however, for black holes and gravitation, because as I said before gravitation is not a force in General Relativity, but an alteration of ambient geometry, so we cannot talk its effect in terms of things "accelerating up to the speed of light" in the sense of "acceleration" the theory prefers. In fact, it's kind of misleading to talk of an "escape speed" at all in this context: to talk of such (and to call gravity an "acceleration") you need to set up a global spatial (not space-time) coordinate system, and it is a property of the theory that there is great freedom in choosing such coordinates. That is, there is no non-arbitrary standard of purely spatial or purely temporal "distance" and thus not also "speed" or "acceleration", either.

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No, this will not happen. For example, if oyu have a black hole in which you put more and more charge the Schwarzschild radius will increase untill there is so much charge that it has disappeared. No black hole will be there anymore. You can imagine that because of this, a Schwarzschild radius will not appear if you cram more and more charge into a small volume of space. It should be noted though that pushing charge together in a small volume will require much energy. But for a charged black hole this is not taken into consideration. Only the electric field of the charge inside the hole is taken into consideration. See this article about charged black holes.

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