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A fairly common misconception is that 'most' of the charge
will collect on the more sharply curved surface, (needle tip).
A closer analysis will indeed show that the charge DENSITY
is quite large on the highly curved surface, but the small
area involved does NOT lead to a preponderance of charge
being on the tip. Corona discharge has been ruled out in this
discussion of an 'isolated' object. Lightning rod tips are NOT
isolated and do indeed exhibit coronal discharge.