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Opus_723
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Homework Statement
A cigar-shaped static charge distribution is situated at the origin of coordinates. The long dimension of the "cigar" extends along the z-axis. The total charge is q. The field at point P on the z-axis outside the charge distribution will be called E. If q were concentrated at the origin, the field at point P would be E'. Is E greater than, equal to, or less than E'?
Homework Equations
Coloumb's Law.
I'm assuming that a "cigar" can be thought of as a rotation of some "radius function" so that it can be thought of as an infinite amount of very thin rings centered on the same line segment, bulging in the center and narrowing to a point at the ends. If this is not what is usually meant by a "cigar," please correct me.
The Attempt at a Solution
I first tried to think of this qualitatively. If you take a ring of charge and condense it to a point at its center, the electric field along the axis of the original ring grows larger, since there are no longer components of the field perpendicular to the axis canceled out by symmetry. The entire field now acts along the axis. So if we were to condense our entire cigar into a line charge, the field would grow stronger. The factor by which it grows stronger would be determined by the original radius of the rings that composed the cigar.
If you take a line charge and condense it to a point at its center, the field off the end of the line charge grows weaker. This is because for any two point charges equidistant from the center, the field due the charge being moved away from point P decrease more than the field from the charge being moved closer increases, due to the inverse square nature of coloumb's law. Again, this overall decrease will be more significant for longer line charges.
So it seems to me that there is no way to tell what will happen to the field at point P without knowing the dimensions of the cigar. Condensing it to a line increases the field, while condensing that line to a point decreases it. The magnitude of each change will depend on the original dimensions of the cigar.
Presumably there's a right answer to this question, though, although I don't have access to it. Where am I wrong?