A particle of charge -q and mass m moves in a circular orbit of radius r around a very long, uniformly charged wire. If V is the orbital speed of the particle, which statement about the V vs r relationship is correct?

V vs r graph is a straight line through origin with positive slope
V vs r graph is a curve rising upward and concave up
V vs r graph is a horizontal line
V vs r graph is not given

Correct answer: V vs r graph is a horizontal line

Solution

The electric field of an infinite line charge falls as 1/r. The centripetal force needed also goes as V²/r. When you equate them, the r cancels, showing that V² = q*lambda/(2*pi*eps0*m) is independent of r. Hence V is constant for all orbital radii, and the V vs r graph is a horizontal line.

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