Problem — Ranking three pairs of charges

Problem

Three pairs of point charges: A: $+q$ and $+q$ at distance $r$; B: $+2q$ and $+q$ at distance $2r$; C: $+2q$ and $+q$ at distance $r/2$. Order the magnitudes of the forces from smallest to largest.

Solution

It suffices to compare the index $q_1 q_2/r^2$ of Coulomb’s law, in units of $q^2/r^2$:

$F_A \propto \frac{q\cdot q}{r^2} = 1;\quad F_B \propto \frac{2q\cdot q}{(2r)^2} = \frac{2}{4} = 0{,}5;\quad F_C \propto \frac{2q\cdot q}{(r/2)^2} = \frac{2}{1/4} = 8$

Ordering $0{,}5 < 1 < 8$ gives $F_B < F_A < F_C$, i.e.:

$\ev{F_C>F_A>F_B}$

Links

Topics: Elettrostatica Concepts: Legge di Coulomb Skills: Ragionamento di ranking