Problem — Ranking of Coulomb forces (identical pairs)

Problem

Four pairs of charges: (A) $q_1 = q_2 = 1\ \mu\text{C}$, $r = 1\ \text{m}$; (B) $q_1 = 1\ \mu\text{C}$, $q_2 = 4\ \mu\text{C}$, $r = 1\ \text{m}$; (C) $q_1 = q_2 = 1\ \mu\text{C}$, $r = 2\ \text{m}$; (D) $q_1 = 1\ \mu\text{C}$, $q_2 = 10\ \mu\text{C}$, $r = 10\ \text{m}$. Rank the magnitudes of the forces from smallest to largest.

Solution

Coulomb’s law. The magnitude of the force is $F = k\,\dfrac{q_1 q_2}{r^2}$. Since $k$ is common to all pairs, it suffices to compare the factor $\dfrac{q_1 q_2}{r^2}$ with the charges in $\mu\text{C}$ and the distances in metres:

$\begin{aligned} A:\ \frac{1\cdot 1}{1^2} = 1 \qquad B:\ \frac{1\cdot 4}{1^2} = 4 \\ C:\ \frac{1\cdot 1}{2^2} = 0{,}25 \qquad D:\ \frac{1\cdot 10}{10^2} = 0{,}1 \end{aligned}$

Ranking. Comparing the values $0{,}1 < 0{,}25 < 1 < 4$:

$\ev{D < C < A < B}$

Links

Topics: Electrostatics Concepts: Coulomb’s law Skills: Ranking reasoning