Chapter 10, Problem 1TC

Shown below are three separate pair of point charges. Assume the pairs only interact with each other. Compare and rank the magnitudes of the forces between each pair from highest to lowest order.

To determine

Compare and rank the magnitudes of the forces between each pair from highest to lowest.

Answer to Problem 1TC

The maximum force is acting between $+2q$ and $-2q$ charges, than between $+2q$ and $-4q$ charges and minimum force acting between $+2q$ and $+2q$ charges.

Explanation of Solution

Write the expression for the Coulomb’s force.

$F=k\frac{q_1{q}_2}{r^2}$

Here,

$k$ is proportionality constant

$q_1$ is magnitude of first charge

$q_2$ is magnitude of other charge

$r$ is the distance between two charges

Substitute $+2q$ for $q_1$, $-4q$ for $q_2$ and $x$ for the distance between them in the above expression to get $F$.

$\begin{array}{c}{F}_1=k\frac{2q\left(-4q\right)}{x^2}\\ =-k\frac{8q^2}{x^2}\mathrm{........................}(1)\end{array}$

Now, substitute $+2q$ for $q_1$, $+2q$ for $q_2$ and $x$ for the distance between them in the above expression to get $F_2$.

$\begin{array}{c}{F}_2=k\frac{2q\left(+2q\right)}{x^2}\\ =k\frac{4q^2}{x^2}\mathrm{........................}(2)\end{array}$

Now, substitute $+2q$ for first charge, $-2q$ for second charge and $\frac{x}{2}$ for the distance between them in the above expression to get $F_3$.

$\begin{array}{c}{F}_3=k\frac{2q\left(-2q\right)}{{\left(\frac{x}{2}\right)}^2}\\ =-k\frac{16q^2}{x^2}\mathrm{........................}(3)\end{array}$

From equation (1), (2) and (3) the magnitude forces from highest to lowest order.$F_3>F_1>F_2$

Conclusion:

Therefore, the maximum force is acting between $+2q$ and $-2q$ charges, than between $+2q$ and $-4q$ charges and minimum force acting between $+2q$ and $+2q$ charges.