Problem — Ranking of electric fields

Problem

A point charge of $1\ \mu\text{C}$ generates an electric field. Four situations at different distances: (A) $r = 0{,}5\ \text{m}$; (B) $r = 1\ \text{m}$; (C) $r = 5\ \text{m}$; (D) $r = 10\ \text{m}$. Rank the magnitudes of $E$ from smallest to largest.

Solution

Field of a point charge. The magnitude is $E = k\,\dfrac{q}{r^2}$. With $q$ fixed, the field depends only on $1/r^2$: it suffices to compare the values of $1/r^2$ (in $\text{m}^{-2}$):

$\begin{aligned} A:\ \frac{1}{0{,}5^2} = 4 \qquad B:\ \frac{1}{1^2} = 1 \\ C:\ \frac{1}{5^2} = 0{,}04 \qquad D:\ \frac{1}{10^2} = 0{,}01 \end{aligned}$

Numerical values (with $k = 8{,}99\cdot 10^9$ and $q = 1\cdot 10^{-6}\ \text{C}$), as a check:

$E_A = 8{,}99\cdot 10^9 \cdot \frac{1\cdot 10^{-6}}{0{,}25} = 3{,}6\cdot 10^4\ \text{N/C}, \qquad E_D = 8{,}99\cdot 10^9 \cdot \frac{1\cdot 10^{-6}}{100} = 90\ \text{N/C}$

Ranking. From $0{,}01 < 0{,}04 < 1 < 4$:

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

Links

Topics: Electric field and potential Concepts: Electric field Skills: Ranking reasoning