Exercise EE1-01.05 – Faulty calculation

Prof. Dr. Thorsten Jungmann (last updated 2025-09-12)

Relates to EE1-01 Introduction

Task

A student calculates the force between two electric charges $Q_1=5\mu \mathrm{C}$ and $Q_2=-3\mu \mathrm{C}$ at a distance of $r=2\mathrm{cm}$ as follows:

$$F=\frac{1}{4\pi {\epsilon }_0}\cdot \frac{Q_1\cdot Q_2}{r^2}=9\cdot 10^9\cdot \frac{5\cdot (-3)}{0.02^2}=-337.5\cdot 10^9\mathrm{N}$$

a) Identify all the errors in this calculation.
b) Carry out the calculation correctly.
c) Interpret the sign of the result physically.

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Solution

Identifying the errors

• The student completely ignored the units.
• In addition, the order of magnitude of the electric charge was not taken into account in the calculation.
• The result of the calculation, had it been written down correctly ($-337.5\cdot 10^{12}\mathrm{N}$), is too large by a factor of $10^{12}$. Here, a wrong result was additionally noted with the wrong precision.

Correct calculation

$$F=\frac{1}{4\pi {\epsilon }_0}\cdot \frac{|Q_1|\cdot |Q_2|}{r^2}$$

With $\pi \approx 3.142$ and ${\epsilon }_0\approx 8.854\cdot 10^{-12}\frac{\mathrm{F}}{\mathrm{m}}$:

$$F=9\cdot 10^9\frac{\mathrm{m}}{\mathrm{F}}\cdot \frac{5\cdot 10^{-6}\mathrm{C}\cdot 3\cdot 10^{-6}\mathrm{C}}{(0.02\mathrm{m}{)}^2}$$

With $1\mathrm{F}=\frac{{\mathrm{C}}^2}{\mathrm{N}\cdot \mathrm{m}}$:

$$F=9\cdot 10^9\frac{\mathrm{N}\cdot {\mathrm{m}}^2}{{\mathrm{C}}^2}\cdot \frac{15\cdot 10^{-12}{\mathrm{C}}^2}{4\cdot 10^{-4}{\mathrm{m}}^2}$$

Work out the numbers and simplify the units:

$$F=9\cdot 10^9\cdot 3.75\cdot 10^{-8}\frac{\mathrm{N}\cdot {\mathrm{m}}^2}{{\mathrm{C}}^2}\cdot \frac{{\mathrm{C}}^2}{{\mathrm{m}}^2}=0.3375\mathrm{N}$$

What remains is newton ($\mathrm{N}$):

$$\underline{F\approx 0.3\mathrm{N}}$$

Note on precision

The precision with which results were stated depends on the precision of the source data, here: one significant digit for $Q_1$, $Q_2$ and $r$. The calculations were carried out with the calculator values, but always stated in such a way that the result has no more than one significant digit.

Interpreting the sign

Opposite charges ($Q_1>0$, $Q_2<0$) attract each other. The force is attractive.