⊕Read Tipler and Mosca (2004) chapters 21 and 22 (The electric field I and II); Pelcovits and Farkas (2024) chapter 10 (Electrostatics). See also https://www.youtube.com/watch?v=pyMmRRIB5yw.

Coulomb’s law

The magnitude of the electrostatic force between two stationary point charges is given by Coulomb’s lawFirst equation on your AP Physics C E&M equation sheet. Published in 1785 by French officer, physicist, and engineer Charles-Augustin de Coulomb. See https://en.wikipedia.org/wiki/Coulomb\%27s_law

.

\[|\vec{F}_E| = k \dfrac{q_1 q_2}{r^2}\qquad{(1)}\]

where \(q_1\) and \(q_2\) are the charges in coulombs (\(\unit{\coulomb}\))defined as the amount of charge delivered by a current of \(\qty{1}{\ampere}\) in \(\qty{1}{\second}\). A coulomb is a fairly large unit; the charge of a single electron is \(q=\qty{1.6e-19}{\coulomb}\). See https://en.wikipedia.org/wiki/Coulomb

; \(r\) is the distance between them, and \(k=\qty{9.0e9}{\newton\meter\squared\per\coulomb\squared}\) is the Coulomb constantFirst constant given on your AP Physics C E&M equation sheet.

. The form of eq. 1 is an inverse square law.

An alternative form of eq. 1 is \[|\vec{F}_E| = \dfrac{1}{4 \pi \epsilon_0} \dfrac{q_1 q_2}{r^2}\qquad{(2)}\]

where here the inverse square law form is the same, but the constant \(\epsilon_0=\qty{8.85e-12}{\coulomb\squared\per\newton\per\meter\squared}\) is used instead. \(\epsilon_0\) is the permittivity of free space or vacuum permittivityhttps://en.wikipedia.org/wiki/Vacuum_permittivity

.

The charges (\(q_1\) and \(q_2\)) can be positive or negative. If the charges are the same sign, the force between them repels them apart. If the charges are of the opposite sign, the force between them attractsOpposites attract, like the Paula Abdul song from the late 80s https://www.youtube.com/watch?v=xweiQukBM_k.

them together.

Coulomb obtained eq. 1 by making measurements of the force between two charged spheres using a torsion balance. Prior workers had shown the force of attraction was likely an inverse square law. Cavendish later used a torsion balance similar to Coulomb’s in gravity experiments to estimate \(G\).

Analogy with gravitational force

Recall that the gravitational force between two objects was given bySee https://en.wikipedia.org/wiki/Newton\%27s_law_of_universal_gravitation

\[F = \dfrac{G M_1 M_2}{r^2}\] which was also an inverse square law, with universal gravitation constant \(G=\qty{6.67e-11}{\newton\meter\squared\per\kilo\gram\squared}\). This will lead to similarity in how we calculate forces and also energy and potential later.

Force due to a system of charges

We can use superposition when applying eq. 1. Thus for a system of multiple point charges, we might repeat the calculation for each pair and add the resulting force vectors together to get the sum of the forces on a charge of interest.

Similarly, we can distribut charge of lines, areas, or volumes. These result in charge densities \(\lambda\ \unit{\coulomb\per\meter}\), \(\sigma\ \unit{\coulomb\per\meter\squared}\), or \(\rho\ \unit{\coulomb\per\meter\cubed}\), respectively. Of course, if we wish to find the force resulting from such distributed charges, we can use superposition and do an integral to sum up the contributions of each tiny piece \(dq=\lambda ds\) or \(\sigma dA\) or \(\rho dV\), respectively, as we shall see.

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References

Pelcovits, Robert A, and Joshua Farkas. 2024. Barron’s AP Physics c Premium. Kaplan North America.

Tipler, Paul A, and Gene Mosca. 2004. Physics for Scientists and Engineers. 5th ed. W H Freeman; Company.