> [!summary] Coulomb's law describes the electric field produced by a point charge and can explain the interactions of point charges > **Key equation:** > Electric field by a point charge: $E = \frac{q}{4\pi R^2 \epsilon _0}$ >[!info]+ Read Time **⏱ 3 mins** # Definition Coulomb’s Law describes the electrostatic force between two point [[Charge|charges]]. It shows that the force is proportional to the product of the charges. This law also underlies how point charges generate [[Electric Fields|electric fields]] and interact through those fields. Mathematically it is defined below. $ E = \frac{q}{4 \pi \epsilon_0 r^2} $ # Deriving Coulomb's Law Using Experimental Foundations & Dimensional Analysis Historically and physically, Coulomb's law was derived from experiment, but we can justify the result from experiment using [[Dimensional Analysis|dimensional analysis]] and some reasoning. >[!warning] Assumptions Assume the field from a point charge only depends on: >- Distance ($r$) >- Charge ($q$) >- Vacuum permittivity ($\epsilon_0$) Using the dimensional analysis, we want to get some equation in the form of an electric field. $ E = \frac{[F]}{[q]} = \frac{[M][L]}{[T^2] [Q]} $ >[!note] $[\epsilon_0] = [Q^2][M]^{-1}[L]^{-3}[T]^{2}$ $ \begin{array}{c} \text{Assume the form of our electric field is in this form} \\ E =\frac{q}{\epsilon_0 r^n} \\ [E] = \frac{[q]}{[\epsilon_0][] r^n]} \\ \frac{[M][L]}{[T^2] [Q]} = \frac{[Q]}{[Q^2][M]^{-1}[L]^{3}[T]^{2} [L]^n } \\ \\\frac{[M][L]}{[T^2] [Q]} = \frac{M}{[Q][T]^2 [L]^{3-n}} \\ \\ \text{All dimensions work upset [L]} \\ 1 = 3-n \\ 2 = n \\\\ \text{So now the dimensions work out returning to our equation:} \\ E = \frac{q}{\epsilon_0 r^2} \end{array} $ Note that the actual equation is the following below, but the constants don't have dimensions by dimensional analysis and are found from experimental analysis $ E = \frac{q}{4 \pi \epsilon_0 r^2} $ # Deriving Coulomb's Law from Gauss's Law >[!warning] Assumptions In order to derive the general principle for Coulomb's Law well find the [[Electric Fields|electric field]] produced at any point from a stationary point [[Charge|charge]]. > We will do this using [[Gauss's Law]] and [[Flux]] > Well assume the following: >- There is a point charge >- Gaussian surface is symmetric (assume its a spherical Gaussian surface) >- Total area is $4\pi R^2$ (sphere) >- Normal vector from electric field and $\vec{dA}$ are in the always same direction ![[gau_3.png|500]] [^1] >[!note] Explanation Point charge with a Gaussian surface surrounding it. There is a electric field produced by the point charge. $\begin{array}{c} \Phi = \oint E\cdot dA \\ \Phi = E\oint dA \\ \Phi = EA \\ \Phi = E(4\pi R^2) \\ \\ \text{By gauss's Law} \\ \Phi = \oint E\cdot dA = \frac{q}{\epsilon_0} \\ E(4\pi R^2) = \frac{q}{\epsilon_0} \\ E = \frac{q}{4\pi R^2 \epsilon _0} \end{array}$ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/X_CHPTZfUGo?si=VbfHVYBLbLEiuwj1" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen> </iframe> --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidian-MathMatter) to support the project and help others discover it! --- [^1]: Taken from https://tikz.net/electric_field_sphere/ by Izaak Neutelings (Februari, 2020)