Electric Force - Yousef's Notes

The electric force is a fundamental interaction between charged particles. Quantitatively, this force is determined by Coulomb’s law (Experimental law discovered by Charles Augustin de Coulomb in the XVIII century), $$ \vec{F}_{1}=\frac{1}{4 \pi \epsilon_{0}} \frac{q_{1} q_{2}}{r_{12}^{2}} \frac{\vec{r}_{12}}{r_{12}}=\frac{1}{4 \pi \epsilon_{0}} \frac{q_{1} q_{2}}{r_{12}^{2}} \hat {r}_{12} $$ $$ \vec{r}_{12}=\vec{r}_{1}-\vec{r}_{2} $$

where $q_1$,$q_2$ are point charges. $\mathbf{F}_{1}$ reads “force applied on $q_1$ by $q_2$”.

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What do we mean by a point charge? A point charge is a model: a charged object small enough to be considered a mathematical point in space (i.e. without internal structure).

$$ \epsilon_{0}=8.854 \times 10^{-12} \mathrm{C}^{2} / \mathrm{N} \cdot \mathrm{m}^{2} $$ $$ k=1 / (4 \pi \epsilon_{0})=8.987551787 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} \cong 8.988 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2} $$

Coulomb’s constant k is universal constant of nature. $\epsilon_0$ is the vacuum permittivity. Why do we define like that? it will simplify other formulas later on. Some properties:

Charges are supposed to be in vacuum. Electrically speaking, air is like vacuum.
Magnitude of electric force is $F=k \frac{\left|q_{1} q_{2}\right|}{r^{2}}$, this is, proportional to the squared inverse of the distance between the charges. Experimentally tested to an accuracy of $10^{-16}$!!
Most of everyday forces are the result of electric forces at the level of atoms and molecules, like the normal force, the tension force or the friction force.
It obeys Newton’s third law: the force applied on charge 1 by 2 is equal in magnitude and opposite in direction to the force applied on charge 2 by 1.
Formally analogous to Newton’s gravitation law.
It satisfies the superposition principle: The net electric force applied on a charge is the vector sum of all electric forces acting on it.

$$ \mathbf{F}_{i}=q_{i} \sum_{j \neq i}^{N} \frac{q_{j}}{4 \pi \epsilon_{0}} \frac{\mathbf{r}_{i j}}{r_{i j}^{3}} $$

#Generalization to extensive bodies

Let us define the charge density per unit of area ($\sigma$) and the charge density per unit of volume ($\rho$) (think: are we allowed to do this? why?) to describe a continuous charge distributed within a volume or on a surface.

$$ \rho=\lim _{\Delta V \rightarrow 0} \frac{\Delta q}{\Delta V} $$ $$ \sigma=\lim _{\Delta S \rightarrow 0} \frac{\Delta q}{\Delta S} $$ Then, as a direct application of the superposition principle and the meaning of an integral, we have the force applied on charge q is $$ \vec{F}_q= \frac{q}{4 \pi \epsilon_{0}} \sum_{i=1}^{N} q_{i} \frac{\vec{r}-\vec{r}_{i}}{|\vec{r}-\vec{r}_{i}|^{3}}+\frac{q}{4 \pi \epsilon_{0}} \int_{V} \frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^{3}} \rho\left(\vec{r}'\right) d V'\\ +\frac{q}{4 \pi \epsilon_{0}} \int_{S} \frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^{3}} \sigma\left(\vec{r}'\right) d S' $$

$\vec{r}’$ is used to identify a point within the body.

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