Point Charge

A point charge is a (hypothetical) 1 dimensional point in space with a given Charge, $q$. A system consisting of only point charges is said to be a Discrete Charge Distribution.

Electrical Field §

Given a charge, $q$, the Electric Field of a point charge is:

$$\vec{E}=\frac{k_{e\times }q}{r^2}\times \hat{u}(\hat{u}\text{ directed radially outwards})$$

• $\vec{E}$ = Electric field (in $N/C$ or $V/m$)
• $q$ = Charge of point (in $C$)
• $r$ = Distance from point
• $k_e$ = Coulomb’s Constant

This is directly obtained from using Coulomb’s Law.

Electric Potential §

The Electrical Potential generated by a Point Charge with charge $q$ at a distance of $r$ is given by:

$$V=\frac{q{k}_e}{r}$$

$$V\propto \frac{1}{r}$$

In a 3D space, it results in Equipotential concentric spheres.

Finding the change in electric potential, $\Delta V$, can be simplified using these Equipotentials:

$$\Delta V_{direct}=-{\int }_i^j\vec{E}\cdot d\vec{l}$$

Different $\vec{E}$ for each $d\vec{l}$, tricky to differentiate. But:

$$\Delta V_{direct}=\Delta V_{blue}=-{\int }_i^k|\vec{E}|dl$$

Derivation Of Electric Potential §

Let’s use a test charge of $+1C$ placed at a radius, $r$, away from the point charge with charge $q$:

Due to the Electrostatic Force being conservative, the path, $\vec{L}$ , can be expressed as a straight-line distance, $\Delta r$. Furthermore, the $\vec{r}\parallel \vec{E}$

$$\Delta V=-{\int }_i^f\vec{E}\cdot \text{d}\vec{L}=-{\int }_{r_i}^{r_f}E\text{d}r$$

Using the formula for the Electric Field of a point charge:

$$\Delta V=-{\int }_{r_i}^{r_f}\frac{q\times k_e}{r_2}\text{d}r$$

Pulling out $q$ and $k_e$, which remain constant:

$$\Delta V=-q\times k_e{\int }_{r_i}^{r_f}\frac{1}{r^2}\text{d}r=-q{k}_e[-(\frac{1}{r_f}-\frac{1}{r_i})]$$

$$=q\times k_e\times (\frac{1}{r_f}-\frac{1}{r_i})$$

$$V_f-V_i=q\times k_e\times (\frac{1}{r_f}=\frac{1}{r_i})$$

Now, we need to define a $V_f$ for the equation to be useful. Let’s set $V_f=0$ when $r_f=\infty$. What this means is that, if we move the test charge infinitely far away, there is essentially 0 force acting on it, and hence zero electric potential:

#wtf