Provide expressions for the electric the electric potential V(r) at a positionr relative to a point charge q. Draw a figure showing the charge and the vector r clearly. positio

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ELECTRIC FIELD

Let us consider the source point with a charge q  at the position A $\left(x\text{'},y\text{'},z\text{'}\right)$ and the field at point P $\left(x,y,z\right)$.

They are arranged as shown in the figure:

O is the origin.

The position vectors are given as:

$\overrightarrow{r}\text{'}=x\text{'}\hat{x}+y\text{'}\hat{y}+z\text{'}\hat{z}$

$\overrightarrow{r}=x\hat{x}+y\hat{y}+z\hat{z}$

By vector addition we have,

$$\begin{gathered} \overrightarrow{r}\text{'}+\overrightarrow{R}=\overrightarrow{r} \\ \Rightarrow \overrightarrow{R}=\overrightarrow{r}-\overrightarrow{r}\text{'}=\left(x-x\text{'}\right)\hat{x}+\left(y-y\text{'}\right)\hat{y}+\left(z-z\text{'}\right)\hat{z} \end{gathered}$$

$\left|\overrightarrow{R}\right|=\sqrt{{\left(x-x\text{'}\right)}^2+{\left(y-y\text{'}\right)}^2+{\left(z-z\text{'}\right)}^2}$

Thus electric field at P is given by 

$\overrightarrow{E}\left(P\right)=\frac{1}{4\pi {\epsilon }_0}\frac{q}{{\left|\overrightarrow{R}\right|}^2}$$\hat{R}$