# Electric Potential (V)

Electric field intensity $\overrightarrow{E}$ can be calculated by using either Coulomb’s law/Gauss’s law . when the charge distribution is symmetric another way of obtaining $\overrightarrow{E}$ is from the electric scalar potential V

Assume a test charge $Q_{t}$ at A in an Electric field, let points A and B are located at $r_{A}and&space;r_{B}$ units from the origin O,from Coulomb’s law the force acting on a test charge $Q_{t}$ is $\overrightarrow{F}=&space;Q_{t}\overrightarrow{E}$

The work done in moving a point charge $Q_{t}$ along a differential length $\overrightarrow{dl}$ is $dW$ is given by $dW&space;=&space;-\overrightarrow{F}.\overrightarrow{dl}$

$dW&space;=&space;-Q_{t}\overrightarrow{E}.\overrightarrow{dl}$

so the total work done in moving a point charge $Q_{t}$ from A to B is $W=-Q_{t}\int_{A}^{B}\overrightarrow{E}.\overrightarrow{dl}$

the direction of work done is always opposite to the direction of displacement.

where A is the initial point and B is the final point. Dividing the work done by the charge $Q_{t}$ gives the potential energy per unit charge denoted by $V_{AB}$,this is also known as potential difference between  the two points A and B.

Thus $V_{AB}&space;=&space;\frac{W}{Q_{t}}=&space;-\int_{A}^{B}\overrightarrow{E}.\overrightarrow{dl}$

if we take B as initial point and A as final point , then $V_{BA}&space;=&space;\frac{W}{Q_{t}}=&space;-\int_{B}^{A}\overrightarrow{E}.\overrightarrow{dl}----Equation(1)$

To derive the expression for V in terms of charge Q and distance r , we can use the concept of Electric field intensity $\overrightarrow{E}$ produced by a charge Q, which is placed at a distance r

i.e, $\overrightarrow{E}&space;=&space;\frac{Q}{4\pi&space;\epsilon&space;_{o}r^{2}}\overrightarrow{a_{r}}$

from Equation(1) $V_{BA}&space;=&space;-\int_{B}^{A}\overrightarrow{E}.\overrightarrow{dl}$

$V_{AB}=&space;-\int_{r_{A}}^{r_{B}}\frac{Q}{4\pi&space;\epsilon&space;_{o}r^{2}}\overrightarrow{a_{r}}.dr&space;\overrightarrow{a_{r}}$  since $\overrightarrow{dl}=dr.\overrightarrow{a_{r}}$

$V_{AB}=&space;-\int_{r_{A}}^{r_{B}}\frac{Q}{4\pi&space;\epsilon&space;_{o}r^{2}}dr$

$V_{AB}=&space;-\frac{Q}{4\pi&space;\epsilon&space;_{o}}\left&space;[&space;\frac{-1}{r}&space;\right&space;]_{r_{A}}^{r_{B}}$

$V_{AB}=&space;-\frac{Q}{4\pi&space;\epsilon&space;_{o}}[\frac{1}{r_{B}}-\frac{1}{r_{A}}]$

$V_{AB}=V_{B}-V_{A}$

similarly, $V_{BA}=V_{A}-V_{B}$

where $V_{A}$ and $V_{B}$ are the scalar potentials at the points A and B respectively. If A is  located at $\infty$ with respect to origin ,with zero potential $V_{A}&space;=0$ and B is located at a distance r with respect to origin. then the work done in moving a charge from  A (infinity) to B is given by

$V_{AB}&space;=&space;\frac{Q}{4\pi&space;\epsilon&space;_{o}r_{B}}$  here $r_{B}&space;=&space;r$

$\therefore&space;V&space;=&space;\frac{Q}{4\pi&space;\epsilon&space;_{o}r}$ volts.

hence the potential at any point is the potential difference between that point and a chosen point at which the potential is zero. In other words assuming Zero potential at infinity .

The potential at a distance r from a point charge is the work done per unit charge by an external agent in transferring a test charge from infinity to that point.

i.e, $V&space;=&space;-\int_{\infty&space;}^{r}\overrightarrow{E}.\overrightarrow{dl}$

So a point charge $Q_{1}$ located at a point P with position vector $\overline{r_{1}}$ then the potential at another point Q with a position vector $\overline{r}$ is

$V_{at&space;\overline{r}}&space;=&space;\frac{Q_{1}}{4\pi&space;\epsilon&space;_{o}\left&space;|&space;\overline{r}&space;-\overline{r_{1}}\right&space;|}$

As like $\overrightarrow{E}$ superposition principle is applicable to V also that is for n point charges $Q_{1},Q_{2},Q_{3},Q_{4},........Q_{n}$ located at points with position vectors  $\overline{r_{1}},\overline{r_{2}},\overline{r_{3}},.......\overline{r_{n}}$

then the potential at $\overline{r}$ is

$V_{at&space;\overline{r}}&space;=&space;\frac{Q_{1}}{4\pi&space;\epsilon&space;_{o}\left&space;|&space;\overline{r}&space;-\overline{r_{1}}\right&space;|}+\frac{Q_{2}}{4\pi&space;\epsilon&space;_{o}\left&space;|&space;\overline{r}&space;-\overline{r_{2}}\right&space;|}+\frac{Q_{3}}{4\pi&space;\epsilon&space;_{o}\left&space;|&space;\overline{r}&space;-\overline{r_{3}}\right&space;|}+........+\frac{Q_{n}}{4\pi&space;\epsilon&space;_{o}\left&space;|&space;\overline{r}&space;-\overline{r_{n}}\right&space;|}$

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