5.1 Electric Potential

In physics, Potential generally refers to the ability to do work on a test particle by virtue of its position in the field of the source charge. And It's generally the property of the system and not of an individual body or particle. The definition of potential is as follows,

 "The electric potential at a point($r$) in the free space is defined as the work done in bringing a unit positive charge from infinity to the point($r$). It is a scalar quantity and it's SI unit is the Volt(V) or Joule per Coulomb (JC$^{-1}$)."

$$V=\frac{Q}{4\pi \epsilon_{0} r}$$It is important to note that in the above definition the potential due to the source charge at the infinity is by definition taken to be zero. For the positive source charge the potential is positive and negative for the negative source charge. 

The signs can be understood with regard to the work done by the field or work done against the field when we place a positive test charge at infinity to the point $r$. With negative source charge its clear that the field does the work to move the positive test charge from infinity to point $r$ (i.e attractive, thus by def$^{n}$ negative sign). And for the positive source charge, we do work against the field to move positive test charge from infinity to the point $r$ (i.e repulsion,  thus by def$^{n}$ positive sign)

Potential Gradient/Electric Field strength: 
"It is the rate of change of electric potential with respect to displacement in the direction of the field." $$P.G=\frac{\Delta V}{\Delta r}$$
Where, $\Delta V$= change in potential or potential difference, in $V$
             $\Delta r$= displacement in the direction of the field, in $m$

Electric Potential Energy: The electric potential energy of a charge($q$) placed at a given point in the field is given by,$$E_{p}=Vq=\frac{Qq}{4\pi \epsilon_{0}r}$$
Here's a solved problem for practice. 

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