# Electric potential

Table of Content

## Electric Potential

• We now move towards the electric potential which is potential energy per unit charge.
• Thus electrostatic potential at any point of an electric field is defined as potential energy per unit charge at that point.
• Electric potential is represented by letter V.
V=U/q' or U=q'V ----(6)
• Electric potential is a scalar quantity since both charge and potential energy are scalar quantities.
• S.I. unit of electric potential is Volt which is equal to Joule per Coulomb. Thus,
1 Volt = 1 JC-1
• In equation 4 if we divide both sides by q' we have

where V(r1) is the potential energy per unit charge at point R and V(r2) is potential energy per unit charge at point S and are known as potential at points R and S respectively.
• Again consider figure 1. If point S in figure 1 would be at infinity then from equation 7

Since potential energy at infinity is zero therefore V(∞)=0. Therefore

hence electric potential at a point in an electric field is the ratio of work done in bringing test charge from infinity to that point to the magnitude of test charge.
• Dimensions of electric potential are [ML2T-3A-1] and can be calculated easily using the concepts of dimension analysis.

## Electric potential due to a point charge

• Consider a positive test charge +q is placed at point O shown below in the figure.

• We have to find the electric potential at point P at a distance r from point O.
• If we move a positive test charge q' from infinity to point P then change in electric potential energy would be

• Electric potential at point P is

• here we see that like electric field potential at any point independent of test charge used to define it.
• Here are the graph of potential with respect to distance

## Electric Potential due to a system of charges

• Consider a system of charges $q_1$, $q_2$,...., $q_n$ with position vectors $\mathbf{r_1}$,$\mathbf{r_2}$,..., $\mathbf{r_n}$ relative to some origin
• Potential at any point will be the sum of potential of the individual charges $V=V_1 + V_2 ... + V_n$
• Potential V at any point due to arbitrary collection of point charges is given by

• For continuous charge distributions summation in above expression will be replaced by the integration

where dq is the differential element of charge distribution and r is its distance from the point at which V is to be calculated.

Question -1
(a)Calculate the potential at a point P due to a charge of $1 \times 10^{-7} \ C$ located 9 cm away.
(b) Hence obtain the work done in bringing a charge of $1 \times 10^{-9} \ C$ from infinity to the point P. Does the answer depend on the path along which the charge is brought?
Given $\frac {1}{4 \pi \epsilon _0}=9 \times 10^9 \ Nm^2/C^2$
Solution
(a) Potential is given by the formula
$V= \frac {1}{4 \pi \epsilon _0} \frac {Q}{r}$
Here Q=$1 \times 10^{-7} \ C$ and $r=.09 \ m$
$V= 9 \times 10^9 \times \frac {1 \times 10^{-7}}{.09}=10^4 \ V$
(b)Work done is given by
$W= qV= 1 \times 10^{-9} \times 10^4= 10^{-5} \ J$
Work done will be path independent

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