define electric potential at a point as the work done per unit positive charge in bringing a small test charge from infinity to the point

Electric Potential

Objective

Define electric potential at a point as the work done per unit positive charge in bringing a small test charge from infinity to that point.

Key Definition

The electric potential \$V\$ at a point \$P\$ in an electric field is defined as

\$V = \frac{W_{\infty \to P}}{q}\$

where \$W_{\infty \to P}\$ is the work required to move a positive test charge \$q\$ from a location at infinity (where the potential is taken as zero) to the point \$P\$.

Derivation from the Electric Field

1. Consider a small positive test charge \$q\$ moving an infinitesimal displacement \$d\mathbf{s}\$ against the electric field \$\mathbf{E}\$.
2. The infinitesimal work done on the charge is

   \$dW = -\mathbf{F}\cdot d\mathbf{s}= -q\mathbf{E}\cdot d\mathbf{s}.\$

3. Dividing by \$q\$ gives the infinitesimal change in potential:

   \$dV = \frac{dW}{q}= -\mathbf{E}\cdot d\mathbf{s}.\$

4. Integrating from infinity (where \$V=0\$) to the point \$P\$:

   \$V(P)= -\int_{\infty}^{P}\mathbf{E}\cdot d\mathbf{s}.\$

Units and Dimensions

• SI unit of electric potential: volt (V), where \$1\ \text{V}=1\ \text{J C}^{-1}\$.
• Dimensionally: \$[V]=\frac{ML^{2}}{T^{3}I}\$ (where \$I\$ is electric current).

Relationship to Electric Potential Energy

The electric potential energy \$U\$ of a charge \$q\$ at a point of potential \$V\$ is

\$U = qV.\$

This shows that potential is a property of the field alone, independent of the test charge.

Example Calculation

Find the potential at a distance \$r\$ from a point charge \$Q\$.

1. The electric field of a point charge is \$\displaystyle \mathbf{E} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^{2}}\hat{r}\$.
2. Using \$V = -\int_{\infty}^{r}\mathbf{E}\cdot d\mathbf{s}\$,

   \$\$V(r) = -\int{\infty}^{r}\frac{1}{4\pi\varepsilon0}\frac{Q}{s^{2}}\,ds

   = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}.\$\$

3. Thus \$V\$ is positive for a positive \$Q\$ and falls off as \$1/r\$.

Table of Common Symbols

Suggested Diagram

Key Points to Remember

• Potential is a scalar quantity; it can be added algebraically.
• Zero potential is defined at infinity for isolated charge distributions.
• The sign of \$V\$ follows the sign of the source charge.
• Electric field is the negative gradient of potential: \$\displaystyle \mathbf{E}= -\nabla V\$.