7.1 Electric Potential Energy

The work done to move a charge from point A to B in an electric field is path independent, and the work around a closed path is zero. Therefore, the electric field and electric force are conservative.
We can define an electric potential energy, which between point charges is $U (r) = k_{e} \frac{q Q}{r}$ , with the zero reference taken to be at infinity.
The superposition principle holds for electric potential energy; the potential energy of a system of multiple charges is the sum of the potential energies of the individual pairs.

7.2 Electric Potential and Potential Difference

Electric potential is potential energy per unit charge.
The potential difference between points A and B, $V_{B} - V_{A},$ that is, the change in potential of a charge q moved from A to B, is equal to the change in potential energy divided by the charge.
Potential difference is commonly called voltage, represented by the symbol $Δ V$ :
$Δ V = \frac{Δ U}{q} or Δ U = q Δ V .$
An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,
$1 eV = (1.60 \times 10^{−19} C) (1 V)$
$= (1.60 \times 10^{−19} C) (1 J/C) = 1.60 \times 10^{−19} J .$

Electric potential is a scalar whereas electric field is a vector.
Addition of voltages as numbers gives the voltage due to a combination of point charges, allowing us to use the principle of superposition: $V_{P} = k_{e} \sum_{1}^{N} \frac{q_{i}}{r_{i}}$ .
An electric dipole consists of two equal and opposite charges a fixed distance apart, with a dipole moment $\vec{p} = q \vec{d}$ .
Continuous charge distributions may be calculated with $V_{P} = k_{e} \int \frac{d q}{r}$ .

Just as we may integrate over the electric field to calculate the potential, we may take the derivative of the potential to calculate the electric field.
This may be done for individual components of the electric field, or we may calculate the entire electric field vector with the gradient operator.

An equipotential surface is the collection of points in space that are all at the same potential. Equipotential lines are the two-dimensional representation of equipotential surfaces.
Equipotential surfaces are always perpendicular to electric field lines.
Conductors in static equilibrium are equipotential surfaces.
Topographic maps may be thought of as showing gravitational equipotential lines.

Electrostatics is the study of electric fields in static equilibrium.
In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink jet printers, and electrostatic air filters.