### Summary

## 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 $U(r)={k}_{e}\frac{qQ}{r}$, 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 $\text{\Delta}V$:

$\text{\Delta}V=\frac{\text{\Delta}U}{q}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\text{\Delta}U=q\text{\Delta}V.$ - An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,

$1\phantom{\rule{0.2em}{0ex}}\text{eV}=\left(1.60\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\phantom{\rule{0.2em}{0ex}}\text{C}\right)\left(1\phantom{\rule{0.2em}{0ex}}\text{V}\right)$

$=\left(1.60\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{\phantom{\rule{0.2em}{0ex}}10}^{\mathrm{-19}}\phantom{\rule{0.2em}{0ex}}\text{C}\right)\left(1\phantom{\rule{0.2em}{0ex}}\text{J/C}\right)=1.60\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\phantom{\rule{0.2em}{0ex}}\text{J}\text{.}$

## 7.3 Calculations of Electric Potential

- 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: ${V}_{P}={k}_{e}{\displaystyle \sum _{1}^{N}\frac{{q}_{i}}{{r}_{i}}}$.
- An electric dipole consists of two equal and opposite charges a fixed distance apart, with a dipole moment $\overrightarrow{\text{p}}=q\overrightarrow{\text{d}}$.
- Continuous charge distributions may be calculated with ${V}_{P}={k}_{e}{\displaystyle \int \frac{dq}{r}}$.

## 7.4 Determining Field from Potential

- 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.

## 7.5 Equipotential Surfaces and Conductors

- 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.

## 7.6 Applications of Electrostatics

- 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.