Section Summary

19.1 Electric Potential Energy: Potential Difference

• Electric potential is potential energy per unit charge.
• The potential difference between points A and B, $V_{\mathrm{B}}–V_{\mathrm{A}}$, defined to be the change in potential energy 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 $\text{Δ}V$.
  $$\mathrm{\Delta }V=\frac{\text{ΔPE}}{q}\text{and ΔPE =}q\mathrm{\Delta }V\text{.}$$
• An electron volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,
  $$\begin{array}{lll}\text{1 eV}& =& \left(1.60\times {\text{10}}^{\text{–19}}\text{C}\right)\left(\mathrm{1\; V}\right)=\\ & \left(1.60\times {\text{10}}^{\text{–19}}\text{C}\right)\left(\mathrm{1\; J/C}\right)\\ & =& 1.60\times {\text{10}}^{\text{–19}}\text{J.}\end{array}$$
• Mechanical energy is the sum of the kinetic energy and potential energy of a system, that is, $\text{KE}+\text{PE}.$ This sum is a constant.

19.2 Electric Potential in a Uniform Electric Field

• The voltage between points A and B is
  $$\begin{array}{c}\left.\begin{array}{c}{V}_{\text{AB}}=\mathrm{Ed}\\ E=\frac{V_{\text{AB}}}{d}\end{array}\right\}\text{(uniform}E\text{- field only),}\end{array}$$
  where $d$ is the distance from A to B, or the distance between the plates.
• In equation form, the general relationship between voltage and electric field is
  $$E=–\frac{\mathrm{\Delta }V}{\mathrm{\Delta }s},$$
  where $\mathrm{\Delta }s$ is the distance over which the change in potential, $\mathrm{\Delta }V$, takes place. The minus sign tells us that $\mathbf{\text{E}}$ points in the direction of decreasing potential.) The electric field is said to be the gradient (as in grade or slope) of the electric potential.

19.3 Electrical Potential Due to a Point Charge

• Electric potential of a point charge is $V=\text{kQ}/r$.
• Electric potential is a scalar, and electric field is a vector. Addition of voltages as numbers gives the voltage due to a combination of point charges, whereas addition of individual fields as vectors gives the total electric field.

19.4 Equipotential Lines

• An equipotential line is a line along which the electric potential is constant.
• An equipotential surface is a three-dimensional version of equipotential lines.
• Equipotential lines are always perpendicular to electric field lines.
• The process by which a conductor can be fixed at zero volts by connecting it to the earth with a good conductor is called grounding.

19.5 Capacitors and Dielectrics

• A capacitor is a device used to store charge.
• The amount of charge $Q$ a capacitor can store depends on two major factors—the voltage applied and the capacitor’s physical characteristics, such as its size.
• The capacitance $C$ is the amount of charge stored per volt, or
  $$C=\frac{Q}{V}.$$
• The capacitance of a parallel plate capacitor is $C={\epsilon }_0\frac{A}{d}$, when the plates are separated by air or free space. ${\epsilon }_{\text{0}}$ is called the permittivity of free space.
• A parallel plate capacitor with a dielectric between its plates has a capacitance given by
  $$C={\mathrm{\kappa \epsilon }}_0\frac{A}{d},$$
  where $\kappa$ is the dielectric constant of the material.
• The maximum electric field strength above which an insulating material begins to break down and conduct is called dielectric strength.

19.6 Capacitors in Series and Parallel

• Total capacitance in series $\frac{1}{C_{\text{S}}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\text{.}\text{.}\text{.}$
• Total capacitance in parallel $C_{\text{p}}=C_1+C_2+C_3+\text{.}\text{.}\text{.}$
• If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.

19.7 Energy Stored in Capacitors

• Capacitors are used in a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps, to supply energy.
• The energy stored in a capacitor can be expressed in three ways:
  $$E_{\text{cap}}=\frac{\text{QV}}{2}=\frac{{\text{CV}}^2}{2}=\frac{Q^2}{2C},$$
  where $Q$ is the charge, $V$ is the voltage, and $C$ is the capacitance of the capacitor. The energy is in joules when the charge is in coulombs, voltage is in volts, and capacitance is in farads.