# 01. Concepts and Principles

- Page ID
- 10154

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## An Alternative Approach

The electric field surrounding electric charges and the magnetic field surrounding moving electric charges can both be conceptualized as information embedded in space. In both cases, the information is embedded as *vectors*, detailing both the magnitude and direction of each field. Moreover, when this information is "read" by other moving electric charges, the result is a *force* acting on the charge. These forces can be calculated to allow us to determine the subsequent motion of the charge.

Just as in mechanics, there is an alternative to this force-based approach to analyzing the behavior of electric charges. In this chapter I will define a new field, the *electric potential*, which surrounds every electric charge. This field differs from the electric field because it is a *scalar *field, meaning that this field has only a magnitude at every point in space and no associated direction. Moreover, the information in this field, when read by other electric charges, does not result in a force on the charge but rather determines the *electric potential energy* the charge possesses at that point in space.

Understanding how to calculate this new field, how this field relates to energy, and how this field is related to the electric field will be the focus of this chapter.

## A Gravitational Analogy

Rather than thinking in terms of the gravitional force and Newton's Second Law, an alternative way to examine mechanics scenarios is by using the concept of gravitational potential energy and the conservation of energy.

In the force approach, we envision a vector field surrounding the earth, regardless of whether a second mass is nearby to interact with this field. If a mass is present, the mass interacts with this field and feels a gravitational force. This idea is captured in the equation:

pic 1

In the energy approach, we can envision a scalar field, the *gravitational potential*, which is present regardless of whether a second mass is nearby to interact with this field. If a mass is present, the mass interacts with this field and has gravitational potential energy. Near the surface of the earth, the familiar expression for gravitational potential energy:

pic 2

can be thought of as the product of the mass of the object and this pre-existing gravitational potential, V_{G}:

pic 3

if we define the gravitational potential by:

pic 4

Although we didn't use the concept of gravitational potential while studying mechanics, it will prove to be a very useful concept in our study of electrical phenomenon. The general expression for gravitational potential, valid regardless of distance from a massive object, is:

pic 5

In summary, just as a mass will interact with the vector gravitational field as a force, a mass will interact with the scalar gravitational potential field as potential energy.

The situation is very similar for electrical phenomenon. We can envision a scalar field, the *electric potential*, which is present regardless of whether a second charge is nearby to interact with this field. If a charge is present, the charge interacts with this field and has *electric potential energy*. The electric potential, V_{E}, is defined by the relationship:

pic 6

where

- q is the source charge, the electric charge that creates the field,
- and r is the distance between the source charge and the point of interest.

This leads to an expression for the electric potential energy of:

pic 7

where

- q is the charge on the particle of interest, the charge that is interacting with the field,
- and V
_{E}is the net electric potential at the location of the particle of interest (created by all of the*other*charged particles in the universe).

We will typically leave the subscript off the electric potential and electric potential energy unless the possibility of confusion with the gravitational potential and potential energy are present.

## Relating the Electric Field and the Electric Potential

The electric field and the electric potential are not two, independent fields. They are two independent ways of conceptualizing the effect that an electric charge has on the space surrounding it. Just as problems in mechanics can be analyzed using a force-approach or an energy-approach, problems dealing with electrical phenomenon can be analyzed by focusing on the electric field or on the electric potential.

Additionally, just as it is sometimes necessary in mechanics to transfer between force and energy representations, it is sometimes necessary to transfer between the electric field and electric potential representations. The relationship between two fields can be understood by examining the expression for work, which relates force to transfer of energy.

Utilizing the dot product, the work donw in moving a particle from an initial point, i, to a final point, f, can be written as:

pic 8

where (pic 9) is an infinitesimal portion of the path along which the particle moves. You may also recall that the difference in potential energy between initial and final locations is defined as the opposite of the work needed to move the particles between the two points:

pic 10

Putting these two ideas together yields:

pic 11

We can now use this result to relate the electric potential and the electric field. Substituting in expressions for potential energy and force in terms of the fields that convey them leads to:

pic 12

In English, this final result states that the electric potential difference between any two points is defined as the negative of the integral of the electric field along a path connecting the two points. (I'm sure that doesn't seem like particularly *clear* English, but this idea will become more tangible once you get to work on some of the fun activities in this chapter.) The bottom line is that the electric potential can be determined by intergrating the electric field, and, conversely, the electric field can be determined by differentiating the electric potential.