Chapter V - Electric Force, Field, Energy, and Potential
Chapter Outcomes
- Calculate the mutual force between $2$ point charges.
- Calculate the electric field distribution generated by arbitrary charge distributions using the electric field incrementation method.
- Derive analytical expressions for symmetric charge distributions with finite dimensions which was not possible using the HGS method described in chapter 4.
- Describe the physical process of the development of electric potential in a system of charges.
- Calculate the spatial distribution of electric potential resulting from a given system of charges using the potential field incrementation method.
- Derive the electric field for a given electric potential distribution and vice versa.
- Calculate the energy density in a given system of charges for a given electric potential distribution.
Complete Chapter
Chapter Overview
In this chapter:
Coulomb's Force Law in Free Space
${{\vec{F}}_{2}}=\frac{{{q}_{1}}~{{q}_{2}}}{4\pi ~{{\varepsilon }_{o}}~R_{12}^{2}}{{\vec{a}}_{12}}$ $~and~$ ${{\vec{F}}_{1}}=\frac{{{q}_{2}}~{{q}_{1}}}{4\pi ~{{\varepsilon }_{o}}~R_{21}^{2}}{{\vec{a}}_{21}}$, ${{\varepsilon }_{o}}=8.854187817\times {{10}^{-12}}~\simeq ~\left( 1/36\pi \right)\times {{10}^{-9}}~F/m$
Electric Field Intensity Vector
${{{\vec{E}}}_{{p}}}={{{\vec{F}}}_{{normalized }~{ }\left( {per }~{ unit }~{ charge} \right)}}=\frac{1}{{{{ }\varepsilon{ }}_{{o}}}}{{{\vec{D}}}_{{p}}}{ }~{ }~{ }~{ and }~{ }~{ }~{ }{{{\vec{D}}}_{{p}}}={{{ }\varepsilon{ }}_{{o}}}{ }~{ }{{{\vec{E}}}_{{p}}}$
Electric Field "Incrementation" Scheme:
${{\vec{E}}_{p}}=\underset{Point~Charges}{\mathop \sum }\,\frac{q}{4\pi ~{{\varepsilon }_{o}}~R_{q-p}^{2}}{{\vec{a}}_{{{R}_{q-p}}}}+\underset{Linear~Densities}{\mathop \sum }\,\mathop{\int }^{}\frac{{{\rho }_{\ell }}~d\ell }{4\pi ~{{\varepsilon }_{o}}~R_{dQ-p}^{2}}{{\vec{a}}_{{{R}_{dQ-p}}}}+$
$\underset{Surface~Densities}{\mathop \sum }\,\mathop{\int }^{}\frac{{{\rho }_{S}}~dS}{4\pi ~{{\varepsilon }_{o}}~R_{dQ-p}^{2}}{{\vec{a}}_{{{R}_{dQ-p}}}}+\underset{Volumetric~Densities}{\mathop \sum }\,\mathop{\int }^{}\frac{{{\rho }_\text{v}}~d\text{v}}{4\pi ~{{\varepsilon }_{o}}~R_{dQ-p}^{2}}{{\vec{a}}_{{{R}_{dQ-p}}}}$
Energy of a System of Charges
${{W}_{e,~q,Q}}=\sum\nolimits_{q,Q}{\underset{\infty }{\overset{p}{\mathop \int }}\,{{{\vec{F}}}_{external}}\cdot \overrightarrow{d\ell }}=\sum\nolimits_{q,Q}{\underset{p}{\overset{\infty }{\mathop \int }}\,{{{\vec{F}}}_{Coulomb}}\cdot \overrightarrow{d\ell }}$
${{W}_{e,~q,Q}}=\sum\nolimits_{{q}'s}{q~\underset{p}{\overset{\infty }{\mathop \int }}\,\vec{E}\cdot \overrightarrow{d\ell }}+\underset{{Q}'s}{\mathop \int }\,dQ~\underset{p}{\overset{\infty }{\mathop \int }}\,\vec{E}\cdot \overrightarrow{d\ell }$
${{W}_{e,{{q}_{1}}-{{q}_{2}}}}=\frac{{{q}_{2}}~{{q}_{1}}}{4\pi {{\varepsilon }_{o}}~{{R}_{12}}}$
${{W}_{e,{{q}_{1}}~\ldots ~{{q}_{n}}}}=\frac{1}{2}\underset{j=1}{\overset{n,~j\ne i}{\mathop \sum }}\,{{q}_{j}}\underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{~{{q}_{i}}}{4\pi {{\varepsilon }_{o}}~{{R}_{ij}}}=\frac{1}{2}\underset{j=1}{\overset{n,~j\ne i}{\mathop \sum }}\,{{q}_{j}}~\left[ {{V}_{j}} \right]$
${{W}_{e,Q}}=\frac{1}{2}\underset{all~Q}{\mathop \int }\,\left\{ d{{Q}_{j}}~\left[ \underset{all~Q}{\mathop \int }\,\left( \frac{~d{{Q}_{i}}}{4\pi {{\varepsilon }_{o}}~{{R}_{ij}}} \right) \right] \right\}=\frac{1}{2}\underset{vol}{\mathop \int }\,\left\{ {{\rho }_\text{v}}~\left[ V \right] \right\}d\text{v}$
${{W}_{e}}=\underset{vol}{\mathop \int }\,\frac{1}{2}\left\{ \vec{D}.\vec{E} \right\}~d\text{v}=\underset{vol}{\mathop \int }\,\frac{1}{2}{{\varepsilon }_{o}}{{E}^{2}}~d\text{v}$
${{w}_{e}}=\frac{d{{W}_{e}}}{d\text{v}}=\frac{1}{2}\left\{ {{\rho }_\text{v~}}V \right\}=\frac{1}{2}\left\{ \vec{D}.\vec{E} \right\}=\frac{1}{2}{{\varepsilon }_{o}}{{E}^{2}}$
Electrostatic Potential
${{V}_{p}}=\frac{\Delta {{W}_{e}}}{\Delta Q}=\underset{Position}{\overset{\infty }{\mathop \int }}\,\vec{E}\cdot \overrightarrow{d\ell }~~and~~{{V}_{BA}}=~{{V}_{B}}-{{V}_{A}}=\underset{B}{\overset{A}{\mathop \int }}\,\vec{E}\cdot \overrightarrow{d\ell }=-\underset{A}{\overset{B}{\mathop \int }}\,\vec{E}\cdot \overrightarrow{d\ell }$
Electric Potential "Incrementation" Scheme:
${{V}_{p,~~system~of~charges}}=\underset{all~discrete~{q}'s}{\mathop \sum }\,\frac{q}{4\pi {{\varepsilon }_{o}}~{{R}_{q-p}}}~or~=\underset{all~continuous~{Q}'s}{\mathop \int }\,\frac{dQ}{4\pi {{\varepsilon }_{o}}~{{R}_{dQ-p}}}$
$\vec{E}=-\nabla V=-grad~V$
Cartesian, Cylindrical and Spherical forms for the gradient are:
$\nabla =\left[ \frac{\partial }{\partial x}{{{\vec{a}}}_{x}}+\frac{\partial }{\partial y}{{{\vec{a}}}_{y}}+\frac{\partial }{\partial z}{{{\vec{a}}}_{z}} \right]$--$\nabla =\left[ \frac{\partial }{\partial \rho }{{{\vec{a}}}_{\rho }}+\frac{1}{\rho }~\frac{\partial }{\partial \varphi }{{{\vec{a}}}_{\varphi }}+\frac{\partial }{\partial z}{{{\vec{a}}}_{z}} \right]$--$\nabla =\left[ \frac{\partial }{\partial r}{{{\vec{a}}}_{r}}+\frac{1}{r}~\frac{\partial }{\partial \theta }{{{\vec{a}}}_{\theta }}+\frac{1}{r~sin\theta }~\frac{\partial }{\partial \varphi }{{{\vec{a}}}_{\varphi }} \right]$
Conservative Electrostatic Field:
The Electrostatic Potential is single valued and that value is unique and is solely determined by the location and not the path taken to get there
$\mathop{\oint }^{}\vec{E}\cdot \overrightarrow{d\ell }=0$
Deriving Expressions for $\vec{E}$ and V for a Few Famous Examples
Finite and Infinite Line of Charges
Addenda: Ring of Charges - Disc of Charges - Electric Dipole