Electric Charge, $Q$

Positive and negative - Measured in $Coulombs~(C)$
Electron charge:$~e=-1.6021765\times {{10}^{-19}}~C$
Universe is neutral - Default state of a material body is neutral

Charge Distributions

Point charge (infinitesimally small volume)
Line distribution – linear density ${{\rho }_{\ell }}$ $C/m$
Surface distribution – surface density ${{\rho }_{s}}$ $C/m^{2} $
Volume distribution – volume density ${{\rho }_\text{v}}$ $C/m^{3}$

Electric Flux, $\psi $

Flux equals charge producing the Flux $(Coulombs)$
Emerges out of Positive Charge (source)
Lands on Negative Charge (sink)
${{\psi }_{Closed~surface}}=Total~{{Q}_{enclosed~by~the~surface}}$

Flux Density Vector, $\vec{D}$

$D={{\left\{ d\psi /dS \right\}}_{max}}=d\psi /\left( d{{S}_{normal}} \right)~$$C/m^{2}$
Vector direction along flux lines
$d{{\psi }_{dS}}=\vec{D}\cdot \overrightarrow{dS}~~~and~~~{{\psi }_{s}}=\iint_{s}{d\psi }=\iint_{s}{\vec{D}\cdot \overrightarrow{dS}}$

Gauss' Law (Integral Form)

$\left\{ {{\psi }_{closed~surface~S}} \right\}=\unicode{x222F}_s~ \vec{D}\cdot \overrightarrow{dS} =\iiint_{enclosed~volume}{{{\rho }_\text{v}}d\text{v}}$

Gauss' Law (Point Form)

$\vec{\nabla }\cdot \vec{D}={{\rho }_\text {v}}\left( position \right)$
$\vec{\nabla }\cdot \vec{D}=\left[ \frac{\partial }{\partial x}{{D}_{x}}+\frac{\partial }{\partial y}{{D}_{y}}+\frac{\partial }{\partial z}{{D}_{z}} \right]$
$\vec{\nabla }\cdot \vec{D}=\left[ \frac{1}{\rho }\frac{\partial }{\partial \rho }\left( \rho {{D}_{\rho }} \right)+\frac{1}{\rho }\frac{\partial }{\partial \varphi }{{D}_{\varphi }}+\frac{\partial }{\partial z}{{D}_{z}} \right]$
$\vec{\nabla }\cdot \vec{D}=\left[ \frac{1}{{{r}^{2}}}\frac{\partial }{\partial r}\left( {{r}^{2}}{{D}_{r}} \right)+\frac{1}{r~sin\theta }\frac{\partial }{\partial \theta }\left( sin\theta ~{{D}_{\theta }} \right)+\frac{1}{r~sin\theta }\frac{\partial }{\partial \varphi }{{D}_{\varphi }} \right]$

The Divergence Theorem

$\unicode{x222F}_s ~\vec{D}\cdot \overrightarrow{dS} =\iiint_{volume~enclosed~by~S}{\vec{\nabla }\cdot \vec{D}~d \text {v}}$
$\unicode{x222F}_s ~\vec{A}\cdot \overrightarrow{dS} =\iiint_{volume~enclosed~by~S}{\vec{\nabla }\cdot \vec{A}~d \text {v}}$

Application of Gauss' Law Integral Form

Addendum
Evaluate Flux Density Vector

• ONLY for certain symmetrical charge distributions and combinations there of
  • Charges distributed with spherical symmetries (no variations with $\theta ~or~\varphi $):
    - Point charge
    - spherical surface charge
    - spherical shell charge
    - spherical volume charge
  • Charges distributed with cylindrical symmetries (no variations with$~\varphi ~or~z$):
    - Line charge
    - Cylindrical surface charge
    - Cylindrical shell charge
    - Cylindrical volume charge
  • Charges distributed with planar symmetries (no variations with$~x~or~y$):
    - Planar surface charge
    - Planar "slab" of charge
• Predict flux distribution (map) based on charge symmetry
• Choose appropriate Closed Hypothetical Gaussian Surfaces, HGS, with surface(s) either normal or tangential to expected flux lines
• Perform the closed surface integration $\unicode{x222F}_s ~\vec{D}\cdot \overrightarrow{dS} $ with the integrand $D$ factored outside the integration
• Equate the flux density integral to the charge enclosed by HGS, $\iiint_{HGS}{{{\rho }_\text {v}}d\text {v}}$
• Evaluate $\vec{D}$

Evaluate Charge

Total enclosed in a volume, given flux density distribution
${{Q}_{vol}}=\unicode{x222F}_{S~enclosing~vol} \vec{D}\cdot \overrightarrow{dS} $

Applications of Gauss' Law Point Form

Evaluate Flux Density Vector

• Given charge density distribution
• Divide space in regions as per the charge distribution
• Solve $\vec{\nabla }\cdot \vec{D}={{\rho }_\text {v}}\left( position \right)$ differential equation for each region
• Boundary conditions determine integration constants (Continuity of flux density except at singularities)
• Example: Spherical volume distribution, $ρ_\text {v}$$(r<b)$
  
  $\vec{D}=\left\{ \begin{matrix} {{{\vec{a}}}_{r}}~\left( 1/{{r}^{2}} \right)~\int ~{{r}^{2}}{{\rho }_\text {v}}\left( r \right)~dr~~~~~~~~~0{<}r{<}b \\ {{{\vec{a}}}_{r}}~\left( 1/{{r}^{2}} \right)\overset{r=b}{\mathop \int }\,{{r}^{2}}{{\rho }_\text {v}}\left( r \right)~dr~~~~~~~~~~b{<}r{<} \infty \\ \end{matrix} \right.$

Evaluate charge

• Evaluate volumetric charge density distribution, given flux density distribution.
  ${{\rho }_\text {v}}\left( position \right)=\vec{\nabla }\cdot \vec{D}$
• Total charge in a volume, given flux density distribution.
  ${{Q}_{vol}}=\iiint_{vol}{{{\rho }_\text {v}}~d\text {v}}=\iiint_{vol}{\left( \vec{\nabla }\cdot \vec{D} \right)d\text {v}}$