Conductors

• Conductors have mobile "free" charges – Applied Field forces mobile charge drift (conduction) current – Average drift velocity = (charge carrier mobility)x(electric field)
• Conductors under static conditions:
  • The entire bulk of the conductor is equipotential.
  • Electric fields are nulled within the conductor bulk and can only exist external to the conductor.
  • Charges can only exist on the surface and never within the bulk of the conductor.
• Conductors under dynamic conditions:
  • Charges flow in the body of the conductor causing an electric current
  • An internal electric field sustains the motion of the charge carriers to overcome the conductor's internal resistance due to the carriers' multiple collisions.
  • A potential difference appears across the terminals of the conductor as determined by the current flow/internal electric field.

Electric Current, Current Densities, and Resistance

Line, Surface, and Volume Current ($A$):$~I=dQ/dt$

Linear Density of Surface Current ($A/m$):$~\vec{K}=dI/d{{\ell }_{n}}{{\vec{a}}_{I}}$

Surface Density of Volume Current ($A/m$^$2$):$~\vec{J}=dI/d{{S}_{n}}~{{\vec{a}}_{I}}$

$I=\iint_{S}{\vec{J}\cdot \overrightarrow{dS}}$ --- $\left\{ {{I}_{closed~surface~S}} \right\}=-\frac{d{{Q}_{enclosed~by~S}}}{dt}$ --- $\nabla \cdot \vec{J}=-\frac{\partial {{\rho }_{v}}}{\partial t}~~$[$=0$ for statics]

$\vec{J}=\sigma ~\vec{E}$ --- $\sigma =\left[ \rho _{v}^{+}~{{\mu }^{+}}-\rho _{v}^{-}~{{\mu }^{-}} \right]=1/\rho $ --- $ρ~units~Ω.m$

${{R}_{AB}}=\frac{{{V}_{AB}}}{{{I}_{AB}}}=\frac{\mathop{\int }_{A}^{B}\vec{E}.\overrightarrow{d\ell }}{\iint\limits_{S}{\vec{J}.\overrightarrow{dS}}}=\frac{\mathop{\int }_{A}^{B}\vec{E}.\overrightarrow{d\ell }}{\iint_{S}{\sigma ~\vec{E}.\overrightarrow{dS}}}=1/\iint_{S}{\frac{1}{\mathop{\int }_{A}^{B}\frac{d\ell }{\sigma ~dS~}}}=\underset{A}{\overset{B}{\mathop \int }}\,1/\iint_{S}{\frac{\sigma ~dS}{d\ell }~=\frac{1}{{{G}_{AB}}}}$ --- ${{V}_{AB}}={{R}_{AB}}~{{I}_{AB}}$ [Ohm's Law]

Coaxial TL $\left( per~unit~axial~length \right)$: ${{R}_{z,~TL,~DC}}=\frac{1}{{{\sigma }_{i}}~\pi ~{{a}^{2}}~}+~\frac{1}{{{\sigma }_{o}}~\pi ~\left( {{c}^{2}}-{{b}^{2}} \right)~}$ --- ${{G}_{z,~TL}}=\frac{2\pi {{\sigma }_{d}}~~}{~ln\left( b/a \right)}$

Dielectrics

Dielectrics have bound charges (No mobile charges – No Conduction Currents).
Applied electric field causes bound charge polarization – dipole moments, ${{\vec{p}}_{d}}$

Polarization Vector, $\vec{P}=\underset{\Delta \text {v}\to 0}{\mathop{\lim }}\,\frac{\mathop{\sum }_{\Delta \text {v}}{{{\vec{p}}}_{d}}~}{\Delta \text {v}}=\chi ~{{\varepsilon }_{o}}\vec{E}$ --- $\vec{D}={{\varepsilon }_{o}}\vec{E}+\vec{P}=~{{\varepsilon }_{r}}~{{\varepsilon }_{o}}\vec{E}=\varepsilon ~\vec{E}$

Bound charge volumetric density:$~~{{\rho }_{\text {v}b}}=-\nabla \cdot \vec{P}$ --- Surface density: ${{\rho }_{Sb}}=\vec{P}.{{\vec{a}}_{S~}}$

Volumetric energy density: ${{w}_{e}}=\frac{1}{2}\left\{ \vec{D}.\vec{E} \right\}=\frac{1}{2}\varepsilon {{E}^{2}}~~~\left[ =\frac{1}{2}{{\varepsilon }_{o}}{{E}^{2}}~for~free~space \right]$

Energy stored: ${{W}_{e}}=\underset{vol}{\mathop \int }\,\frac{1}{2}\left\{ \vec{D}.\vec{E} \right\}d\text {v}=\underset{vol}{\mathop \int }\,\frac{1}{2}\varepsilon {{E}^{2}}d\text {v}~~~\left[ =\underset{vol}{\mathop \int }\,\frac{1}{2}{{\varepsilon }_{o}}{{E}^{2}}d\text {v}~~for~free~space \right]$

Capacitance

${{C}_{AB}}=\frac{{{\psi }_{AB}}}{{{V}_{AB}}}=\frac{\iint_{S}{\varepsilon ~\vec{E}.\overrightarrow{dS}}}{\mathop{\int }_{A}^{B}\vec{E}.\overrightarrow{d\ell }}=\iint_{S}{\frac{1}{\mathop{\int }_{A}^{B}\frac{d\ell }{\varepsilon ~dS~}}}=1/\underset{A}{\overset{B}{\mathop \int }}\,\frac{1}{\iint_{S}{\frac{\varepsilon ~dS}{d\ell }}~}$ --- $~{{Q}_{AB}}={{C}_{AB}}~{{V}_{AB}}$ --- ${{Q}_{i}}=\underset{j=1}{\overset{n}{\mathop \sum }}\,{{C}_{ij}}{{V}_{ij}}$

Parallel Plate capacitance: ${{C}_{AB}}=\frac{\varepsilon ~{{S}_{AB}}}{{{d}_{AB}}}$ --- Coaxial TL capacitance: ${{C}_{z,TL}}=\frac{2\pi ~\varepsilon }{\ln \left( {}^{b}/{}_{a} \right)}\left( per~unit~axial~length \right)$

Boundary Conditions

Dielectric- Dielectric BC:
$E_{1t}=E_{2t}$ --- $D_{1n}-D_{2n}=ρ_{s}$ --- If $ρ_{s}=0, tan~θ_{1}/ε_{1} = tan~θ_{2}/ε_{2}$ --- ${{\rho }_{Sb1}}=-\frac{{{\varepsilon }_{r1}}-1}{{{\varepsilon }_{r1}}}{{D}_{1n}}$ --- ${{\rho }_{Sb2}}=\frac{{{\varepsilon }_{r2}}-1}{{{\varepsilon }_{r2}}}{{D}_{2n}}$ --- ${{\rho }_{Sb}}={{\rho }_{Sb1}}+{{\rho }_{Sb2}}$

Conductor-Conductor:
Static: ${{E}_{1}}={{E}_{2}}=0$
Dynamic: ${{E}_{1t}}={{E}_{2t}}$ --- ${{J}_{1n}}\text{ }\!\!~\!\!\text{ }-{{J}_{2n}}\text{ }\!\!~\!\!\text{ }=-\frac{d{{\rho }_{s}}}{dt}$ --- If $\frac{d{{\rho }_{s}}}{dt}$$=0, tan~θ_{1}/σ_{1} = tan~θ_{2}/σ_{2}$

Conductor-Dielectric:
Static: ${{E}_{dt}}={{E}_{ct}}=0$ --- ${{E}_{cn}}=0$ --- ${{D}_{dn}}={{\rho }_{s}}$ --- ${{\rho }_{Sbt}}={{\rho }_{Sb1}}=-\frac{{{\varepsilon }_{r1}}-1}{{{\varepsilon }_{r1}}}{{\rho }_{s}}$
Dynamic:$\text{ }\!\!~\!\!\text{ }{{E}_{dt}}={{E}_{ct}}$ --- $E_{cn}=0$ --- ${{D}_{dn}}={{\rho }_{s}}$ --- ${{\vec{J}}_{c}}={{\sigma }_{c}}\text{ }\!\!~\!\!\text{ }{{\vec{E}}_{ct}}$

Resistors and Capacitors as Circuit Elements

${{i}_{AB}}={{G}_{AB}}~{{v}_{AB}}=\frac{1}{{{R}_{AB}}}{{v}_{AB}}$ --- ${{\bar{I}}_{AB}}={{G}_{AB}}{{\bar{V}}_{AB}}=\frac{1}{{{R}_{AB}}}{{\bar{V}}_{AB}}~$--- ${{P}_{AB,av}}=\frac{1}{2}{{G}_{AB}}~{{\left| {{{\bar{V}}}_{AB}} \right|}^{2}}=\frac{1}{2}{{R}_{AB}}~{{\left| {{{\bar{I}}}_{AB}} \right|}^{2}}$

${{R}_{series~combination}}=\underset{all~series}{\mathop \sum }\,{{R}_{individual~}}~~$--- $\frac{1}{{{R}_{parallel~combination}}}=\underset{all~parallel}{\mathop \sum }\,\frac{1}{{{R}_{individual~}}}$

${{i}_{AB}}={{C}_{AB}}\frac{d\left[ {{v}_{AB}} \right]}{dt}$ --- ${{\bar{I}}_{AB}}=j\omega {{C}_{AB}}~{{\bar{V}}_{AB}}$ --- ${{P}_{AB,av}}=0$ --- ${{W}_{e}}=\frac{1}{2}{{Q}_{AB}}~{{V}_{AB}}=\frac{1}{2}{{C}_{AB}}~V_{AB}^{2}=\frac{1}{2}~\frac{Q_{AB}^{2}}{{{C}_{AB}}}$

$\frac{1}{{{C}_{series~combination}}}=\underset{all~series}{\mathop \sum }\,\frac{1}{{{C}_{individual~}}}$ --- ${{C}_{parallel~combination}}=\underset{all~parallel}{\mathop \sum }\,{{C}_{individual~}}$