PX157 - B13b - resistance

resistivity

ohm's law:

\vec{E} = ρ \vec{J}

where, $ρ =$ electrical resistivity

units: $V m A^{- 1}$ or $Ω m$
expect: $ρ \propto | {\vec{v}}_{e^{-}} - {\vec{v}}_{i o n} | < ν_{e i} >$ , where $ν_{e i} =$ collision frequency
electrical conductivity: $σ = \frac{1}{ρ}$
- units: $Ω^{- 1} m^{- 1}$

\begin{aligned} \frac{V}{L} & = ρ \frac{I}{A} \\ ∴ V & = \frac{ρ L}{A} I = R I \end{aligned}

where, $R =$ resistance

resistors in series

same current, $I$ , flowing though both the resistors
potential drops across $R_{1}$ and $R_{2}$ are $V_{1}$ and $V_{2}$ respectively:

\begin{aligned} V_{1} & = R_{1} I \\ V_{2} & = R_{2} I \\ V & = V_{1} + V_{2} \\ ∴ V & = (R_{1} + R_{2}) I \end{aligned}

the equivalent resistance is: $R_{e q} = R_{1} + R_{2}$
if both resistors have the same $ρ$ and $A :$

R_{e q} = \frac{ρ}{A} (L_{1} + L_{2})

resistors in parallel

same potential drop, $V$ , across both the resistors:

\begin{aligned} I_{1} & = \frac{V}{R_{1}} \\ I_{2} & = \frac{V}{R_{2}} \end{aligned}

total current:

I = I_{1} + I_{2} = V (\frac{1}{R_{1}} + \frac{1}{R_{2}})

the equivalent resistance is: $\frac{1}{R_{e q}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$
if both resistors have the same $ρ$ and $L :$

\frac{1}{R_{e q}} = \frac{1}{ρ L} (A_{1} + A_{2})

if both resistors are the same:
- in series: $R_{e q} = 2 R ⟹ I = \frac{V}{2 R}$
- in parallel:

\frac{1}{R_{e q}} = \frac{2}{R} ⟹ \frac{2 V}{R}