Electrical Resistance

Current Density $\vec{J}$

$$\vec{J}=qn\vec{v}$$

$$I=\frac{dQ}{dt}={\int }_S\vec{J}\cdot {\vec{u}}_{N}dS$$

Resistivity

When an electron flows in a conductor, it will (on average) collide with a particle and zero its velocity every $2\tau$ seconds, where $\tau$ is called the lifetime. When colliding, $K_E$ is transferred to the material, heating it.

$$\begin{array}{rl}& v_{max}=2\tau a=2\tau \frac{q}{m}E\\ & ⟹{\vec{v}}_{\text{avg}}=\frac{q}{m}\vec{E}\tau \end{array}$$

And

$$\vec{J}=\sigma \vec{E}=\frac{\vec{E}}{\rho }$$

Where $\sigma$ is called electrical conductivity of the material, and $\rho$ is the electrical resistivity.

We therefore get (in a uniform wire of length $d$ and cross-sectional area $S$):

$$\begin{array}{rl}& I=\sigma ES\\ & \Delta V=Ed=\frac{d\rho }{S}I=IR\\ & R=\frac{d\rho }{S}=\frac{d}{\sigma S}\end{array}$$

Electrical conductance:

$$G=\frac{1}{R}=\frac{\Delta V}{I}$$

Its unit is a Siemen.

Heating Effect

$$P=I\Delta V=R{I}^2=\frac{\Delta V^2}{R}$$

Warning

Resistance varies with temperature!

Resistors

In Series:

$$R_{tot}=\Sigma R_i$$

In Parallel:

$$\frac{1}{R_{tot}}=\Sigma \frac{1}{R_i}$$