The current is the same at any two points in a wire.

At a junction,

This is Kirchhoff’s junction law.

Conduction electrons move freely around the positive ions that form the atomic lattice.

An electric field causes a slow drift at speed $v_{\text{d}}$ to be superimposed on the rapid but random thermal motions of the electrons.

Collisions of electrons with the ions transfer energy to the atoms. This makes the wire warm and lightbulbs glow. More collisions mean a higher resistivity $\rho$ and a lower conductivity $\sigma$.

The drift speed is $v_{\text{d}}={\displaystyle \frac{e\tau }{m}}E$, where $\tau$ is the mean time between collisions.

The electron current is related to the drift speed by

where $n_{\text{e}}$ is the electron density.

An electric field $E$ in a conductor causes a current density $J=n_{\text{e}}e{v}_{\text{d}}=\sigma E$, where the conductivity is

The resistivity is $\rho =1\text{/}\sigma$.

A potential difference $\text{Δ}{V}_{\text{wire}}$ between the ends of a wire creates an electric field inside the wire:

The electric field causes a current in the direction of decreasing potential.

The size of the current is

where $R={\displaystyle \frac{\rho L}{A}}$ is the wire’s resistance.

This is Ohm’s law.