# What is drift speed in electricity and how is it related to the cross sectional area of the conductor?

In fluids dynamics, I learnt that as rate of flow is constant for an ideal fluid, Area(cross sectional area of the tube) x velocity is constant.I had a doubt whether the same relation exists between drift speed and cross sectional area of conductor.

The relation between drift speed and cross sectional area in a conductor with constant current and different cross section is the same as for an incompressible fluid flow in a tube with changing cross section, i.e., $$v_1 A_1=v_2 A_2$$ as long as the charge carrier density $n$ (e.g. electrons in a metal) is constant. This follows from the conduction current continuity for stationary currents: The conduction current $I$ stays constant along the conductor $$I=I_1=A_1 n_1 e v_1=A_2 n_2 e v_2=I_2$$ This follows from the law of charge conservation which in the stationary case reads $$\nabla · \vec j=-\frac {\partial \rho}{\partial t}=0$$ In its integral form this means that the closed surface integral $$\int_{\Sigma} \vec j d \vec a=0$$ where the current density is given by $\vec j=n e \vec v$.

In the case of a pipe, what is conserved is $$\frac{dm}{dt} = \rho A v.$$

In a wire, the current is conserved:

$$\frac{dq}{dt} = nAu$$ where $u$ is the electron drift speed and $n$ is the charge density.