# Drift velocity of electrons with changing area

What would happen with the drift velocity of a cylindrical resistor's diameter increases, with a given voltage between its terminals? According to the expression:

\begin{align} R&=\rho\frac{L}{A}\\ I&=neAv_d\\ \Delta V &= IR\\ v_d&=\frac{\Delta V}{\rho L n e}\\ \end{align}

The resistivity does not change, neither does the length of the resistor nor the term $ne$ but the resistance does change as well as the current, so the area is eliminated from the expression. I wonder if the drift velocity would be the same after increasing the diameter or if my derivation is wrong.

• Possibly more interesting question (but not one you asked) - what happens to the drift velocity in a conductor of non-uniform cross section? I think there the answer is that it will increase in the narrower section - resistance is higher so field strength is higher, hence greater drift. Alternatively if you think of the electrons as a "gas", continuity says that if it has to travel through a constriction it must be flowing faster. Commented Oct 3, 2014 at 2:27
• I believe it depends, if you make infinitesimals changes in the leght, the cross section would be the same for each division and if for a given voltage the drift velocity is the same, then the changing cross sectional area wouldnt matter, but for a given current it would Commented Oct 3, 2014 at 12:02
• I was thinking more of non uniform changes - the wire changing into an hour glass shape. Now there is a section with higher resistance per unit length. In that region the drift velocity would be higher. Commented Oct 3, 2014 at 12:18

The drift velocity is the average velocity due to an applied electric field. In a conductor, electrons scatter around at the Fermi velocity but have a net zero average (i.e., equal scattering in all directions). When the electric field is applied, the electrons are given a small velocity in one direction. Thus, we can say, $$v_\textrm{drift}=\eta E$$ where $\eta$ is some constant. Since the electric field comes from a gradient in a potential, which changes as a function of the length of the bar, $L$. This approximates to $$v_\textrm{drift}\simeq\eta\frac{V}{L}$$ which is similar to what you have. Since there is no factor of $A$ in the latter equation (not in my $\eta$ here), then increasing the area (by increasing the diameter) should not change the drift velocity.