# Resistance and electricity

strong text In the answer to this question, it stated that the current would remain unchanged and I was wondering why the current would remain the same for the portion of the wire even though the radius has decreased which would restrict flow and therefore current. Any help much appreciated

• Compare with a water pipe. – Pieter Oct 11 '19 at 22:05
• I am curious about something so I'll give it a try in the comments rather than an answer for now. If you think about this in terms of say, a four-lane road full of traffic suddenly narrowing down to a two-lane road for some distance due to, e.g., road construction, does it not seem intuitive to you that the two-lane restriction not only reduces the flow of cars through the construction section but also through the wider, four-lane sections before and after? – Alfred Centauri Oct 12 '19 at 0:16
• @AlfredCentauri The traffic analogy is interesting. I would think that for a steady flow of traffic, on average the number of vehicles passing any point per unit time along the entire stretch of highway has to be the same. Since the 2 lane portion restricts the number of vehicles passing a point in that section of the road, the average speed of those vehicles needs to be greater than on the 4 lane stretch. We see this happen. Counter intuitively cars go faster in the restricted lanes. – Bob D Oct 12 '19 at 11:35

In terms of particle electrons, consider a cross section of the wire. The number of electrons passing any cross section per unit time is the same - assuming that the electrons are not pooling up anywhere. For this to occur, the density of the electrons in the thinner section has to be greater than in the thicker section. Think in terms of the flow lines of the electrons down the wire - they have to bunch up to get through the thin section. Hence, the current density is greater in the narrow section. And heating is related to current density.

Alternatively, think of it as three resistors. A big resistance between two smaller resistances, in series. The current is conserved, so the current is the same in each resistor because it has no where to go to or come from other than the other resistors. The (linear) heating is proportional to resistance for a given current.

I was wondering why the current would remain the same for the portion of the wire even though the radius has decreased which would restrict flow and therefore current.

Electric current $$i(t)$$ through a surface is defined as the rate of charge transport through that surface, or

$$i(t)=\frac{dq(t)}{dt}$$

where $$q(t)$$ denotes instantaneous charge.

If the current through the "restricted" (decreased radius) cross sectional surface of wire was less than the current through the cross sectional surface of the wire before the restricted section, then charge would build up (accumulate) in the non restricted section.

Hope this helps