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Why is the current before and after a resistor exactly the same? I understand the same amount of charge that enters the resistor leaves, but current is defined to be charge per time. The way I understand it, resistors slow down the speed of electrons, so even though the same amount of charge that enters, leaves, the speed is different, so the current must be different. What is going on here?

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    $\begingroup$ You can't remove current from a circuit. Where would it go? Heck, in your "head model", it seems like either electrons would have to somehow magically disappear from the resistor, or pile up. $\endgroup$
    – Luaan
    Jul 10 at 17:15
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    $\begingroup$ Imagine a circular highway filled with cars. If you reduce the number of lanes to 1 at a point it affects the speed of all cars, because the effects of the bottleneck cascade through the whole system $\endgroup$
    – eps
    Jul 10 at 18:29
  • $\begingroup$ You've gotten a lot of good answers, but IMO the one by Farcher is the most succinct $\endgroup$
    – Bob D
    Jul 10 at 21:49
  • $\begingroup$ Related: simple Drude model $\endgroup$ Jul 12 at 8:10
  • $\begingroup$ This question can only be asked from the simplified model that assigns zero resistance to the conductors, which is plain wrong. All (regular) conductors are resistors. Electrons are slowed down in every wire by collisions, and yet they flow everywhere at the same speed. You probably know what's counteracting the "braking force" of the collisions. Now go from there and construct a reality where the braking force is at every point perfectly counteracted by the accelerating force. That must be the "reality" as it is. $\endgroup$ Jul 12 at 12:15

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Some teachers use the hydraulic analogy which likens electrons flowing through a wire to water flowing through a pipe. The analogy for a resistor is a narrow orifice or a partially closed valve that makes it harder to pump water through the pipe.

Q: Why is the amount of water flowing out of the valve the same as the amount of water flowing in to the valve?

A: It's because there's no place else that the water can go.

It's the same with electrons entering and leaving a resistor. There's no place else that they can go.

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  • $\begingroup$ For this analogy the water goes faster through the "resistor" than to the rest of the pipe and not slower. The OP's asumption is that the electrons slow down in the resistor. It cannot come from this model but maybe from another analogy. $\endgroup$
    – nasu
    Jul 10 at 16:24
  • $\begingroup$ @nasu, That is true. The analogy is far from perfect. Maybe a better analogy would be to a long freight train, with the electrons being like wagons in the train. I'm not sure what a "resistor" would look like in that case. Maybe a small change in elevation. $\endgroup$ Jul 10 at 19:40
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    $\begingroup$ @nasu It can be explained by this model. The issue is that electrical current is not analogous to water velocity, but water flow rate (eg. liters per second). The water flow rate is the same everywhere in the system, both inside and outside the valve. However the flow rate is slower with the value than if the valve weren't there. $\endgroup$ Jul 10 at 19:58
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    $\begingroup$ @nasu you could also imagine this analogy working if you made the pipe bigger but stuffed it with steel wool, so that the overall cross section was larger (so the water slows down), but the drag from the steel wool causes pressure loss. Or, you could imagine the pipe transitions into thousands of microtubes that overall have a much bigger cross section.... $\endgroup$
    – Joel Keene
    Jul 11 at 0:18
  • $\begingroup$ I just said that this model does not justify the OP's miconception about the electrons slowing down in the resistor. I did not say that the analogy, if used corectly and carefully is not helpful. But whole point of the OP's question starts from a false premise. The water model does not seem to have anything to do with it. He asks how can the current be the same given that the electrons slow down in the resistor. $\endgroup$
    – nasu
    Jul 11 at 0:29
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The OP premise is false. The resistor does not "slow down" the motion of electrons. Actually the drift velocity in some resistors can be tens or hundreds of times higher than in the metallic connecting wires. I cannot see anything that can be said that is going slower in the resistor. On both sides of the resistor the drift velocity is the same if the resistor does not change cross section.Also, is the same in the connecting wires before and after the resistor.

The magnitude of the current density is related to the drift velocity by $j=n\cdot e\cdot v_d$ where n is the carrier concentration and e is the charge of the carriers (electrons in metals). The difference between different materials is mostly in the carrier concentration as long as we don't have ions that can carry multiple elementary charges. Now, the current is the current density times the cross section of the resistor or wire. So, we cannot assume that we have the same current density all over the circuit. The current will be $I=nev_d A$ and we have two parameters (n and A) that will determine the distribution of the drift velocity in the circuit. But, for example, in a carbon resistor the value of n can be many orders of magnitude lower than in the metal of the connecting wires. So even if the cross section area is larger than that of the wire it is safe to expect that the drift velocity will be much higher in the resistor than in the wires, for the same current.

If the current in the circuit is increased the drift velocity increases in all parts of the circuit by the same factor so it is still larger in the resistor.

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  • $\begingroup$ Since the answer is correct if a bit terse, I'd point to the goold old Drude model, where the drift velocity is $e/f$, with $e$ the electron charge and $f$ a "friction" coefficient representing the action of the metallic cristal on the free electrons. No matter the intensity of the current or the voltage, this drift velocity remains the same (within the limits of usual electricity of course). $\endgroup$
    – Miyase
    Jul 10 at 18:43
  • $\begingroup$ The drift elocity depends on the current, for a gien conductor (or resistor, whaever you want to call it). You can see this from the formula forcurrent density: $j=nev_d $. The carrier density (n) is constant for a given material. e is the charge of the carrier so it is also independent of the current. So the drift elocity increases as you increase the current density. Which, for a given geometry (cross section) means it increases with the current. $\endgroup$
    – nasu
    Jul 11 at 0:07
  • $\begingroup$ Materials with higher specific resistance surely have slower drift velocity, ceteris paribus. You get high drift velocities only with small cross sections (and correspondingly better conducting materials). $\endgroup$ Jul 12 at 12:22
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    $\begingroup$ The drift velocity is not a material property. It depends on the current through the circuit. For the same current and same cross section, the current density is the same in the circuit. But in the zones with high resistance the drift velocity is higher, as the formula shows. The higher drift velocity has to compensate for the lower carrier density. Same is true in semiconductors. The drift velocitiesin in S.C. are orders of magnitudes higher than in metals, for comparable current densities. $\endgroup$
    – nasu
    Jul 12 at 13:25
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Actually the resistor is decreasing the drift velocity because of more collisions and interaction between electrons and the lattice. However, think about what happens if an electron suddenly slows down when entering the resistor. A still faster electron in the wire approaching the resistor will feel the negative charge of the slowed down electron in front of it, which repels the faster electron and slows it down while still being in the wire. Vice versa the faster electron will push the electron in front of it. An equlibrium forms and in summary, the resistor will slow down the drift velocity in the entire circuit.

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The (average) drift velocity of the electrons is constant because although collisions with the lattice slow them down and change their direction an electric field accelerates them between collisions.

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  • $\begingroup$ The drift velocity is an average, by definition. It is not the instantaneous velocity of an electron at a given time. $\endgroup$
    – nasu
    Jul 10 at 12:34
  • $\begingroup$ This implies (and the implication is correct) that within the resistor the field must be stronger, since there are more collisions. Lo and behold: Since most of the voltage drops at the resistor, indeed, in equilibrium state the field is much stronger there. Not only that, miraculously the field adjusts always exactly according to the relative resistances! $\endgroup$ Jul 12 at 12:10
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If the amount of charge per unit time leaving the resistor were different than the amount of charge per unit time entering it then the resistor would build up a net positive or negative charge over time (think of water flowing into and out of a tank). But it doesn’t. Therefore the amount of charge per unit time entering and leaving the resistor are the same.

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Let's be quantitative. Assume a 1 kΩ resistor. Apply a current of 1 mA, that is, one millicoulomb per second.. By Ohm's law, that means you have a drop of 1 V. Kirchoff's current law tells you that the current in must equal the current out, except...

When you use Kirchoff's current law, you are implicitly pledging to account for both displacement current and physical current. The connections at the ends of the resistor have capacitance to the environment, ~1 pF for a practical small resistor. So, if you apply 1 V to one end of the resistor, keeping the other end at the potential of the environment (ground), ~1 pC flows into displacement current before the circuit comes to its steady state. That flows into the 1 V end of the the resistor, but it bypasses the grounded end, going directly to the environment.

That's one billionth of the flow per second, and it doesn't continue to flow once the circuit is at a steady state. So, yes, you are right that the resistance causes the current to be different going in and out, but only very briefly. For many circuits, this makes no difference. For microwave circuits, with the current changing on time scales of nanoseconds or less, it's a serious consideration.

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Imagine a lot of people all walking in the same direction through a hallway that is just wide enough for all the people to walk undisturbed. Now someone will place a restriction in the hallway so less people fit through the hallway at that point. Since less people can fit through, the stream of people has to slow down before passing the obstacle. As a consequence the people walking behind them will also have to slow down. It takes a while to settle but eventually everyone will be walking with the same speed again. The new speed will of course be slower because the obstacle is restricting the flow.

If people would speed up or slow down only at a single point in the hallway it would mean a lot people would bunch up. So the only stable configuration is a constant speed throughout.

Here, the resistor is the same as the obstacle. It will cause the electrons to slow, which means the electrons in the entire wire will slow down. In general for a circuit that is wired in series the current will be the same throughout the entire circuit.

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  • $\begingroup$ Paradoxically, people even speed up in your "resistor"! That's the phenomenon nasu commented on in his answer: In smaller cross sections (one resistor type is a length of wire with a small cross section) the individual electrons/people must move faster to achieve the same "flow rate"/current. The narrow corridor section is a "good conductor" (no obstacles) where a narrow cross section facilitates the resistance. One could also leave the corridor width as-is and fill the "resistor part" with sand or mud; that would imitate a bad conductor and indeed slow down individual people/"electrons". $\endgroup$ Jul 12 at 12:29
  • $\begingroup$ @Peter-ReinstateMonica I guess it depends on how hard people are forced to move through the hallway. Do we assume the velocity of the stream at some distance from the obstruction is constant? In that case people would have to speed up to get through the obstruction. Do we assume people move at some maximal velocity? In that case people moving through the obstruction slow down and so does the entire hallway. The first case is similar to constant current supply and the last case is somewhat similar to a constant voltage supply. $\endgroup$ Jul 12 at 16:12
  • $\begingroup$ @Peter-ReinstateMonica I didn't even think of the first case so it's good that you point it out. $\endgroup$ Jul 12 at 16:13
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Why is the current before and after a resistor exactly the same?

For a combination of two reasons.

(1) The electrons flowing into the resistor have nowhere to flow but out of it.

(2) If more electrons were to flow into the resistor than out of it, a negative charge would quickly build up inside. The electrostatic force created by this charge would, in turn, "push backward" against inbound electrons and "push forward" the outbound ones. This would continue until the incoming flow of electrons equaled the outgoing flow again.

The same logic applies, with signs reversed, when fewer electrons flow into than out of the resistor. Either way, "inbound flow equals outbound flow of electrons" is the only stable equilibrium. Temporary deviations from this equilibrium are possible, but they're self-correcting as described in point #2.

I hope that helps.

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The velocity of the electrons is irrelevant to the operation of a resistor. The key feature of a resistor is a substantial voltage / current ratio: i.e. it takes energy to shove electrons through the resistor. Different materials & geometries may constrain the electrons in different ways, and may yield different drift velocities.

If a circuit element requires a lot of energy to shove an electron from one end of the element to the other, it will be a "resistor" - regardless of the speed at which the electrons move. The energy input to force the electrons through the material is commonly dissipated as heat:

  • Hotter metals have a higher resistivity than cool ones, since the electrons (and associated EM waves) scatter off the mobile atoms and the less-regular crystal structure. Different types of materials may have different temperature-dependence.
  • A narrow sample of a material will have a higher resistance than a thicker wire: Fine-gauge wires can thus function as resistors or even fuses. The electrons interfere with and repel each other, limiting their motion through the bottleneck.
  • Superconductors allow all the electrons to move together as a coordinated whole, without scattering off atoms or each other - but they needn't move at an especially high speed.
  • Different materials have different resistivities because their energy bands have different structures in physical and momentum space.
  • Magnetic and electric fields can perturb the atomic energy levels, changing the number of electrons which can travel simultaneously in the material without interfering with each other or the atomic lattice.
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  • $\begingroup$ If, under the same electric field, an electron moves fast through a material compared to other materials, that material is not a good resistor. Materials of high specific resistance always reduce the drift velocity compared to those with lower resistance. $\endgroup$ Jul 12 at 12:33
  • $\begingroup$ True, but the per-element electric field is set collaboratively with all the other elements in a circuit, by balancing the speed of electron flow, the number of flowing electrons, and the electric field. Holding the external field constant was not part of OP's description, nor is it part of typical circuit design. At best, the voltage will often be approximately constant, but in that case, the physical size of the resistor (and the number of charge carriers) varies. $\endgroup$ Jul 12 at 13:02
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I would say that we should look up the real mechanism of current flow Here's the brief view Excess and deficient charges appear to the sides of the wire having connected to the battery ( the wire connected to the positive end will have a positive charge gradient ) Now the battery does continuous work to maintain these potential difference We know that drift velocity is constant in a given circuit so there must be more charge gradient (potential gradient to set up electric field) near the resistor so the electric field intensity is more near the resistor so more number of electrons will enter the cross-section of resistor per second causing charge coming out of resistor still same

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