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I was informed that in a circuit, the current will stay the same, and this is why the lightbulbs will light up (because in order for the current to stay the same, the drift speed of the electrons need to get faster). However, I do not understand why the current needs to stay the same from point to point.

Why does the current stay the same from point to point in a circuit?

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Electrons are present everywhere in an electric circuit. When a potential difference is applied to the circuit, an electric field is set up throughout the circuit, almost with the speed of light. Electrons in every part of the circuit begin to drift under the influence of this electric field and a current begins to flow in the circuit immediately.

You have to note here that if the potential difference you're applying is constant as with a D.C battery the electric field remains constant, and thus the current remains constant.

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The person with lowest number of  \$ $ \$  was the only one who wrote something on topic. Not so surprising for me… – Incnis Mrsi Oct 25 '14 at 7:43

The original poster explained unambiguously that the question is about spatial dependence, not temporal, i.e. a partial case of the Kirchhoff’s current law was interested about.

Tiny electrons are not things providing high-level integrity of electric circuits; they are only charge carriers. It is a good metaphor: when you send a mail, you do not care about carriers and their speed; you care about integrity of mail and terms of delivery. For an explanation how this integrity is ensured along a non-branching circuit read Why isn't resistance proportional to distance squared

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The current flows in the wires of a circuit, carried by the movement of electrons. At any particular time, if you measure the current at two different places in the same wire, you will get the same reading. This is Kirchhoff's current law in action: all the current entering a point in a circuit must leave that point. Any point on your wire can be seen as a 'node' with two current paths leaving it.

One way to look at this is to think of the flow of current through a wire similar to water flowing in a pipe. (I actually detest this analogy, but it's simple enough.) If you measure the flow at two different points in the same pipe, the readings will be the same, as long as you are looking at the volume of water passing per unit time, in whatever units you want, cubic meters per second, for example. Unless there's a leak, the water has to go through the pipe. The same goes for the electrons: unless there's a fault or a short, the electrons have to go through the wire.

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Look into Kirchoff's current law.

It is simply charge conservation. If a current of 2 C/s flows in, then 2 C/s must flow out. Because charge is not accumulated anywhere. Inflow must equal outflow at every single point in a steady circuit. Mathematically:

$$\sum I=0 \quad \Leftrightarrow \quad \sum I_{in}-\sum I_{out}=0$$

  • When you in a series circuit have only one path, then the whole current has to leave along this path. So the current entering the next point on the path is still the original current. Through the entire circuit, the current is therefore the same at any point.
  • In a parallel circuit a path might split into two, so the incoming current can be split and smaller portions take each path - in total the sum must be zero at any point, so now the current along each path is not necessarily the same. This is how current flow can be controlled.
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