A Restatement of Kirchoff’s Current Law

A Restatement of Kirchoff’s Current Law

Kirchoff’s Current Law is often phrased as all the currents entering a node sum to 0.  So, in the figure below showing a node of a circuit where three wires come together, i1 + i2 + i3 = 0.  If any individual current entering a node is positive, then at least one other of the currents must be negative (meaning that the current is traveling in the opposite direction – exiting the node rather than entering it).


Keep in mind that electric current is just movement of electrons.  So all Kirchoff’s Current Law says is that all the electrons entering a node in a circuit must equal all the electrons exiting the node.  Similarly, all the protons entering a node must equal all the protons exiting the node.  This seems perfectly reasonable.  Let’s consider what happens when we apply Kirchoff’s Current Law to another system.

For any object in motion, momentum is conserved (Newton’s Laws of Physics).  Suppose you were in outer-space, and you were holding a rock.  If you threw the rock, it would travel through space in the direction you threw it.  However, since momentum is conserved, you travel in the direction opposite to the rock.  So the rock goes one way, and you go the other.  You are much larger, though, so you will travel slower than the rock you threw.

This is the main idea behind an ion-thruster, a deep space engine.  But where does the engine get the rocks?  It doesn’t use rocks; instead, it uses ions (electrically charged molecules).  The ion thruster shoots ions out in one direction, which propels the ship in the other direction.  Each individual ion causes very little acceleration for the ship.  But velocity is the integration of acceleration; so after the ion thruster is turned on for several years, the ship’s velocity can get very high.  (The main figure of this article is an artist’s depiction of an ion thruster [2].)  NASA just demonstrated an ion thruster that operated continually for over five years [3].

Let’s look at this system from the point of view of Kirchoff’s Current Law.  If we think of the entire ship as the node, then there are ions leaving that system.  Suppose the ion thruster is ejecting positively charged ions out; then the ship/node has a net loss of protons.  Doesn’t this violate Kirchoff’s Current Law?

The answer is, of course, that it doesn’t.  But it’s because we weren’t careful when we stated the law.  If you re-read my first sentence, I wrote that Kirchoff’s Law “is often phrased as …”  But what it’s often phrased as and what it states are two different things.  A complete statement of Kirchoff’s Current Law is

In the lump element model of a circuit, all the currents entering a node sum to 0.

We (humans) almost always leave off the conditional when stating the law.  And what is the lump element model of a circuit?  Part of its definition is that charge is conserved [4].  With this understanding, we could re-phrase the complete version of Kirchoff’s Current Law as follows:

When charge is conserved then charge is conserved.

It’s hard to argue with that.


[1]  https://en.wikipedia.org/wiki/Ion_thruster

[2]  https://sservi.nasa.gov/articles/how-will-nasas-asteroid-redirect-mission-help-humans-reach-mars/

[3]  https://www.extremetech.com/extreme/144296-nasas-next-ion-drive-breaks-world-record-will-eventually-power-interplanetary-missions

[4]  https://en.wikipedia.org/wiki/Lumped_element_model



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