Kirchoff's Laws

Created by Aditya Kuntamukkula - Spring 2018

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The node (or junction) rule states that the current flowing in is equal to the current that flows out. I₁ + I₄ = I₂ + I₃ + I₅

Kirchoff's Laws are two rules used to "solve" simple circuits, or find out different values for the different components involved in the circuit.

Kirchoff's Node Rule, also known as Kirchoff's First Law or Kirchoff's Junction Rule, further exercises the law of Conservation of Charge and states that if current is constant, all the current that flows through one junction must be equal to all the current that flows out of the junction. This rule can be applied both to conventional and electron currents. This rule is not a fundamental principle, but rather a consequence of the fundamental principle of conservation of charge and the definition of steady state.

Kirchoff's Second Law, also known as Kirchhoff's Loop Rule or Kirchhoff's Voltage Law states that the sum of potential differences around a closed circuit is equal to zero. More simply, in a completed circuit, the voltages around a loop will sum to 0. This is because voltage is just energy per unit charge, and both energy and charge are conserved by fundamental laws.

Note that is only true when the magnetic field is neither fluctuating nor time varying. If a changing magnetic field links the closed loop, then the principle of energy conservation does not apply to the electric field, causing the Loop Rule to be inaccurate in this scenario.

Kirchoff's Node Rule

The node rule states that at any junction in an electrical circuit, the amount of current flowing into the junction is equal to the amount of current flowing out of the junction in steady state.

In the steady state, for many electrons flowing into and out of a node,

electron current: [math]\displaystyle{ net\ i_{in} = net\ i_{out}, }[/math] where [math]\displaystyle{ i = nA\mu }[/math][math]\displaystyle{ E }[/math]
conventional current: [math]\displaystyle{ net\ I_{in} = net\ I_{out}, }[/math] where [math]\displaystyle{ I = |q|nA\mu }[/math][math]\displaystyle{ E }[/math]

Conservation of Charge

This rule is an application of the conservation of electric charge, basically that charge within a circuit cannot be created or lost. During the flow process around the circuit, there is no loss of any charge, thus the total current in any cross-section of the circuit is the same. If there are no nodes in the loop, the conventional current is the same throughout the loop.

A Mathematical Model

The node rule can be stated as:

[math]\displaystyle{ \sum \Delta{I} = 0 }[/math]
where [math]\displaystyle{ I }[/math]stands for the current of the individual parts or wires in a circuit and the sign of the current that flows into the junction is opposite of the current that flows out of the junction. This simply supports the idea that energy cannot be created or destroyed because the total current must remain equal regardless of the path it takes.
[math]\displaystyle{ \sum_{k=1}^n I_k = 0 }[/math]
where [math]\displaystyle{ n }[/math] is the total number of branches with current flowing towards or away from the node considering the direction due to signed quantity assigned to the current.
[math]\displaystyle{ \sum_{k=1}^n \tilde{I_k} }[/math] = 0
for complex currents.

A Computational Model

- In an electric circuit in series, electrons flow from the negative end of a power source, creating a constant current. This current remains consistent at each point in the circuit in series. Sometimes, a circuit is not simply one constant path and may include parts that are in parallel, where the current must travel down two paths such as this:
- File:Noderule.jpg
- In this case, when the current enters a portion of the circuit where the items are in parallel, the total amount of current in must equal the total amount of current out. Therefore, the currents in each branch of the parallel portion must sum up to the amount of current at any other point in series in the circuit.
- People also call this the "Junction Rule"
- Another important point is that this comes from the Kirchoff's Circuit Laws

Examples

Ex. 1

Figure 1 displays a node in a circuit. I₁ is equal to 10 amps. I₂ is equal to 4 amps. What is I₃?

The current flowing into the node: [math]\displaystyle{ I_1 = 10A }[/math]
The current flowing out of the node: [math]\displaystyle{ I_2 + I_3 }[/math]
We know that the current flowing in must equal the current flowing out, so [math]\displaystyle{ 10A = 4A + I_3 }[/math]
Therefore [math]\displaystyle{ I_3 }[/math] must equal 6A.

Ex. 2

In Figure 2, I₁ equal 23 amps, I₂ equals 5 amps and I₃ equals 42 amps. What is I₄?

The current flowing into the node: [math]\displaystyle{ I_1 + I_2 = 23A + 5A = 28A }[/math]
The current flowing out of the node: [math]\displaystyle{ I_3 + I_4 = 42A + I_4 }[/math]
Applying the node rule by using substitution, [math]\displaystyle{ I_4 = -14A }[/math]
But how could we get a negative current? Getting a negative current is physically (no pun intended) impossible, but when our current comes out negative, it simply lets us know that we guessed the direction of I₄ incorrectly. We assumed that the unknown current flows out instead of into the node when, in fact, the current flows into the node.

Limitations

Time-Varying Currents

Kirchhoff's law is based off the conservation of charge along with the nature of conductors. This law assumes that current will immediately flow from one end of the conductor to the other, which may not be true for time-varying currents, especially with higher frequencies.

Regions vs Circuits

Throughout a region, the charge can vary and be non-uniform, unlike in a wire. According to the law of conservation of charge, the only way to have a non-uniform charge density is if there is a net flow of current in or out of the region, which clearly violates the Node Rule. Therefore the node rule cannot be applied to regions with non-uniform charge densities. When looking at a junction in an electric circuit, we are looking at a point and therefore the point (which is infinitesimally small) must have a uniform charge distribution. In general, wires should have a uniform charge distribution across their length, because they are conductors and allow for the movement of charge.

Non-Steady State

The node rule can only be applied to the steady state. Thus, when considering the capacitor, the node rule is applied to the capacitor as a whole not just one plate of the capacitor.

Other Topics

Solving Circuits

In order to solve circuits, we must first define a couple characteristics of circuits.

The voltage of a (perfect, or non-resistive) wire is equal to its length (in meters) times its electric field.

The voltage of an ohmic resistor (including a resistive wire) is equal to current times resistance.

The voltage of a capacitor is equal to the charge on the capacitor divided by its capacitance.

There are other characteristics and equations that may be useful, but these two are the most important and used. If you are confused by any of the components of circuits mentioned above, visit the "Components" page. Below is a circuit. Using the node and loop rules we will find the current at points a, b, c, d, and e, and the charge on the capacitor after the switch has been closed for a very long time.

First you must realize that when the capacitor is fully charged, no current will flow through the capacitor, which is an important characteristic of a capacitor. From this we can say that the current through point c is 0.

Using the node rule, we can see that the current through resistor 1 and resistor 2 must be the same because, since no current flows through the wire connected the capacitor, all of the current must flow through one loop containing both resistors. So the current at a, b, d and e must all be the same. Due to the loop rule, the emf of the battery must be equal to I*R₁ + I*R₂. Therefore, I = emf/(R₁ + R₂); this is the current through points a, b, d and e. Using the loop rule, we can look at the loop containing the capacitor and resistor 2. We know the voltage in resistor 2 is equal to I*R₂. We know the voltage of the capacitor is equal to its charge divided by C (its capacitance). Because of the loop rule, we know that these two voltages must be equal, so I*R₂ = Q/C. Therefore, Q = I*R₂*C. Replacing I with the current we found above, Q = (emf/(R₁ + R₂))*R₂*C. As you can see, in order to solve this circuit, we had to use the node rule. In fact, we used the node rule at the very beginning of solving this circuit and there was no way possible we could have solved this problem without the node rule.

Here is a link to a helpful video explaining the node rule with the following circuit and the case where the switch has just been closed so the capacitor is not fully charged: https://youtu.be/nyYA0d7rQzE

History

Gustav Kirchhoff was a German physicist who lived during the 19th century. There are many equations and laws named after him that he helped to discover, one of them being the node rule. His circuit laws (the node rule and loop rule) were the first laws that he conceived, and he actually did this during his years in school and later wrote his doctoral dissertation on them. He created these laws in a time where electricity was a fairly new concept and was not widely used. His other achievements included coining the term "black body" radiation in 1862, which would later lead to important findings in the field of thermal radiation. In addition to his circuit laws, he is also known for his law of thermochemistry and three laws of spectroscopy, the latter of which helped lead to quantum mechanics.

Connectedness

The Node Rule is connected to a lot of other topics in physics. The loop rule is the most important one, as the node rule and loop rule in conjunction allow us to solve circuits. The node rule is also connected to other concepts such as voltage, current and electricity. The Node Rule is also supports the law of conservation of energy because in essence, current is simply the flow of electric charge, and since you cannot (at least as of now) create energy or electrons out of nothing, everything that is put into the system must come out somehow. Therefore, all the current that is applied, must come out the other end.

References

Matter & Interactions 4th Edition by Ruth W. Chabay & Bruce A. Sherwood
http://www.regentsprep.org/Regents/physics/phys03/bkirchof1/
https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff.27s