### Further Reading

- Example: A complex network
- Example: Batteries in series and parallel
- Example: Resistors in parallel
- Example: Resistors in series

# Kirchhoff's principles

#### Prerequisites

- The electric potential
- Electric currents
- Resistive electric flow -- Ohm's law
- Ways to think about current -- a toolbox of models

The basic ideas that we have developed about how electric charges move in matter serve as a basis for analyzing a wide variety of electric circuits and devices and for modeling the electrical behavior of biological systems. But these circuits, devices, and models can quickly become quite complex.

It's useful to establish a set of *foothold ideas* — principles that we can hold on to and refer back to in order to organize our thinking in a complex situations. — to provide "stakes in the ground" that we can trust and use to support our safety net of coherent and linked ideas.

The foothold principles for understanding electric currents were developed by the 19th century German physicist, Gustav Kirchhoff (yes, two "h"s) are called *Kirchhoff's laws (or principles).* (He also formulated laws of spectroscopy and thermochemistry.)

### The (idealized) context for Kirchhoff's principles

Kirchhoff's principles are restrictions of more general electromagnetic laws (Maxwell's equations, conservation of charge) to standard situations in electrical circuits. We'll talk about them and use them in the context of analyzing connected networks of electrical devices — batteries, resistors, capacitors, and wires. Here's how we will represent and idealize them:

— devices that maintain a constant electrical pressure difference (voltage) across their terminals: like a water pump that raises water to a certain height. We use the symbol shown at the right with the longer line corresponding to the end of the battery with the higher potential.Batteries |
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— devices that have significant drag and oppose current. Pressure will drop across them when current is flowing through them. The resistor shown in the figure is of the kind used in hand-built electrical circuits. The stripes are color-code that tell the size of the resistor and its precision. We indicate them with the zig-zag symbol shown on the right.Resistors |
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— devices that can maintain a charge separation in response to a pressure differential (voltage) applied across its plates. The symbol for a capacitor is the pair of parallel lines shown on the right. Be sure to distinguish between the symbol for a capacitor (parallel lines of equal length) and a battery (parallel lines of different length)!Capacitors |
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— conductors that connect the other devices. We treat them as perfect conductors; that is, as having zero resistance. This approximation only works when there are other resistors in the circuit that have significant resistance, compared to which the resistance of the wires are negligible. The symbol for a wire is just a line and it could be straight or curved — makes no difference.Wires |

### Kirchhoff's 1^{st} (Flow) Principle

The first principle is basically a combination of two ideas:

- conservation of charge (the total amount of positive charge minus the total amount of negative charge is a constant)
- in electrical circuits, due to the strong repulsive forces between like charges, electrical elements remain neutral — there is no build-up of charge anywhere.

The principle is often called "the flow rule" and is stated as follows:

*The total amount of current flowing into any volume in an electrical network equals the amount flowing out.*

From our analysis of how a capacitor and a resistor both work, we know that this idea doesn't hold when things are just getting started.

For example, when we charge a capacitor, charge is flowing into one side of the capacitor and out of the other: charge (of opposite sign) is building up on each plate of the capacitor in violation of the flow rule. But if we put a box around the capacitor and don't look inside, the rule still works. It also works when the system is in the steady state and things have stabilized.

A similar thing holds for a resistor. When a current just starts to build up through the resistor, a build-up of equal and opposite charges at the two ends of the resistor are what is responsible for establishing the electric field in the resistor (creating a potential drop across the resistor) that keeps the charges moving through the drag of the resistor at a constant velocity, consistent with Newton's laws of motion.

In both these cases, when we hook up these devices to a circuit, the violations of the flow principle (the build-up of unbalanced charges) taking place in the interior of the device happen fast — in nanoseconds or less. And if we consider the whole device instead of just a part of it, the principle still works even on that time scale.

### Kirchhoff's 2^{nd} (Resistance) principle

The second principle tells what happens when there is a current in a resistor: there is a potential drop in the direction of the current which is proportional to the current times a property of the resistor. This is *Ohm's law* and it hold for any device in which the drag resisting the flow is proportional to the velocity. (See Resistive electric flow -- Ohm's law.) We can even stretch its validity by letting the resistance be a function of the current. Mostly, we wont need to do this.

*In a resistor, the current through is proportional to the potential drop across it*: $ΔV = IR$*.*

Note that we have to be a bit careful here. This is ONLY true for a resistor. It is NOT necessarily true for a cluster of connected resistors. If we want to deal with a cluster of resistors, and put a box around it pretending it is a single resistor, we have to figure out what the effective resistance of the combination is using the combined principles. (See Example: Resistors in series and Example: Resistors in parallel.)

### Kirchhoff's 3rd (Loop) principle

Where Kirchhoff's first principle controls the current in an electrical network, the second principle deals with the voltage drops in the network.

We can understand it by using the water analogy. (See Ways to think about current: A toolbox of models.) The electric potential is analogous in the water model to the height that the water has been raised. One of the things we know about heights is that if you make a loop and come back to the same point you started from, you will be at the same height at which you started. Whatever drops (descents) you made had to be cancelled by and equal sum of rises (climbs) in order to get back to your starting point.

The same thing is true of electric potential (voltage). As we travel through a circuit, we may have rises, say if we go through a battery from its low end to its high end, and we may have drops, say if we go through a resistance in the direction of the current flow. Kirchhoff's third principle states:

*Following around any loop in an electrical network the potential has to come back to the same value (sum of drops = sum of rises).*

This can be a bit tricky to apply! Just as if you go up a hill you are rising, but if you walk down that same hill you are descending, whether you have a rise or a drop in electric potential as you go through a device depends on which way you are following your loop If you go through a battery from the positive end to the negative end, it gives you a drop! If you go through a resistor in the direction opposite the direction a current is flowing you get a rise! When you evaluate the 2^{nd} principle around some loop, you have to pay careful attention to what direction you're going in.

### Useful heuristics

Applying Kirchhoff's principles to a complex circuit is sometimes complicated. There are two variables to be solved for — the voltage (electrical pressure) and the current. These are independent variables. They affect each other, but your intuitions as to what is happening sometimes refers to one, sometimes to the other — but it's easy to get confused!

A useful way to think of the voltage throughout the circuit is as analogous to pressure (in the air flow model) or height (in the water flow model). Moving throughout the circuit there are different values of this variable — the voltage (electric pressure) — but it doesn't move or change. it is the *difference* between voltages (say at opposite ends of a resistor) that drives current through a resistor.

One of the best ways to start analyzing an electrical network is by figuring out what you know about the voltage. And here's a corollary to Ohm's law that helps a great deal in analyzing complex networks:

*A conductor in a circuit that can be treated as having 0 resistance (for example, a wire), is an equipotential (it has the same value of the potential everywhere along it) even if there is current flowing through it. For that wire, the voltage drop across the wire is correctly given by Ohm's law: $*ΔV=IR=0$*, even if I is NOT zero.*

The best advice in handling the current in an electrical circuit problem is to choose some directions for the directions you think the currents are flowing in and take those as positive. Then just apply Kirchhoff's principles to generate relationships (equations) among the various variables. If you have chosen wrong some signs might come out negative. No problem! That just tells you that your initial assumption about the direction was wrong and that the current is flowing in the opposite direction from the one you expected.

An excellent way of getting a feel for how these works is to play with the PhET simulation, Circuit Construction Kit - DC in the Workout below. Following through the examples in the follow-ons will give you an idea of how to apply these ideas mathematically.

Workout: Kirchhoff's principles

Joe Redish 2/28/12

#### Follow-ons

Last Modified: May 2, 2019