# Symmetry in Electrical Circuit Analysis

How is symmetry used to solve electrical circuits? I have seen several problems in books in which currents in two resistors are said to be equal due to 'symmetry'. That is a concept that I fail to understand and thus cannot apply. In class, we were shown a few circuit diagrams which were symmetrical about their perpendicular bisectors etc. We were told to note that the potential of all circuit elements lying on a line of symmetry would be the same. While this is not too difficult to grasp and use, I don't understand the correlation between the two: symmetry and potential. Sure I can prove it for one or two circuits using Kirchhoff's Laws and other methods of circuit analysis. But won't that proof be circumstantial? Without understanding the reason behind it, how can I proceed to build my understanding of the subject?

In a nutshell, my question is: what is the correlation between symmetry (as in symmetry in geometrical shapes) and current and/or voltage and how do I utilise it in understanding circuit analysis?

## Answers and Replies

phinds
Gold Member
2021 Award
How is symmetry used to solve electrical circuits? I have seen several problems in books in which currents in two resistors are said to be equal due to 'symmetry'. That is a concept that I fail to understand and thus cannot apply. In class, we were shown a few circuit diagrams which were symmetrical about their perpendicular bisectors etc. We were told to note that the potential of all circuit elements lying on a line of symmetry would be the same. While this is not too difficult to grasp and use, I don't understand the correlation between the two: symmetry and potential. Sure I can prove it for one or two circuits using Kirchhoff's Laws and other methods of circuit analysis. But won't that proof be circumstantial? Without understanding the reason behind it, how can I proceed to build my understanding of the subject?

In a nutshell, my question is: what is the correlation between symmetry (as in symmetry in geometrical shapes) and current and/or voltage and how do I utilise it in understanding circuit analysis?
The point generally is that if the potential at two points in a circuit are the same then you could just put a short circuit between the two points and it would have no effect on the behavior of the circuit. This can help simplify circuits.

anorlunda
Staff Emeritus
We were told to note that the potential of all circuit elements lying on a line of symmetry would be the same.

It's not clear what that means, or what you are thinking about.

1. In physics, we talk about the symmetries of natures, such as space translation invariance.
2. If you have a bulk material, like a sheet of steel, and pass a current through it, there will be equipotential lines depending on the shape which might be symmetrical.
3. In circuit analysis, we can have equal branches (as @phinds pointed out), but symmetry is not the word I would use.
Can you elaborate on what you mean by symmetry?

The point generally is that if the potential at two points in a circuit are the same then you could just put a short circuit between the two points and it would have no effect on the behavior of the circuit. This can help simplify circuits.
I understand that. My question is: how does symmetry fit into this? What has a line of symmetry through a network of circuit elements got to do with the potential at different points along the line?

I understand that if the potential of two points is the same, current will not flow between them. But what has symmetry (as in symmetry in geometrical shapes) got anything to do with it? How do I get from 'A line divides this circuit symmetrically, i.e., it is a line of symmetry' to saying 'The potential is the same along this line'. Where does that come from?

Borek
Mentor
If the circuit is symmetrical it means it can get transformed into itself by some operation (reflection, rotation by 180 degrees, whatever). Parts of the circuit that get transformed each into other must have exactly the same properties, the same current and the same voltages must be present at the points that get transformed into each other. You can solve such a circuit ignoring the symmetry, but it often simplifies the problem.

Capt1801
It's not clear what that means, or what you are thinking about.

1. In physics, we talk about the symmetries of natures, such as space translation invariance.
2. If you have a bulk material, like a sheet of steel, and pass a current through it, there will be equipotential lines depending on the shape which might be symmetrical.
3. In circuit analysis, we can have equal branches (as @phinds pointed out), but symmetry is not the word I would use.
Can you elaborate on what you mean by symmetry?

Sure.
The symmetry I'm referring to is the one in geometrical shapes, like a square or a rectangle.
Consider this circuit:

This circuit is symmetrical about the red line - a perpendicular bisector. I have been taught that since it is symmetrical about that line, the potential of each point along that line would be the same, and thus no current would flow through the resistor placed on that line. I am aware that it forms a balanced Wheatstone Bridge and it is therefore obvious that no current would flow through the middle resistor; but that is beside the point. My teacher simplified more complex circuits using the same principle: that no current would flow through any circuit element that lies on a line of symmetry of the circuit. I just don't understand the correlation between symmetry and this. How can I get to the conclusion that the potential of all points on the red line is the same, starting with the fact that the red line is a line of symmetry? How does symmetry fit into this? I guess it could have something to do with equipotential lines, as you said. But without anything to link the two statements, I can't possibly begin to apply this concept in questions. Of course, it might not be called 'symmetry' (that is the heading under which we were taught this little concept).

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Borek
Mentor
This circuit is symmetrical about the red line - a perpendicular bisector.

There is another symmetry, horizontal one.

I have been taught that since it is symmetrical about that line, the potential of each point along that line would be the same, and thus no current would flow through the resistor placed on that line.

Doesn't sound correct to me.

Doesn't sound correct to me.
Exactly! Which is why I wanted to know the reason.

anorlunda
Staff Emeritus
Are you aware that in circuit analysis, we specifically ignore the length and shapes of all components including connecting wires? That circuit you showed in post #6 could be redrawn so that it appears to be totally asymmetrical, yet the solution is the same.

See the PF Insights article https://www.physicsforums.com/posts/5872068/

Also, most circuits do not contain any kind of symmetry. That circuit in #6 is what we call a bridge. such as https://en.wikipedia.org/wiki/Bridge_circuit#/media/File:Wheatstonebridge.svg. It is a special purpose circuit specifically designed to take advantage of those balanced paths and identical resistors. So yes, you can use symmetry in a few special cases, but not in most cases.

CWatters
Homework Helper
Gold Member
The classic example problem is the 3D cube with a resistor R on every edge (12 resistors in total). They ask you to calculate the resistance from diagonally opposite corners. Look it up.

Borek
Mentor
Exactly! Which is why I wanted to know the reason.

Actually I think we can be both wrong. My first instinct was that there can be a current flowing through the vertical resistor, but I am unable to construct an example of such a circuit. Hardly a proof, still.

CWatters
Homework Helper
Gold Member
In the circuit in #6 what matters is the horizontal (eg top to bottom) symmetry.

Imagine that point A is at say 10V and B is at 0V then there are two identical paths from A to B. The top path and the bottom path. Symmetry says that the voltage of any point on the top path equals the voltage on the corresponding point on the bottom path. No current flows through the vertical resistor.

Capt1801
jim hardy
Gold Member
Dearly Missed
that circuit in post #6 is symmetric ONLY IF the R's are equal (or at least have same ratios....)
It's a Wheatstone Bridge.
https://www.grc.nasa.gov/www/k-12/airplane/tunwheat.html

I dislike shortcuts .
Solve the circuit using Kirchoff's Laws .
That makes it obvious why a balanced Wheatstone Bridge has no current through its measuring leg.

My advice is forget about 'symmetry' . It's a useless complication IMHO.

Capt1801
In the circuit in #6 what matters is the horizontal (eg top to bottom) symmetry.

Imagine that point A is at say 10V and B is at 0V then there are two identical paths from A to B. The top path and the bottom path. Symmetry says that the voltage of any point on the top path equals the voltage on the corresponding point on the bottom path. No current flows through the vertical resistor.

That makes much more sense. In fact, I think that is the concept. It has just been given the wrong title. When we were made to solve the 3D cube problem, we used what you stated above.

I dislike shortcuts .

So do I. Especially when they seem so forced, like this one. The reason we're being taught shortcuts and tricks is because we're preparing for the IIT JEE, a competitive examination for admission to undergraduate courses in IITs. Speed matters in the exam, so we're taught a lot of tricks. Most of them make perfect sense, unlike this one. Thanks for all the help!

jim hardy
Gold Member
Dearly Missed
Well, good luck with your studies.

Wheatstone bridge is such a fundamental building block it'll become second nature to you.

I hope the test goes well for you.

old jim