Most general method of distinguishing between series and parallel resistors?

Question

My textbook says that these two circuits are equivalent. The first is not easily solvable, the second is. I have looked around the site at a number of questions and this situation is not addressed anywhere, the answers mostly revert you to using Kirchhoff's laws. My textbook says that this circuit can be solved through series/parallel relationships. Of the two equivalent circuits in the image, what is the general method (for solvable cases, don't revert to other methods) for getting from the first to the second?

It is solved by combining 3,5 in series, that with 4 in parallel, all of that with 2 in series, and putting all of that in one resistance in parallel with 6,7,8. What is the logic here if you were confronted with just the first diagram?

Answer 1

What is the logic here if you were confronted with just the first diagram?

I recommend that you label all unique nodes (two junctions that are connected together by a wire are the same node) and redraw.

There are 5 unique nodes

A: the junction of the positive cell plate, R2, and R6

B: the junction of R2, R3, and R4

C: the junction of R4, R5, R6, R7, and R8

D: the junction of the negative cell plate, R7, and R8

E: the junction of R3 and R5

So, redraw the circuit in a sane way so that you can see the series and parallel connections. For example:

Answer 2

Two two-terminal elements are in parallel if they are connected together on both sides. One pin of element A is connected to a pin of element B, and the other pin of element A is connected to the other pin of element B. This means that the potential across element A is the same as the potential across element B.

Two two-terminal elements are in series if they are connected together at one terminal, and nothing else is connected to the node that connects them. This means that the current through element A is the same as the current through element B.

Your two example circuits are not equivalent. In your first example, R7 and R8 are in parallel (because their terminals are connected together). But R6 is not in parallel with R7 and R8, since its second terminal is connected to the battery instead of to a second terminal of R7 and R8.

Answer 3

I agree with Samuel Weir : the two circuits are not equivalent. After colour-coding and re-drawing the circuit so that there is only one vertical line of each colour/potential, I get the same as Hal Hollis :

I made an adjustment to the first diagram to avoid wires crossing over. Note that in the 2nd diagram $R_4$ has flipped around, and the connection between $R_4, R_6$ has been stretched considerably.