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I am learning electricity and quite often see loops that are connected in one node.

This is a simplified example:

enter image description here

But I found many cases, with more elements.

The thing that makes me question their usefulness is that (so far) they have always behaved as independent circuits.

So, I wonder if circuits are ever designed this way in real life and if this concept (two loops connected at one node) has its own name or is used at all with any purpose.

I also wonder what would happen if one of the loops does not have any voltage source in it, like this:

enter image description here

Would current behave as shown in A or in B or a mix of both and why?

enter image description here

Teacher says: "current is smart and always takes the laziest road" (in this case A), but I don't see how that can be a solid reasoning. Maybe with a clear why.

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  • $\begingroup$ Consider upvoting and accepting the answer(s) which were helpful to you. If you still have any confusion regarding your question then ask one of the answerers about it in the comments. If you are satisfied with an answer, then accept it. Saying this just because it will be classified as "Unanswered" if you don't accept any answer. But (I suppose) it is well answered (3 answers are quite enough to answer a single question). $\endgroup$ – user243267 Dec 29 '19 at 18:34
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    $\begingroup$ @FakeMod Thanks for your kind reminder. The fact that I took time in this case is because I was thinking about them profoundly to understand them and doing some extra research. All of the three answers (including yours) were very helpful and appreciated. But I feel bad about accepting only one because that makes the other answers seem less appreciated. Do you know if it has already been discussed in meta, to generate some sort of mixed-answers accepted answer? I will keep re-reading all three answers and accept one as soon as I fully understand them. Thanks a lot for helping. $\endgroup$ – Alvaro Franz Dec 30 '19 at 9:57
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    $\begingroup$ That dilemma is quite common :). But accepting any one answer is better than not accepting any. And I strongly believe that you should profoundly think and ponder upon the ideas present in the answers, which you said you were doing. If I had known that before I won't have dropped a comment. You should only accept an answer once you understand it fully. I usually drop that comments because many people just forget about rewarding the answerers once they have seen the answers. However you were completely justified to do that as you were pondering on the question and that's essential for learning. $\endgroup$ – user243267 Dec 30 '19 at 11:33
  • $\begingroup$ @FakeMod Thank you. Yes, I see your intention. This is anyways a wonderful community and already works much better than good enough. So yay, good day! $\endgroup$ – Alvaro Franz Dec 30 '19 at 11:51
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I might not be the right person to answer the first part of your question, but I can surely answer the second one.

In the second circuit(which I have redrawn and labelled), let's assume the the points $A$ and $B$ are two distinct points and not the same ones.

enter image description here

Now let's assume that the resistance of wire $AB$ is $R$. So $R$ will be some function of the length $AB$. But whatever that function be, it will always follow the given result,

$$\lim_{{l(AB)} \to 0} R =0$$

where $l(AB)$ is the length of the segment $AB$ and $R$ is the resistance. So what I am saying is that as the length of $AB$ reduces to $0$, the resistance also goes to zero.

So, when $A$ and $B $ get close the resistance $R_2$ is short-circuited. And thus no current flows through it.


Mathematical explanation :-

If you calculate the total resistance of this circuit hy using laws of combination of resistors in series and parallel, then you will obtain,

$$ R_{equivalent} = R_1 + \frac{R R_2}{R+R_2}$$

Now if you decrease the value of $R $ such that it approaches $0$, then $R_{equivalent}$ approaches $R_1$. But $R_1$ is the resistance it faced by the current when it travels like it does in the image A(in your question). Thus the current must be travelling in the way it does in the image A. Then only would the resistance of the circuit be $R_1$.

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Behavior is as shown in A. There can be no current in B because there is no potential difference (voltage) across the two series connected resistors. In the first diagram you simply have two independent circuits with currents due only to the independent voltage sources, with a common point of connection.

While the analysis of such circuits is trivial it doesn’t necessarily mean they do not occur in real life (though I can’t, for the moment, see any for the last diagram). For example, in a building you can have multiple branch circuits that are independent of one another but have a common point of connection where all the grounded circuit conductors of the different circuits are connected together at the neutral bus of the distribution panel.

Hope this helps.

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Your circuit does not make sense. The resistances (both of them) on the right will not draw current. A is correct. Simply because if you notice the node at the centre of the '8' links to two resistances - one which leads to a positive terminal and the other to a negative terminal.

It does not matter to think which path the current would travel because the voltage in that starting terminal of one resistor and the ending terminal of the second resistor is the same. It means energy is not lost by the charge that travels through it - implying that no charge goes there. Because charge has to dissipate energy while travelling through a wire and resistors or else we would have no energy shortage in the world!

enter image description here

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