The existence of a voltage source between the two points indicates that there is a potential difference, so why would the two loops be independent and zero current flow through the connecting wire?
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2\$\begingroup\$ Currents flow in loops. Can you point out a loop that involves the 5 V voltage source? I cannot. Indeed the potential voltage at \$y\$ is 5 V higher than at \$x\$ but does that mean a current flows? \$\endgroup\$– BimpelrekkieCommented Sep 22, 2020 at 9:58
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\$\begingroup\$ @Bimpelrekkie Thanks, I get it now! \$\endgroup\$– ConfusedCommented Sep 22, 2020 at 10:00
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1\$\begingroup\$ Can can flow under transient conditions. But, once enough charges have accumulated on one side, the potential rises and flow eventually stops. The question implicitly assumes that the solution under steady state is what is being asked. Though the answers say that current always flows in loops, the loop need not be made of a physical wire. A typical example is a one wire antenna. See also Displacement current where current completes the loop even though a physical conductor is absent in a section of the loop. \$\endgroup\$– AJNCommented Sep 22, 2020 at 12:03
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\$\begingroup\$ @AJN Yes, that was the first thought in my mind - the implicit condition that the circuit must be in a balanced state wrt potential and the loops must then become independent, however I couldn't exactly understand the end of such a transient state - the answers here have helped clear that confusion up. \$\endgroup\$– ConfusedCommented Sep 23, 2020 at 4:49
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2 Answers
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Current always flows in a circuit; what goes in must return equally out the other side.
The connection between your loops has no return path; no current can get out, so no current can get in.
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If you are not 100% convinced from answers and you are more mathematics oriented than physics try to consider an additional loop covering the XY segment:
you can see that the current in XY is I3-I3=0.
answered Sep 22, 2020 at 12:25