Imagine a simple circuit consisting of a battery with potential $V$ connected to a resistor with resistance $R$ by a loop of copper wire with $0$ resistance. It is obvious that a current $I=\frac VR$ will flow through the loop. Now, say that the section of wire between the battery and the resistor bifurcates into two strands which come back together at the resistor, giving the current two different ways to travel through the circuit, like this:


Intuitively, it seems that a current of $\frac I2$ will flow through both the upper and lower strands. Now, one can imagine the wire being shaped into a complicated "spiderweb" arrangement between the battery and the resistor, giving the current many different paths that it can take:


My question is, how can we calculate from first-principles how the current will reach the resistor (i.e. what fraction of the current travels through each segment of wire, keeping in mind that the total current must sum to $I$)? Kirchhoff's loop law is clearly of no use here, since the voltage over an ideal wire is always $0$.


1 Answer 1


The lumped circuit approximation with the assumption of ideal wires isn't adequate to predict how the current will flow in this scenario.

If the wires are not made of superconductors, you have to consider the actual non-zero resistance of the wires to find the current distribution in the DC steady state.

Or if the wires are superconductors, you need to consider the inductance of the wires, which depends on their physical arrangement, to determine how the current distributes during the initial application of the source.

In either case you might have to do a FEM or similar analysis of the physical structure to obtain an accurate result.


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