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In the following formulation of part of the infinite grid problem

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Why can one assume that $1$ (of current) entering a node must equal $2 \alpha + \beta$ current?

Why aren't all the three other quantities separate?

I.e. that

$$\alpha + \beta + \gamma = 1$$

or same

$$3 \alpha = 1~?$$

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1 Answer 1

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Kirchhoff's Current Law:

At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.

I presume that the central node is one electrode, and the other is connected in a ring at infinity. Then the grid has circular symmetry about the central node. The 4 nearest nodes are identical if rotated, or if reflected about a diameter.

So all the currents marked $\alpha$ are identical and those marked $\beta$ are identical. But $\alpha$ and $\beta$ are not the same because the $\alpha$ are perpendicular to the radius while the $\beta$ are along the radius.

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  • $\begingroup$ Exactly that Kirchhoff's law that the OP claimed here to be slightly useless ;-) $\endgroup$
    – Massimo Ortolano
    Commented Feb 4, 2018 at 17:39
  • $\begingroup$ This does not answer why it's $2 \alpha + \beta$. $\endgroup$
    – mavavilj
    Commented Feb 4, 2018 at 17:42
  • $\begingroup$ @mavavilj I have updated my answer. $\endgroup$
    – sammy gerbil
    Commented Feb 4, 2018 at 17:43

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