While finding the resistance for an infinite square grid of resistors streching in all direction for two adjacent points. It is assumed that the current coming out from the wire is same in all directions. I am not sure why this should be true up and down direction are symmetric but what about the others,
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It is because of the general superposition principle in electric circuits with only linear elements (such as resistors): if you have a source (ideal battery) $1$ creating current $I_1$ in some wire and a source (ideal battery) $2$ creating current $I_2$ in the same wire when they are plugged separately, then if you plug them together, they will create current $I_1+I_2$.
Now, usually we do not treat source and sink of the current separately, because having a separate source would make electric charge build up in our system. However, in infinite systems this is not a problem, because extra charge can go to infinity which presumably has an infinite capacity. So if we consider separately just a source providing current $I$ into an infinite grid, from symmetry it is clear that this current will be distributed equally over all lines of the grid adjacent to the vertex where the current is supplied (in the case of a square grid - each of the 4 adjacent lines will carry current the $\dfrac{I}{4}$). Now, we could also separately consider the sink of the current as a source of current $-I$ which will also create a symmetric distribution if it was alone. Now using the superposition principle we could find the total current between the source and the sink.
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$\begingroup$ its the symmetry part i dont understand the tp and bottom wires are completely symmetric hence must have the same current but what about the others maybe i am missing something here $\endgroup$ Commented Apr 9, 2021 at 4:25
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$\begingroup$ There is complete symmetry if you consider just the source of the current separately: imagine that you just have one wire providing current $I$ attached to some node of the grid. For this one wire every direction is similar. $\endgroup$– VikingCommented Apr 9, 2021 at 14:41
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$\begingroup$ Sorry I didon't completely understand your argument before I understand now thanks $\endgroup$ Commented Apr 13, 2021 at 10:35