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When a metal is made of different conductivity parts, current passes the least resistive percolation path. My question is “does this path depend on the amount of current?” I think the path doesn’t depend on the current, but I am not sure.

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When a metal is made of different conductivity parts, current passes the least resistive percolation path.

Generally you shouldn't think of the "path of least resistance" rule as a strictly binary thing. Current passes through all paths in a circuit, but if there are two paths in parallel, the current in them will be proportional to the conductance of each path.

If the difference in conductance is very large, then this can be very close to a case of the current following only the path of least resistance.

If the difference in conductance is small, then the current in both paths will be significant.

My question is “does this path depend on the amount of current?” I think the path doesn’t depend on the current, but I am not sure.

So long as the materials behave linearly, the ratio of currents in two parallel paths will be equal to the ratio of the conductances of the paths.

But real materials are not strictly linear. Even if we're talking about two metal wires (or a wire with grains of two different elements) the conductivity of each material will vary slightly as the material heats itself up through Joule heating, and the amount of the change will be different between materials. This can result in either better segregation of current into the "path of least resistance" or better current sharing between two paths.

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