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I understand that by using multiple wires, we could increase the surface area of the conductor and thus, allowing more current to pass from one point to another. Does that apply to contacts on terminal blocks as well? Here is an example:

Say if I had a terminal block where each contact is rated at 15A, and wish to pass 20A current through it. So if I were to use cable(s) rated at 20A, to connect/jump 2 or more separate contacts on the terminal block, will those contacts then able to support 20A of current due the increased surface area?

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While increasing the surface area of a unit length of conductor will increase its diameter and, therefore its cross sectional area, wire sizes are generally specified by "Gauge" sizes, by their diameters, or by their cross-sectional area. For DC and low frequency AC, current travels almost exclsively through the center of the wire, so to increase the current through the wire without increasing the power dissipated by the wire its cross-sectional area must be increased, not its external surface area.

To answer your question: Since, if each terminal block is rated to pass 15 amperes, if you jump them (connect them in parallel) with a conductor of a proper size, then you have effectively increased the cross-sectional area of the connection by a factor of two and you should be able to push 30 amperes through the pair with no problem.

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