Re: 13 amp fuse for 1.5mm mains cable
It will be fine, certainly under 3 metres. In fact, you could probably get away with 1mm² cable over that run (unless you plan to use this extension lead in a daisy-chain).
The bulk resistivity of copper is 16.8e-9Ωm (that's ohms times metres; when we divide that by metres**2, we get ohms per metre). With an area of 1.5e-6m², that gives us (16.8 / 1.5)e(-9+6) = 1.12e-3Ω/m. So over the length of the cable, there will be 2.8mΩ in each conductor, for a total of 5.6mΩ. With 13A flowing through it, the voltage developed across the cable will be 0.0728V, giving a power dissipation of less than a watt (0.9464W, to be precise).
If we used 1.0mm² cable, we would get 1.68e-3Ω/m, for a total of 8.4mΩ in the two conductors, dissipating 1.419W at 13A. That's still not much, considering the total surface area over which it is distributed (compare it to the 3W resistor any good engineer would choose to dissipate that much in a hypothetical circuit).
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