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Are there any equations/relevant explanations that can link resistivity of the "wiring"* (or other components) on a phone's PCB to its battery life, i.e. can I prove that using silver wires would extend battery life compared to copper wires, and by how much?

*I know its not actual wires, but I have no idea what the metal line things are called.

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    \$\begingroup\$ Most people call them traces, and no, they will not have any measurable effect. Silver has only a few percent better resistivity than copper, and the majority of losses on a phone is definetly not in the traces. \$\endgroup\$ – PlasmaHH Feb 11 '15 at 16:30
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    \$\begingroup\$ The PCB has negligible effect on the battery life. Even if it was superconducting the battery life would not be measurably affected. \$\endgroup\$ – Spehro Pefhany Feb 11 '15 at 16:30
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    \$\begingroup\$ ... but in any case, decreasing the resistance will generally (with a few exceptions) increase current flow, which will decrease battery life. \$\endgroup\$ – Dave Tweed Feb 11 '15 at 16:32
  • \$\begingroup\$ @DaveTweed, I don't know if you are just trying to make a joke, but it is certainly not true in any meaningful way that adding resistance to PCB traces will help battery life. Parasitic resistances generally reduce battery life. \$\endgroup\$ – mkeith Feb 11 '15 at 17:40
  • \$\begingroup\$ @mkeith: No, I'm not joking. Forget about the traces -- just take any circuit and increase the resistances by, say, 10%. Does the battery life go up or down? \$\endgroup\$ – Dave Tweed Feb 11 '15 at 17:49
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As you can see from the comments, the answer is conditional.

In most cases, in a modern piece of electronics, using a more conductive element in place of copper will reduce power losses across that trace. It is worth pointing out, though, that with the miniscule currents used in a typical mobile device, there shouldn't be any meaningful gain in battery life, especially not when compared to cost.

Heating power losses are linearly related to resistance, but are geometrically related to current flow. Silver only has about 5 percent better conductivity than copper, so, for the same amperage, could reduce those heating power losses by about 5 percent.

Keep in mind, though, that a mobile phone doesn't employ a fan - it has to dissipate all heat generated through it's case. So some very clever people already had to reduce those heating power losses to get it to work at all. They really are quite small in comparison to the amount of power consumed by the active electronics, especially a big, pretty touchscreen or a radio that passes high-speed data to a tower miles away - that's where all the power is going. A five percent reduction in heating power just from the pcb wouldn't be noticed.

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