Say you have a network of resistors. Thanks to Ohm's law and Kirchhoff's laws (or, if you're a mathematical physicist & more generally, thanks to Ohm's law and some minimal action principle), you don't need to know the structure of the network to analyse a larger network in which it is a building block!

  1. Attach a given voltage V across the source and the sink of the network.
  2. Measure the induced current I from source to sink.
  3. The effective resistance (sometimes equivalent resistance) of the network is R=V/I.

This is actually quite remarkable: if you're never interested in points on the network apart from the source and the sink, you need never know anything about its internal structure! It also has some interesting mathematical uses.

Well-known formulae exist for the effective resistance of networks joined in series and in parallel. These can be used to calculate the effective resistance of any network that is built using only these operations. Unfortunately, not every network is a series/parallel network! (For example, the Wheatstone bridge is not, which explains its top ranking on Physics tests...)

Alternatively, note that Kirchhoff's laws and Ohm's laws give linear equations; just solve the linear system for an applied voltage of 1v (it's an interesting mathematical exercise to prove there is always a unique solution) to get the effective resistance.

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