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An obscure but interesting quantity of resistance.

Technically, surface resistance is a measurement of the amount of electrical resistance inherent in a coating of electrically resistive film on a nonconducting surface, such as the coating of chrome on a set of sunglasses.

What makes it interesting is the units: ohms per square. That is to say, a square piece of the material will always have the same amount of resistance, regardless of whether it is a one millimeter square or a 1 kilometer square. Thus, determining the resistance of a shape of the material is a matter of breaking the shape down into its component squares, then adding the equivalent resisters in series and parallel to get the resistance.

Thus, a piece of material with a surface resistance of one ohm per square that is shaped like this:

 |       |2
4|    ___|
 |   | 3
Will have a resistance of 2.25 ohms (3/4 square plus 3/2 square) if you connect the leads at the east and west sides, and a resistance of 1 ohm (1/3 square plus 2/3 square) if you connect the leads at the north and south sides.

Exercise for the reader:

Show that the shape will have infinite resistance if you connect the (perfectly small) leads at the southwest and northeast corners.

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