Did you know you can compute square roots using a supply of resistors and an ohmmeter? Here's how.

If you have an infinite array of resistors

it is straightforward to show that the effective equivalent resistance, R_{eq}, of the
combination is given by the expression

.

Furthermore, by considering the array truncated after N units and computing R_{eq}
using the usual parallel and series combinations, you can show that as long as R_{1}
and R_{2} are not too far different, the value converges to the inifinite case
even for a relatively small N. Here is a table of results obtained numerically.

R_{2}/R_{1} | N for 1% | N for 0.1% |
---|---|---|

0.1 | 1 | 1 |

1 | 2 | 2 |

5 | 4 | 6 |

10 | 5 | 8 |

20 | 8 | 12 |

50 | 13 | 19 |

100 | 18 | 26 |

Hence, you can create an array with an equivalent resistance which is the
square root of Y kiloohms, Y > 1, by choosing R_{1} = 1 kiloohm and

kiloohms
using resistors of sufficient precision and enough sections to obtain the desired
accuracy. In practice this will be very expensive for an accuracy greater
than 0.1%. For values outside the range 1 < Y < 100, you can scale the value
into that range by multiplying by powers of 100, and then adjust the answer
using the same power, but of 10.

Example: This array uses N= 3 and has a resistance equal to the square root of 3 kiloohms
(to better than 0.1% if ideal resistors are used).

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