Number of resistors for potential divider?

[cubzar]

Not sure whether to post in mathematics or physics section.

A circuit contains a cell of unit E.M.F. Use any combination of identical resistors in series and/or parrallel in order to create a voltage of x across a voltmeter connected anywhere on the circuit. Assume that there is no internal resistance or resistance in the wires, and the voltmeter does not draw any current. What is the easiest way to find the minimum number of resistors required for a given value of x (x<1 and x is rational, otherwise there is no solution)?

 

[gsal]

You lost me

 

[willem2]

I don't think there is an easy way to find the minimum numbers. You can make any fraction p/q, with q resistors in series, where you connect the voltmeter across p of them, but this is not always the smallest number.

with four resistances you can make a voltage of 1/5 by putting 2 in series to get a 2 ohm resistance, and 2 in parallel to get 1/2 ohm. If you make a voltage divider from 2 ohm and 1/2 ohm you get 1/2 / (2 + 1/2) = 1/5.

 

[cubzar]

The only pattern I could be sure of is that the numbers of resistors for x=a/b is the same as for x=(b-a)/b. It also looks like b resistors are required when x=1/b and b is prime.
A few values I have found:
voltage no. of resistors
1/2 2
1/3 3
1/4 4
1/5 5
2/5 4
1/6 4
1/7 7
2/7 5
3/7 5
1/8 8
3/8 7
1/10 5
2/13 8
7/13 7
2/19 19
19/23 13
2/31 17

 

[PJC01]

The number of resistors needed for a potential divider can be determined using the formula N = E/x - 1, where N is the number of resistors, E is the unit E.M.F., and x is the desired voltage. This formula applies to both mathematics and physics, as it involves principles of both fields.

To find the minimum number of resistors required for a given value of x, we can rearrange the formula to solve for x. This will give us the minimum voltage required for a given number of resistors. Then, we can compare this voltage to the desired voltage, x, and determine the minimum number of resistors needed.

In terms of posting in the mathematics or physics section, both would be suitable as this topic involves concepts from both fields. However, the physics section may provide more specific and relevant answers as it directly relates to circuits and potential dividers.