Question

You are given 6 identical resistors, and allowed to combine them in any possible configuration to produce different equivalent resistances. What is the ratio of the largest possible equivalent resistance, Rₑᵩ,ₘᵢₙ ​​, that you can make to the smallest possible equivalent resistance, Rₑᵩ,ₘᵢₙ​​, that you can make with these resistors?

Answer 1

The largest possible equivalent resistance of six identical resistors in series is 6R, and the smallest in parallel is R/6. The ratio of maximum to minimum resistance is therefore 36:1.

Step-by-step explanation:

When trying to achieve the largest possible equivalent resistance (Rᵇᵛ,ₘᵉₙ) with six identical resistors, you would connect them all in series. The total resistance in series is simply the sum of all individual resistances. If the resistance of one resistor is R, then Rᵇᵛ,ₘᵉₙ = 6R.

For the smallest possible equivalent resistance (Rᵇᵛ,ₘᵉₙ), you would connect them all in parallel.

According to the formula for resistors in parallel (1/Rᵇᵛ ≈ 1/R₁ + 1/R₂ + ... + 1/Rₙ), the equivalent resistance would then be R/6, when R is the resistance of one resistor.

Therefore, the ratio of the largest to the smallest possible equivalent resistance made with these six resistors is

Rᵇᵛ,ₘᵉₙ / Rᵇᵛ,ₘᵉₙ = (6R) / (R/6) = 36.

The largest possible equivalent resistance is 36 times greater than the smallest.