A connection is made consisting of resistance A in series with a parallel combination of resistances B and C. Three resistors of value 10 Ω, 5 Ω, 2 Ω, are provided. Consider all possible permutations of the given resistors into the positions A, B, C, and identify the configurations with maximum possible overall resistance, and also the ones with minimum possible overall resistance. The ratio of maximum to minimum values of the resistances (up to second decimal place) is __________.

In the range: 2.12 - 2.16

Concept:

The Net resistance when two resistances are in parallel is smaller than the smallest resistance in parallel.

The parallel combination of resistances R₁ & R₂ is given by:

$R_{net}=\frac{R_1{R}_2}{R_1+R_2}$

The series combination of resistances of R₁ and R₂ is given by:

Rₙₑₜ = R₁+ R${}_{2.}$

Calculation:

The given configuration is as shown:



Given resistances: 10 Ω, 5 Ω, and 2 Ω.

To get the maximum value of net resistance, A needs to be maximum, and 5 and 2 ohms will be in parallel.

Taking A = 10 Ω, B = 5 Ω and C = 2Ω

(Rₙₑₜ)${}_{max}$ = 10 + (5||2)

=10 + 10/7 = 10 + 1.4 = 11.42 Ω

To get the minimum value of net resistance, A needs to be minimum.

Taking A = 2 Ω, B = 10 Ω and C = 5 Ω

(Rₙₑₜ)${}_{min}$ = 2 + (5||10)

$=2+\frac{50}{18}$

$=2+\frac{10}{3}=2+3.33$

= 5.33 Ω

$\frac{{\left({\text{R}}_{\text{net}}\right)}_{\text{max}}}{{\left({\text{R}}_{\text{net}}\right)}_{\text{min}}}=\frac{11.428}{5.33}=2.14$