Question

Let there be n resistors R₁............R_n with R_max = max (R₁......... R_n) and R_min = min {R₁..... R_n}. Show that when they are connected in parallel, the resultant resistance RP <  R min and when they are connected in series, the resultant resistance R_S > R_max. Interpret the result physically.

Answer 1

Parallel grouping: Same potential difference appeared across each resistance but current distributes in the reverse ratio of their resistance, i.e. `i  oo 1/R`

Series grouping: Same current flows through each resistance but potential difference distributes in the ratio of resistance, i.e. `V  oo  R`

In parallel combination: When all resistances are connected in parallel, the equivalent resistance R_ρ is given by

`1/R_ρ = 1/R_1 + ... + 1/R_n`

By multiplying both sides by R_min, we have

`R_(min)/R_ρ = R_(min)/R_1 + R_(min)/R_2 + ... + R_(min)/R_n`

Here, in RHS, there exists one term `R_(min)/R_(min)` = 1 and other terms are positive, so we have

`R_(min)/R_ρ = R_(min)/R_1 + R_(min)/R_2 + ... + R_(min)/R_n > 1`

This shows that the resultant resistance R_ρ < R_min.

Thus, in parallel combination, the equivalent resistance of resistors is even less than the minimum resistance available in a combination of resistors.

In series combination: When all resistances are connected in series, the equivalent resistance R_s is given by

R_s = R₁ + ... + R_n

Here, in RHS, there exist one term having resistance R_max.

So, we have

or R_s = R₁ + ... + R_max ... + ... + R_n

R_s = R₁ + ... + R_max ... + Rn = R_max + ... (R₁ + ... + )R_n

or R_s ≥ R_max

R_s = R_max(R1 + ... + R_n)

Thus, in series combination, the equivalent resistance of resistors is greater than the maximum resistance available in a combination of resistors.

Physical interpretation:

(a)

(b)

In figure (b), Rmin provides an equivalent routine as in figure. (a) for current. But in addition, there are (n – 1) routes by the remaining (n – 1) resistors. Current in figure. (b) is greater than current in figure (a) Effective resistance in figure. (b) < R_min. Second circuit evidently affords a greater resistance.

(c)

(d)

In figure (d), R_max provides an equivalent route as in figure. (c)  for current. Current in figure (d) < current in figure (c). Effective resistance in figure. (d) > R_max. Second circuit evidently affords a greater resistance.