Question

The resistance of the series combination of two resistances is $S$ . When they are joined in parallel the total resistance is $P$ . If $S = nP$ , then the minimum possible value of $n$ is
A. $4$
B. $3$
C. 1
D. $2$

Answer 1

Hint: Let the two resistances be ${R_1}$ and ${R_2}$ . In a series combination, the net resistance is ${R_1} + {R_2} = S$ , and in parallel combination the net resistance is $\dfrac{1}{P} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}}$ . Then put these values in the given equation, i.e. $S = nP$ . For the minimum value of $n$ , ${R_1} = {R_2} = R$ use this value to reach the solution.

Complete answer:
Resistance - Resistance is the property of a material due to which it opposes the flow of current through it. Every material has some resistance, conductors (metals, etc.) have a very low resistance, and insulators (wood, rubber,etc.) have very high resistance.
In the problem, we are given two resistances that when connected in series combination have a net resistance of $S$ and when they are connected in parallel combination have a net resistance of $P$ .
We are also given that $S = nP$ .
Let the two resistances be ${R_1}$ and ${R_2}$ .
In a series combination the same current flows through all of the resistances but in parallel combination, the current flowing through the resistances can be the same or different.
So, when these two resistances are connected in a series combination
${R_1} + {R_2} = S$
When these two resistances are connected in parallel combination
$\dfrac{1}{P} = \dfrac{1}{{{R_1}}} + \dfrac{1}{{{R_2}}}$
$\Rightarrow \dfrac{1}{P} = \dfrac{{{R_1} + {R_2}}}{{{R_1}{R_2}}}$
$\Rightarrow P = \dfrac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}$
So, for the relation $S = nP$
${R_1} + {R_2} = n\dfrac{{{R_1}{R_2}}}{{{R_1} + {R_2}}}$
$\Rightarrow {\left( {{R_1} + {R_2}} \right)^2} = n{R_1}{R_2}$
$\Rightarrow R_1^2 + R_2^2 + 2{R_1}{R_2} = n{R_1}{R_2}$
$\Rightarrow \dfrac{{R_1^2 + R_2^2}}{{{R_1}{R_2}}} = n - 2$
For the minimum value of $n$ , the value of ${R_1} = {R_2} = R$
$\therefore \dfrac{{{R^2} + {R^2}}}{{R \times R}} = n - 2$
$ \Rightarrow n - 2 = 2$
$ \Rightarrow n = 4$

So, the correct answer is “Option A”.

Note:
Series and parallel combination of resistances are used in the circuit to alter the net resistance of the circuit using the same resistances. Aside from resistances, all electrical parts have some resistance which is to be taken into account when designing a piece of electrical equipment.