Question

Explain with diagrams what is meant by the "series combination " and "parallel combination" of resistances.
In which case the resultant resistance is (i) less, and (ii) more, than either of the individual resistances?

Answer 1

Resistors in Series and in Parallel:

Resistors are probably the most commonly occurring components in electronic circuits. Practical circuits often contain very complicated combinations of resistors. It is, therefore, useful to have a set of rules for finding the equivalent resistance of some general arrangement of resistors. It turns out that we can always find the equivalent resistance by repeated application of two simple rules. These rules relate to resistors connected in series and in parallel.

Figure 18:Two resistors connected in series.

Consider two resistors connected in series, as shown in Fig. 18. It is clear that the same current flows through both resistors. For, if this were not the case, charge would build up in one or other of the resistors, which would not correspond to a steady-state situation (thus violating the fundamental assumption of this section). Suppose that the potential drop from point to point is . This drop is the sum of the potential drops and across the two resistors and , respectively. Thus,

(135)

According to Ohm's law, the equivalent resistance between and is the ratio of the potential drop across these points and the current which flows between them. Thus,

(136)

giving

(137)

Here, we have made use of the fact that the current is common to all three resistors. Hence, the rule is The equivalent resistance of two resistors connected in series is the sum of the individual resistances. For resistors connected in series, Eq. (137) generalizes to .

Figure 19:Two resistors connected in parallel.

Consider two resistors connected in parallel, as shown in Fig. 19. It is clear, from the figure, that the potential drop across the two resistors is the same. In general, however, the currents and which flow through resistors and , respectively, are different. According to Ohm's law, the equivalent resistance between and is the ratio of the potential drop across these points and the current which flows between them. This current must equal the sum of the currents and flowing through the two resistors, otherwise charge would build up at one or both of the junctions in the circuit. Thus,

(138)

It follows that

(139)

giving

(140)

Here, we have made use of the fact that the potential drop is common to all three resistors. Clearly, the rule is The reciprocal of the equivalent resistance of two resistances connected in parallel is the sum of the reciprocals of the individual resistances.
In parallel combination, the resultant resistance will be less than either resistances.

In series combination, the resultant combination will be larger than both resistances.

suppose there are resistances $R_1,R_2\dots ..R_n$
and eqivalent resistance be $R_q$….

so $\frac{1}{R_q}=\frac{1}{R_1}+\frac{1}{R_2}+\dots ..\frac{1}{R_n}$
and let the smallest resistance be $R_1$(suppose)………so…..$\frac{1}{R_q}>\frac{1}{R_1}$

since $\frac{1}{R_q}=\frac{1}{R_1}+\frac{1}{R_2}\dots ..\frac{1}{R_n}$ which is always greater than $\frac{1}{R_1}$

so, $R_q<R_1$…………hence smaller than the smallest resistance.