Question

(a) What is meant by series and parallel combination of resistances?
(b) In which case, series combination or parallel combination, the combined resistance is less than any of the individual resistances?
(c) How should two resistances of 2 Ω each be connected so as to produce an equivalent resistance of 1 Ω?
(d)

In the circuit diagram given here, find:
(i) total resistance of the circuit,
(ii) total current flowing in the circuit, and
​(iii) the potential difference across R₁

Answer 1

(a) In a series combination, resistances are connected by an end to end such that the current flowing through all of them is equal. Whereas in a parallel combination, the resistances are connected in such a manner that they get an equal voltage.

(b) In a parallel combination, the reciprocal of net resistance is equal to the sum of reciprocals of individual resistances. Hence, in a parallel combination of resistances, the net resistance is smaller than any of the individual resistances.

(c) Two resistances of 2 Ω each should be connected in parallel. The net resistance of the parallel combination of both the resistors would be equal to 1 Ω.

$$\begin{gathered} \frac{1}{R_{\mathrm{net}}}=\frac{1}{R_1}+\frac{1}{R_2}=\frac{1}{2}+\frac{{\displaystyle 1}}{{\displaystyle 2}}=1 \\ \Rightarrow R_{\mathrm{net}}=1\mathrm{\Omega } \end{gathered}$$

(d)

(i) As we can see in the figure, Resistances R₂ and R₃ are connected in parallel.

The net resistance of the parallel combination = $\frac{R_2{R}_3}{R_2+R_3}=\frac{8\times 12}{8+12}=\frac{96}{20}=4.8\mathrm{\Omega }$

Now, the equivalent resistance of the whole circuit is the series combination of R₁ and net resistance of parallel combination.

Equivalent resistance = $7.2\mathrm{\Omega }+4.8\mathrm{\Omega }=12\mathrm{\Omega }$

(ii) Current in the circuit = $\frac{\mathrm{Total}\mathrm{voltage}\mathrm{applied}}{\mathrm{Total}\mathrm{resistance}}=\frac{6\mathrm{V}}{12\mathrm{\Omega }}=0.5\mathrm{A}$

(iii) The potential difference across R₁ = Current through R₁ $\times$ Resistance of R₁ = $0.5\mathrm{A}\times 7.2\mathrm{\Omega }=3.6\mathrm{V}$