question_answer

Show four different ways in which four resistors of r ohm each may be connected in a circuit. In which case is the equivalent resistance of the combination.

(i) maximum                (ii) minimum

Answer:

(a)

Resultant resistance, \[R=r+r+r+r\]

\[R=4r\]

(b)

\[\frac{1}{R}=\frac{1}{r}+\frac{1}{r}+\frac{1}{r}+\frac{1}{r}=\frac{4}{r}\]

\[R=\frac{r}{4}\]

(c)

Resistance (AB) \[={{R}_{1}}=r+r=2r\]

Resistance (PQ) \[={{R}_{2}}=r+r=2r\]

Resultant, \[R=?\]

\[\frac{1}{R}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}=\frac{1}{2r}+\frac{1}{2r}=\frac{1+1}{2r}=\frac{2}{2r}=\frac{1}{r}\]

(d)

Resistance (PQ)\[={{R}_{1}}=r+r+r=3r\]

Resistance (AB)\[~={{R}_{2}}=r\]

Resistance, \[R=?\]

\[\frac{1}{R}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}=\frac{1}{3r}+\frac{1}{r}=\frac{1+3}{3r}=\frac{4}{3r}\]

\[R=\frac{3r}{4}\]

(i) Maximum resistance = Case (a) where all the resistors are combined in series.

(ii) Minimum resistance = Case (b) where all the resistors are combined in parallel combination.