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.
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