| (i) Establish a relationship to determine the equivalent resistance R of a combination of three resistors having resistances \[{{\mathbf{R}}_{\mathbf{1}}}\mathbf{,}\,{{\mathbf{R}}_{\mathbf{2}}}\] and \[{{\mathbf{R}}_{\mathbf{3}}}\] connected in parallel. |
| (ii) Three resistors are connected in an electrical circuit as shown. Calculate the resistance between A and B. |
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Answer:
(i) Three resistances \[{{R}_{1}},\,{{R}_{2}}\] and \[{{R}_{3}}\] are connected in parallel to one another between the same two points. In this case, the potential difference across the ends of all the resistance will be the same. \[V={{V}_{1}}={{V}_{2}}={{V}_{3}}\] ?(i) If the total current flowing through the circuit is I, then the current passing through \[{{R}_{1}}\] will be \[{{I}_{1}}\] through \[{{R}_{2}}\] will be \[{{I}_{2}}\] and through \[{{R}_{3}}\] will be \[{{I}_{3}}\] 
Then \[I={{I}_{1}}+{{I}_{2}}+{{I}_{3}}\] ?(ii) \[{{I}_{1}}=\frac{V}{{{R}_{1}}},{{I}_{2}}=\frac{V}{{{R}_{2}}},{{I}_{3}}=\frac{V}{{{R}_{3}}}\] If R is the effective resistance of the circuit, connected across a battery of V voids, through which I current flows, then \[I=\frac{V}{R}\] Substituting the values in eq. (i), we get \[\Rightarrow \] \[\frac{V}{R}=\frac{V}{{{R}_{1}}}+\frac{V}{{{R}_{2}}}+\frac{V}{{{R}_{3}}}\] \[\Rightarrow \] \[\frac{V}{R}=V\left[ \frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}} \right]\] \[\Rightarrow \] \[\frac{1}{R}=\frac{1}{{{R}_{1}}}+\frac{1}{{{R}_{2}}}+\frac{1}{{{R}_{3}}}\] (ii) Given: \[{{R}_{1}}=4\,\Omega ,\,\,{{R}_{2}}=4\,\Omega ,\,\,{{R}_{3}}=8\,\Omega \] Let, resultant resistance between a and c be R? 
Then, \[R'={{R}_{1}}+{{R}_{2}}\] (Series combination) \[R'=4+4=8\,\Omega \] If R is the effective resistance between A and B, then \[\frac{1}{R}=\frac{1}{R'}+\frac{1}{{{R}_{3}}}\] (R? and \[{{R}_{3}}\] are in parallel combination) \[\frac{1}{R}=\frac{1}{8}+\frac{1}{8}=\frac{2}{8}\] \[\Rightarrow \] \[R=\,4\,\Omega \]
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