Explanation:
Resistance, \(R=\dfrac{\rho l}{A}\) or \(R \propto l\)
\(\therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{R_1}}}{{{R_2}}} = \frac{{{l_1}}}{{{l_2}}} = \frac{L}{{L/4}} = 4\)
\( \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{R_2} = \frac{R}{4}\;\;\;{\mkern 1mu} {\kern 1pt} \left( {\because {R_1} = R} \right)\)
In parallel combination of such four
resistances,
\(\frac{1}{{{R^\prime }}}{\text{ }} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}} + \frac{1}{{{R_4}}}\)
\({\text{ or }}\;\;\;{\mkern 1mu} {\kern 1pt} \frac{1}{{{R^\prime }}}{\text{ }} = \frac{1}{{R/4}} + \frac{1}{{R/4}} + \frac{1}{{R/4}} + \frac{1}{{R/4}}\)
\( \Rightarrow \;\;\;{\mkern 1mu} {\kern 1pt} \frac{1}{{{R^\prime }}}{\text{ }} = \frac{4}{R} + \frac{4}{R} + \frac{4}{R} + \frac{4}{R}\)
\(\therefore \;\;\;{\mkern 1mu} {\kern 1pt} \frac{1}{{{R^\prime }}} = \frac{{16}}{R} \Rightarrow {R^\prime } = \frac{R}{{16}}\)
