A) \[4:9\]
B) \[27:32\]
C) \[16:9\]
D) \[27:128\]
E) \[1:2\]
Correct Answer: B
Solution :
The resistance of one wire \[{{R}_{1}}=\rho \frac{{{l}_{1}}}{{{A}_{1}}}\] and the resistance of second wire \[{{R}_{2}}=\rho \frac{{{l}_{2}}}{{{A}_{2}}}\] Ratio of their resistances \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{{{l}_{1}}}{{{A}_{1}}}\times \frac{{{A}_{2}}}{{{l}_{2}}}\] \[\because \] \[mass=density\times volume\] \[\because \] \[mass=density\times area\times length\] Or \[\frac{{{R}_{1}}}{{{R}_{2}}}={{\left( \frac{{{l}_{1}}}{{{l}_{2}}} \right)}^{2}}\times \frac{\rho {{A}_{2}}{{l}_{2}}}{\rho {{A}_{1}}\times {{l}_{1}}}\] Or \[\frac{{{R}_{1}}}{{{R}_{2}}}={{\left( \frac{{{l}_{1}}}{{{l}_{2}}} \right)}^{2}}\times \frac{{{m}_{2}}}{{{m}_{1}}}\] Or \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{9}{16}\times \frac{3}{2}\] Or \[\frac{{{R}_{1}}}{{{R}_{2}}}=\frac{27}{32}\] \[{{R}_{1}}:{{R}_{2}}=27:32\]
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