A) \[\frac{{{i}_{1}}}{{{i}_{2}}}=\frac{1}{4}\]
B) \[\frac{{{i}_{1}}}{{{i}_{2}}}=4\]8
C) \[\frac{{{W}_{2}}}{{{W}_{1}}}=4\]
D) \[\frac{{{V}_{2}}}{{{V}_{1}}}=\frac{1}{4}\]
Correct Answer: C
Solution :
From Faraday?s Law, the induced voltage \[V\propto L\] rate of change of current is constant \[\left( V=-L\frac{di}{dt} \right)\] \ \[\frac{{{V}_{2}}}{{{V}_{1}}}=\frac{{{L}_{2}}}{{{L}_{1}}}=\frac{2}{8}=\frac{1}{4}\] Þ \[\frac{{{V}_{1}}}{{{V}_{2}}}=4\] Power given to the two coils is same, i.e., \[{{V}_{1}}{{i}_{1}}={{V}_{2}}{{i}_{2}}\] Þ \[\frac{{{i}_{1}}}{{{i}_{2}}}=\frac{{{V}_{2}}}{{{V}_{1}}}=\frac{1}{4}\] Energy stored \[W=\frac{1}{2}L{{i}^{2}}\] Þ \[\frac{{{W}_{2}}}{{{W}_{1}}}=\left( \frac{{{L}_{2}}}{{{L}_{1}}} \right)\,{{\left( \frac{{{i}_{2}}}{{{i}_{1}}} \right)}^{2}}=\left( \frac{1}{4} \right)\,{{\left( 4 \right)}^{2}}=4\] Þ \[\frac{{{W}_{1}}}{{{W}_{2}}}=\frac{1}{4}\]
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