A) \[{{t}_{1}}+{{t}_{2}}\]
B) \[{{t}_{1}}{{t}_{2}}\]
C) \[\frac{{{t}_{1}}+{{t}_{2}}}{2}\]
D) \[\frac{{{t}_{1}}-{{t}_{2}}}{2}\]
E) \[\frac{{{t}_{1}}{{t}_{2}}}{{{t}_{1}}+{{t}_{2}}}\]
Correct Answer: E
Solution :
Let\[{{R}_{1}}\]and\[{{R}_{2}}\]be the resistances of the coils \[H=\frac{{{V}^{2}}{{t}_{1}}}{{{R}_{1}}}\]and \[H=\frac{{{V}^{2}}{{t}_{2}}}{{{R}_{2}}}\] \[\Rightarrow \] \[\frac{{{t}_{1}}}{{{R}_{1}}}=\frac{{{t}_{2}}}{{{R}_{2}}},\]i.e., \[\frac{{{R}_{2}}}{{{R}_{1}}}=\frac{{{t}_{2}}}{{{t}_{1}}}\] ?? (1) Now in parallel \[R=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}}={{R}_{1}}\] \[\therefore \] \[H=\frac{{{V}^{2}}t}{R}\] ?.. (2) Now \[\frac{{{V}^{2}}t}{R}=\frac{{{V}^{2}}{{t}_{1}}}{{{R}_{1}}}\] \[\Rightarrow \] \[\frac{t\times ({{R}_{1}}+{{R}_{2}})}{{{R}_{1}}{{R}_{2}}}=\frac{{{t}_{1}}}{{{R}_{1}}}\] ?? (3) On using Eqs. (1) and (3), we get \[t=\frac{{{t}_{1}}{{t}_{2}}}{{{t}_{1}}+{{t}_{2}}}\]
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