A) \[\frac{{{V}^{2}}}{\omega L-\frac{1}{\omega C}}\]
B) \[{{I}^{2}}\,C\,\omega \]
C) \[{{I}^{2}}R\]
D) \[\frac{{{V}^{2}}}{\omega C}\]
Correct Answer: C
Solution :
Key Idea: In a series L-C-R circuit, resonance occurs when capacitive reactance becomes equal to inductive reactance. In series L-C-R circuit at resonance, capacitive reactance \[({{X}_{C}})=\] inductive reactance \[({{X}_{L}})\] i.e., \[\frac{1}{\omega C}=\omega L\] Total impedance of the circuit \[Z=\sqrt{{{R}^{2}}+{{({{X}_{L}}-{{X}_{C}})}^{2}}}\] \[=\sqrt{{{R}^{2}}+{{\left( \omega L-\frac{1}{\omega C} \right)}^{2}}}\] i.e., Z = R So, power factor \[\cos \phi =\frac{R}{Z}=\frac{R}{R}=1\] Thus, power loss at resonance is given by \[P={{E}_{rms}}\,{{I}_{rms}}\,\cos \,\phi \] \[={{E}_{rms}}\,{{I}_{rms}}\times 1\] \[=({{I}_{rms}}\,R)\,{{I}_{rms}}\] \[={{({{I}_{rms}})}^{2}}\,R\] \[={{I}^{2}}R\]
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