question_answer

impedance of circuit when a resistance R and a inductor of inductance L are connected in series in an AC circuit of frequency v, is :

A) \[\sqrt{{{R}^{2}}+2{{\pi }^{2}}{{v}^{2}}{{L}^{2}}}\]            

B) \[\sqrt{{{R}^{2}}+4{{\pi }^{2}}{{v}^{2}}{{L}^{2}}}\]

C) \[\sqrt{R+4{{\pi }^{2}}{{v}^{2}}{{L}^{2}}}\]           

D) \[\sqrt{R+2{{\pi }^{2}}{{v}^{2}}{{L}^{2}}}\]

Correct Answer: B

Solution :

Key Idea: Impedance 15 the effective resistance of L-R series circuit. Let alternating emf be applied to a circuit containing L and R in series. \[\therefore \]     \[{{E}^{2}}=V_{R}^{2}+V_{L}^{2}\] \[{{E}^{2}}={{i}^{2}}{{R}^{2}}+X_{L}^{2}\] \[\Rightarrow \]        \[i=\frac{E}{\sqrt{{{R}^{2}}+X_{L}^{2}}}\] Applying Ohm's law we see that \[\sqrt{{{R}^{2}}+X_{L}^{2}}\]is the effective resistance of the circuit. It is known as impedance Z \[Z=\sqrt{{{R}^{2}}+X_{L}^{2}}\] Also,      \[{{X}_{L}}=\omega L\] \[\therefore \]     \[Z=\sqrt{{{R}^{2}}+{{(\omega L)}^{2}}}\] Here    \[\omega =2\pi v\] \[\therefore \]     \[Z=\sqrt{{{R}^{2}}+{{(2\pi vL)}^{2}}}\] \[Z=\sqrt{{{R}^{2}}+4{{\pi }^{2}}{{v}^{2}}{{L}^{2}}}\]