Question

At resonance the peak value of current in L-C-R series circuit is:
A) \[{{E}_{0}}/R\]
B) \[\dfrac{{{E}_{0}}}{\sqrt{{{R}^{2}}+{{(wL-\dfrac{1}{wc})}^{2}}}}\]
C) \[\dfrac{{{E}_{0}}}{\sqrt{2}\sqrt{{{R}^{2}}+{{({{w}^{2}}L-\dfrac{1}{{{w}^{2}}{{c}^{2}}})}^{2}}}}\]
D) \[\dfrac{{{E}_{0}}}{\sqrt{2R}}\]

Answer 1

Hint: You can check for the correct option by dimension analysis. We need to check the dimension for current. At resonance the impedance of the circuit is equal to the resistance.

Complete step by step solution: An LCR circuit, also known as a resonant circuit or an RLC circuit, is an electrical circuit which consists of an inductor (L), capacitor (C) and resistor (R) connected in series or parallel.
The resonance of a series LCR circuit occurs when the inductive and capacitive reactance are equal in magnitude but cancel each other because they are $180^\circ$ apart in phase. 
Since the inductance and capacitance cancel each other in magnitude hence the net impedance can be given by the following formula:

\[Z=\sqrt{{{R}^{2}}+({{X}_{L}}^{2}-{{X}_{C}}^{2})}\]
Where \[Z\] is the impedance
\[R\] is the resistance
\[{{X}_{L}}\] is the inductance
And \[{{X}_{C}}\] is the capacitance

Since \[{{X}_{L}}={{X}_{C}}\]
\[\Rightarrow Z=\sqrt{{{R}^{2}}+0}\]
\[\Rightarrow Z=R\]
Now current in LCR circuit at resonance will be
\[I=\dfrac{{{E}_{0}}}{Z}\]
\[\Rightarrow I=\dfrac{{{E}_{0}}}{R}\]

Hence Option (A) is the correct option.

Additional details: LCR circuit and RLC circuit means the same thing there really is no difference between them. Do not confuse. In a series LCR circuit, the phase difference between the current in the capacitor and the current in the resistor is equal to \[{{0}^{\circ }}\] because the same current flows through the capacitor as well as the resistor. Hence there is no lag. The voltage across the capacitor lags the current in the circuit by \[{{90}^{\circ }}\]. Hence, the phase difference between the voltage across the capacitor and the current in the circuit is \[{{90}^{\circ }}\] .

Note: Do remember that current does not lag in LCR circuit. And there is lag in the voltage.
The same formula can be used for LR, LC circuits by putting the value of resistance as zero.
Don’t confuse between resistance and impedance.