Study the circuits (a) and (b) shown in figure and answer the following question.

i) Under which conditions would the rms currents in the two circuits be the same?
ii) Can the rms current in circuit (b) be larger than
that in (a)?

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Solution

(i) Formula used:
$I_{r m s} = V_{r m s} Z = \frac{V_{r m s}}{\sqrt{R^{2} + (X_{C} - X_{L})^{2}}}$

Suppose that rms current in circuit (a)= $(I_{r m s})_{a}$

And rms current in circuit (b)= $(I_{r m s})_{b}$
$(I_{r m s})_{a}$ = $\frac{V_{r m s}}{Z} = \frac{V_{r m s}}{R}$

Circuit contains only Resistance.

$I_{r m s} = V_{r m s} Z = \frac{V_{r m s}}{\sqrt{R^{2} + (X_{C} - X_{L})^{2}}}$

According to condition,

$(I_{r m s} a)$ = $(I_{r m s})_{b}$

$R = \sqrt{R^{2} + (X_{C} - X_{L})^{2}}$

Squaring both sides,
$R^{2} = R^{2} + (X_{C} - X_{L})^{2}$

$(X_{C} - X_{L})^{2} = 0$

$X_{L} = X_{C}$ ( This is the condition of resonance)

If $V_{r m s}$ in the two circuits are same, then at resonance the rms current in LCR will be same as that in R circuit.

Final Answer: If rms voltage in the two circuits is same then at resonance, the rms current in LCR will be same as that in R circuit.

(ii) Formula used:
$I_{r m s} = V_{r m s} Z = \frac{V_{r m s}}{\sqrt{R^{2} + (X_{C} - X_{L})^{2}}}$

As per our condition,

$\frac{(I_{r m s})_{b}}{(I_{r m s})_{a}} = \frac{V_{r m s} / Z}{V_{r m s} / R} = \frac{R}{\sqrt{R^{2} + (X_{C} - X_{L})^{2}}}$

$\frac{(I_{r m s})_{b}}{(I_{r m s})_{a}} = \frac{R}{Z} \leq 1$

$\Rightarrow (I_{r m s})_{a} \geq (I_{r m s})_{b}$

$R \leq Z$ . So, rms current in circuit (b) cannot be larger than that in (a).

Final Answer: No, because $R \leq Z$ ,
so $(I_{r m s})_{a} \geq (I_{r m s})_{b}$

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