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  • Study the circuits (a) and (b) shown in Fig 7.2 and answer the following questions. (a) Under which conditions would the rms currents in the two circuits be the same?(b) Can the rms current in circuit (b) be larger than that in (a)?

Study the circuits (a) and (b) shown in Fig 7.2 and answer the following questions.

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

Answers (1)

Explanation:-

I_{rms} \: in (a)=\frac{V_{rms}}{R}
\\I_{rms} in (b)=\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C )}}\\\\ \\(a)Now (I_{rms} )_a=(I_{rms} )_b\\\\ \frac{V_{rms}}{R}=\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C )^2}}\\\\ \\R=\sqrt{R^2+(X_L-X_C )}

Squaring both sides
\\R^2=R^2+\left ( X_L-X_C \right )^{2}\\\\ Or \left ( X_L-X_C \right )^{2}=0\\ \\X_L=X_C
So, I_{rms} in circuits, a and b will be equal if \\ \\X_L=X_C

(b)\: \: For (I_{rms} )_b>(I_{rms })_a

The rms current is inversely proportional to the total impedance Z of the circuit

\\\frac{V_{rms}}{\sqrt{R^2+(X_L-X_C )}}>\frac{V_{rms}}{R}\\ \\ As V_{rms=V} \\\\So\: \: \sqrt{R^2+(X_L-X_C )}<R \\ \\ Squaring\: \: both \: \: sides R^2+(X_L-X_C )^2<R^2

(X_L-X_C )^2<0

The square of a number cannot be negative. 

Therefore, the rms current in a circuit (b) has to be larger than that in (a).

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infoexpert21

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