MHT CET · Physics · Alternating Current
With an alternating voltage source of frequency ' \(\mathrm{f}\) ', inductor ' \(\mathrm{L}\) ', capacitor ' \(\mathrm{C}\) ' and resistance ' \(R\) ' are connected in series. The voltage leads the current by \(45^{\circ}\). The value of ' \(L\) ' is \(\left(\tan 45^{\circ}=1\right)\)
- A \(\left(\frac{1+2 \pi f \mathrm{CR}}{4 \pi^2 \mathrm{f}^2 \mathrm{C}}\right)\)
- B \(\left(\frac{1-2 \pi \mathrm{fCR}}{4 \pi^2 \mathrm{f}^2 \mathrm{C}}\right)\)
- C \(\left(\frac{4 \pi^2 \mathrm{f}^2 \mathrm{C}}{1+2 \pi \mathrm{fCR}}\right)\)
- D \(\left(\frac{4 \pi^2 \mathrm{f}^2 \mathrm{C}}{1-2 \pi \mathrm{fCR}}\right)\)
Answer & Solution
Correct Answer
(A) \(\left(\frac{1+2 \pi f \mathrm{CR}}{4 \pi^2 \mathrm{f}^2 \mathrm{C}}\right)\)
Step-by-step Solution
Detailed explanation
\(\tan \phi=\frac{X_L-X_C}{R} \)
\( \Rightarrow \tan 45^{\circ}=\left[\frac{2 \pi \mathrm{fL}-\frac{1}{2 \pi \mathrm{fC}}}{R}\right] \ldots .\left(\because \phi=45^{\circ}\right) \)
\( R=\frac{(2 \pi \mathrm{f})^2 \mathrm{LC}-1}{2 \pi \mathrm{fC}} \)
\( \therefore L=\frac{2 \pi \mathrm{fCR}+1}{4 \pi^2 \mathrm{f}^2 \mathrm{C}}\)
\( \Rightarrow \tan 45^{\circ}=\left[\frac{2 \pi \mathrm{fL}-\frac{1}{2 \pi \mathrm{fC}}}{R}\right] \ldots .\left(\because \phi=45^{\circ}\right) \)
\( R=\frac{(2 \pi \mathrm{f})^2 \mathrm{LC}-1}{2 \pi \mathrm{fC}} \)
\( \therefore L=\frac{2 \pi \mathrm{fCR}+1}{4 \pi^2 \mathrm{f}^2 \mathrm{C}}\)
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