A series LCR- circuit containing a resistance of `120 Omega` has angular resonance frequency `4 xx10^(5) rads^(-1)`. At resonance the voltages across resistance and inductance are 60 V and 40 V respectively. Find the values of L and C. At what frequency the current in the circuit lags the voltage by `45^(@)`?
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V At resonance as`X=0,I=(V)/(R)=(60)/(120)=(1)/(2)A` and`V_(L)=IX_(L)=IomegaL,L=(V_(L))/(Iomega)` So`L=(40)/(0.5xx4xx10^(5))=0.25mH` `i.e.,C=(1)/(Lomega_(@)^(2))=(1)/0.2xx10^(-3)xx(4xx10^(5))^(2)=(1)/(32)muF` Now in case of series LCR circuit `tanphi=(X_(L)-X_(C))/(R)` So current will lag the applied voltage by `45^(@) if, tan 45^(@)=(omega-(1)/(omegaC))/(R)` i.e.`1xx120=omegaxx2xx10^(-4)-(1)/(omega(1//32)xx10^(-6)` i.e.,`omega-6xx10^(5)omega-16xx10^(10)=0` i.e.,`omega=(6xx10^(5)pmsqrt((6xx10^(5))^(2)+64xx10^(10)))/(2)` i.e.,`omega=(6xx10^(5)+10xx10^(5))/(2)=8xx10^(5)rad//S`
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