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In a certain series RLC circuit being driven at a frequency of 60.0 Hz, the maximum voltage across the inductor is 2.00 times the maximum voltage across the resistor and 2.00 times the maximum voltage across the capacitor. By what angle does the current lag the generator emf?

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(a)\textbf{(a)} Assume we have an RLCRLC circuit which is driven be a frequency generator, given that Em=30.0\mathcal{E}_m=30.0 V and I=300I=300 mA. The maximum voltage across the inductor VL=2.00VR=2.00VCV_L=2.00V_R=2.00V_C where VCV_C is the voltage across the capacitor and VRV_R is the voltage across the resistor. The phase angle is given by:

tan(Φ)=XLXCR\tan(\Phi)=\dfrac{X_L-X_C}{R}

multiply the right hand side by I/II/I, so:

tan(Φ)=IXLIXCIR\tan(\Phi)=\dfrac{IX_L-IX_C}{IR}

but,

VL=IXLVC=IXCVR=IRV_L=IX_L \qquad V_C=IX_C \qquad V_R=IR

therefore:

tan(Φ)=VLVCVR\tan(\Phi)=\dfrac{V_L-V_C}{V_R}

using VL=2.00VRV_L=2.00V_R and VC=VRV_C=V_R, to get:

tan(Φ)=2.00VRVRVR\tan(\Phi)=\dfrac{2.00V_R-V_R}{V_R}

tan(Φ)=1\tan(\Phi)=1

Φ=45.0\boxed{\Phi=45.0 }

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