Question 144 of 190

When the rms voltages ${V}_{L}, {V}_{C}$ and ${V}_{R}$ are measured respectively across the inductor $L,$ the capacitor $C$ and the resistor $R$ in a series $LCR$ circuit connected to an $AC$ source, it is found that the ratio ${V}_{L}:{V}_{C}:{V}_{R}=1:2:3.$ If the rms voltage of the $AC$ source is $100V,$ then ${V}_{R}$ is close to :

A$50V$B$70V$C$100V$D$90V$

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Answer: (D) $90V$

${V}_{L}:{V}_{C}:{V}_{R}=1:2:3$

${V}_{\text{rms}}=100 \text{V}$ then ${V}_{R}=?$



Let ${V}_{L}=x,$ then ${V}_{C}=2x$ and ${V}_{R}=3x$

using $V=\sqrt{{V}_{R}^{2}+{({V}_{c}-{V}_{L})}^{2}}$
$100=\sqrt{{\left(3x\right)}^{2}+{\left(2x-x\right)}^{2}}$

$100=\sqrt{10{x}^{2}}$




$x=\dfrac{100}{\sqrt{10}}$ $=10\sqrt{10}$

$∴{V}_{R}=3x$$=30\sqrt{10}\approx 90V$