Electromagnetic Induction and Alternating Current 6 Question 9
####9. The instantaneous voltages at three terminals marked $X, Y$ and $Z$ are given by $V_{X}=V_{0} \sin \omega t$,
$V_{Y}=V_{0} \sin \Big(\omega t+\frac{2 \pi}{3})$ and $V_{Z}=V_{0} \sin \Big(\omega t+\frac{4 \pi}{3})$.
An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $X$ and $Y$ and then between $Y$ and $Z$. The reading(s) of the voltmeter will be
(2017 Adv.)
(a) $(V _ {Y Z}) _ {\mathrm{rms}}=V _ {0} \sqrt{\frac{1}{2}}$
(b) $(V _ {X Y}) _ {\mathrm{rms}}=V _ {0} \sqrt{\frac{3}{2}}$
(c) independent of the choice of the two terminals
(d) $(V _ {X Y}) _ {\mathrm{rms}}=V _ {0}$
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Answer:
Correct Answer: 9. (b, c)
Solution:
- $V_{X Y}=V_{0} \sin \omega t+\frac{2 \pi}{3}-V_{0} \sin \omega t$
$$ =V_{0} \sin \omega t+\frac{2 \pi}{3}+V_{0} \sin (\omega t+\pi) $$
$$ \Rightarrow \quad \varphi=\pi-\frac{2 \pi}{3}=\frac{\pi}{3} $$
$\Rightarrow \quad V_{0}{ }^{\prime}=2 V_{0} \cos \frac{\pi}{6}=\sqrt{3} V_{0}$
$\Rightarrow \quad V_{X Y}=\sqrt{3} V_{0} \sin (\omega t+\varphi)$
$\Rightarrow \quad (V _ {X Y}) _ {\mathrm{rms}}= (V _ {Y Z}) _ {\mathrm{rms}}=\sqrt{\frac{3}{2}} V _ {0}$
