Question

The instantaneous potential difference between points $A$ and $B$ is :

• A. $8\sin (50\pi t+\frac{37\pi }{180})$
• B. $8\sin (50\pi t-\frac{37\pi }{180})$
• C. $10\sin \left(50\pi t\right)$
• D. $10\cos \left(50\pi t\right)$

Answer 1

The correct option is B $8\sin (50\pi t-\frac{37\pi }{180})$


Impedance of the given $\text{LCR}$ circuit,

$Z=\sqrt{{(X_L-X_C)}^2+R^2}$

$\Rightarrow Z=\sqrt{{(8-2)}^2+8^2}=10\Omega$

Phase angle,

$\varphi ={\tan }^{-1}\left(\frac{|X_L-X_C|}{R}\right)={\tan }^{-1}\left(\frac{3}{4}\right)={37}^{\circ }$

As $X_L>X_C$, voltage leads the current by ${37}^{\circ }$.

So, current passing through the resistor,

$i=\frac{10\sin (50\pi t-{37}^{\circ })}{10}=\sin (50\pi t-{37}^{\circ })$

The potential difference between $A$ and $B$ is,

$V_{AB}=i\times R=8\sin (50\pi t-{37}^{\circ })=8\sin (50\pi t-\frac{37\pi }{180})$

Hence, option $\left(B\right)$ is the correct answer.