Question

Find the voltage across inductor at $t=\mu \omega$ if an A.C. circuit having supply voltage $E$ consists of a resistor of resistances $3\mathrm{\Omega }$ and an inductor of reactance $4\mathrm{\Omega }$ as shown in the figure.

A. $2volt$
B. $10volts$
C. Zero
D. $4.8volts$

Answer 1

Hint Evaluate the total impedance of series inductive – resistor A.C circuit by the expression –
$Z=\sqrt{R^2+X_L^2}$
where, $R$ is the resistor of resistance and $X_L$ is the inductor of reactance.
Now, put $t=\mu \omega$ in $E=10\sin \omega t$
Then, $\sin ({\omega }^2\times {10}^{-6})$ will be approximately equal to zero. So, current will be equal to zero.

Complete step-by-step solution:Let $R$ be the resistor of resistance and $X_L$ be the inductor of reactance.
So, according to the question, it is given that -
$R=3\mathrm{\Omega }$
$X_L=4\mathrm{\Omega }$
Now, calculating the impedance for this inductive – resistor circuit so, this calculated by using the formula –
$Z=\sqrt{R^2+X_L^2}$
Putting the values of resistance and inductor of reactance in the above equation –
$Z=\sqrt{3^2+4^2}$
$Z=5\mathrm{\Omega }$
Therefore, the value for impedance of this circuit is $5\mathrm{\Omega }$.
Now, it is given that –
$E=10\sin \omega t$
Also, it is given in equation that, $t=\mu \omega$, therefore, putting this value of $t$ in the equation of $E$, we get
$$\begin{gathered} E=10\sin \omega (\mu \omega ) \\ E=10\sin \mu {\omega }^2 \end{gathered}$$
We know that, $\mu ={10}^{-6}$
$\therefore E=10\sin ({\omega }^2\times {10}^{-6})$
If we take the value of ${\omega }^2\times {10}^{-6}$ then, we come to know that its value is approximately very low
${\omega }^2\times {10}^{-6}\approx verylow$
Therefore, $\sin ({\omega }^2\times {10}^{-6})$ becomes approximately equal to zero.
$\sin ({\omega }^2\times {10}^{-6})\approx 0$
Then, current also becomes zero.
So, if we calculate the voltage across inductor, we get that –
$V_L=i{X}_L$
Because, the current is equal to zero
$\therefore V_L=0$

Therefore, option (C), zero volt is the correct option.

Note:- The R – L circuit can be defined as electrical circuit of elements which includes resistor R and inductance L connected together having source as voltage or as current.
The impedance of series R – L circuit is the combined effect of resistance R and inductive reactance $X_L$ of circuit as whole.