Question

A device $X$ is connected across an ac source of voltage $V=V_0\sin \omega t$. The current through $X$ is given as $I=I_0\sin (\omega t+\frac{\pi }{2})$.
(a) Identify the device $X$ and write the expression for its reactance.
(b) Draw graphs showing variation of voltage and current with time over one cycle of ac, for $X$.
(c) How does the reactance of the device $X$ vary with frequency of the ac? Show this variation graphically.
(d) Draw the phasor diagram for the device $X$.

Answer 1

(a) Here $V=V_0\sin \omega t$

$I=I_0\sin (\omega t+\frac{\pi }{2})$

Since current is leading the voltage by $\frac{\pi }{2}$, hence the circuit must be capacitive and the element "X" must be a capacitor.

Its reactance is given by $X_c=\frac{1}{\omega C}$

(b) Graph of voltage and current w.r.t time in fig(b)

(c) Graph showing the variation of $X_C$ with frequency $f$ in fig(c)

Since $X_c=\frac{1}{2\pi fC}$

$X_c\propto \frac{1}{f}$

Hence, reactance is inversely proportional to the frequency.

(d) Phase diagram for device $X$ is shown in fig.(d)