Question

In a black box of unknown elements ($L$ or $R$ or any other combination), an ac voltage $E=E_0\sin (\omega t+\varphi )$ is applied and current in the circuit was found to be $I=I_0\sin [\omega t+\varphi +\left(\frac{\pi }{4}\right)]$. Then the unknown elements in the box may be

• A. Only capacitor
• B. Inductor and resistor both
• C. Either capacitor, resistor and inductor or only capacitor and resistor
• D. Only resistor

Answer 1

The correct option is C Either capacitor, resistor and inductor or only capacitor and resistor

Here,current is leading the voltage by phase angle $\frac{\pi }{4}$.

Let's assume that it is an $RLC$ circuit i.e all 3 elements are present in the black box.


The phasor diagram for the circuit is given below


Case a: If there is only the capacitor $C$, then we can see that the net voltage will lag the current by $\frac{\pi }{2}$

Case b: With an inductor and a resistor, the net voltage will lead the current by some phase angle.

Case c: With only capacitor and resistor, the net voltage will lag the current. So it's possible for voltage to lag by $\frac{\pi }{4}$ for some value of $C$ and $R$. Similarly, with all three elements of $LCR$, voltage can lag current, provided $V_C>V_R$

Case d : Not possible since the voltage and current will be in phase.

Hence option c is correct.