Question

In an $$LCR$$ series a.c. circuit, the voltage across each of the components $$L,C$$ and $$R$$ is $$50V$$. The voltage across the $$LC$$ combination will be:
A) $50V$
B) $50\sqrt{2}V$
C) $100V$
D) $0V$ (zero)

Answer 1

Hint: In $$LCR$$ series a.c circuit, the components $$L,C$$ and $$R$$ are related to each other in a way that the voltage across the inductor $$L$$($V_L$) and voltage across the capacitor $$C$$($V_C$) has a constant phase difference. The voltage across the resistance $$R$$($V_R$) and current($i$) does not have any phase difference.

Complete step by step solution:
According to the question, a $$LCR$$ series a.c. circuit is given. The voltage across $$L,C$$ and $$R$$ is $$50V$$.

We know that in an $$LCR$$ series circuit, the voltage across the inductor $$L$$($V_L$) leads the current($i$) by $${90}^{\circ }$$ and voltage across the capacitor $$C$$($V_C$) lags the current($i$) by $${90}^{\circ }$$. So, the inductance and the capacitance are in opposite phases. In an $$LCR$$ series circuit, the voltage across the resistance $$R$$($V_R$) is in the same phase with current($i$).
So, the voltage across the $$LC$$ combination will be given as:
$$\begin{gathered} V_{LC}=V_L-V_C \\ \Rightarrow V_{LC}=50-50 \\ \Rightarrow V_{LC}=0 \end{gathered}$$
To understand the phase difference in different voltages, we can make a graph which shows the phase difference between voltages.

According to the above graph, $V_R$ and $i$ are in the same phase. $V_L$leads current $i$ by $${90}^{\circ }$$ and $V_C$ lags current $i$ by $${90}^{\circ }$$. So, the voltage across $$LC$$ combination is zero.

Hence, option (D) is correct.

Note: Voltage across the inductor $V_L$ and current $i$ has a phase difference. Voltage across the capacitor $V_C$ and current $i$ has a phase difference. The voltage across the resistance $V_R$ and current $i$ has zero phase difference. According to the graph, current $i$ and voltage across the resistance $V_R$ are plotted on X-axis.