Question

If the readings ${V_1}$ and ${V_3}$ are $100volt$ , each then reading of ${V_2}$ is:

(A) $0volt$
(B) $100volt$
(C) $200volt$
(D) Cannot be determined by given information

Answer 1

Hint : Here, we are given the LRC circuit. We can see that the inductor, the resistor and the capacitor are connected in series connection here. In series connection, the current flowing through all the elements will be the same but the voltage across each element is different. We will use the formula for total voltage in the series connection for LRC circuit to find the required answer.
$V = \sqrt {{V_R}^2 + {{\left( {{V_L} - {V_C}} \right)}^2}}$ , Where, $V$ is the total voltage supply, ${V_R}$ is the voltage across the resistor, ${V_L}$ is the voltage across the inductor and ${V_C}$ is the voltage across the capacitor.

Complete Step By Step Answer:
From the given figure, we can see that ${V_1}$ is the voltage across the inductor, ${V_2}$ is the voltage across the resistor and ${V_3}$ is the voltage across the capacitor. Therefore if we use the formula $V = \sqrt {{V_R}^2 + {{\left( {{V_L} - {V_C}} \right)}^2}}$ for the given case , we will get
$V = \sqrt {{V_2}^2 + {{\left( {{V_1} - {V_3}} \right)}^2}}$
From the given data, we can write that $V = 200volt$ and ${V_1} = {V_3} = 100volt$ .
Now, we will put these values in the equation.
$\Rightarrow 200 = \sqrt {{V_2}^2 + {{\left( {100 - 100} \right)}^2}} = {V_2} \\\ \Rightarrow {V_2} = 100volt \\\$
Thus, the voltage across the resistor in the given case is $200volt$ .
Hence, option C is the right answer.

Note :
In this question, we have used the formula for the total voltage in the series connection of an RLC circuit. We have seen that when the voltage across the inductor and the voltage across the capacitor is same, the voltage across the resistor will be same as the total voltage supply. It is also important to remember that in this series connection of an RLC circuit, the current flowing through all the components is the same.