Question

In an L-R-C series circuit, the rms voltage across the resistor is 30.0 V, across the capacitor it is 90.0 V, and across the inductor it is 50.0 V. What is the rms voltage of the source?

Short answer

The RMS voltage of the source is 50.0 volts

Step-by-step solution

concept

Resistance is measure of opposition to the flow of current in a closed electrical circuit. It is measured in Ohm (Ω).

An inductor is a passive two-terminal device that stores energy in a magnetic field when current passes through it. When an inductor is attached to an AC supply, the resistance produced by it is called inductive reactance (X_L).

A capacitor is a device consisting of two conductors in close proximity that are used to store electrical energy. These conductors are insulated from each other. When a capacitor is attached to an AC supply, the resistance produced by it is called capacitive reactance (X_C).

Root Mean Square (RMS) value is defined as the effective value of an alternating quantity. It is assumed as the equivalent steady value which will give the same effect.

given values

RMS value for individual electric component is

$$\begin{gathered} {\left(V_R\right)}_{rms}=30.0V \\ {\left(V_C\right)}_{rms}=90.0V \\ {\left(V_L\right)}_{rms}=50.0V \end{gathered}$$

Calculation

Thus, the RMS value of the source voltage is given by

$$\begin{gathered} {\left(V_S\right)}_{rms}=\sqrt{{{\left(V_R\right)}_{rms}}^2+{\left[{\left(V_L\right)}_{rms}-{\left(V_C\right)}_{rms}\right]}^2} \\ =\sqrt{30.0^2+{\left[50.0-90.0\right]}^2} \\ =50.0V \end{gathered}$$

Therefore, the RMS voltage of the source is 50.0 volts.