An ac voltmeter with large impedance is connected in turn...

An ac voltmeter with large impedance is connected in turn across the inductor, the capacitor, and the resistor in a series circuit having an alternating emf of 100 $\mathrm{V}(\mathrm{rms})$ ; the meter gives the same reading in volts in each case. What is this reading?

Key Concepts

Voltage Division in Series AC Circuits

Voltage division in a series AC circuit involves the distribution of the total applied voltage among the individual elements based on their impedances. Because the components may have different phase angles, the division must be considered vectorially (phasor addition), not just by magnitude. This principle is essential when predicting the voltage drop across each element in an AC series circuit.

Impedance in AC Circuits

In AC circuits, impedance is the complex quantity that extends the idea of resistance to encompass both the resistive (real) and reactive (imaginary) effects. It is defined as the ratio of the voltage to the current, taking into account both magnitude and phase, and is expressed in ohms. This concept is fundamental to analyzing how various circuit components, including resistors, inductors, and capacitors, respond to an alternating current source.

Inductive and Capacitive Reactance

Inductive reactance and capacitive reactance are the frequency?dependent components of impedance found in inductors and capacitors, respectively. Inductive reactance (given by ?L, where ? is the angular frequency and L is the inductance) causes the current to lag behind the voltage, while capacitive reactance (given by 1/(?C), where C is the capacitance) causes the current to lead the voltage. These reactances determine how much voltage appears across each reactive element in an AC circuit.

Phasor Addition and Resonance in RLC Circuits

Phasor addition is a technique used to add sinusoidally varying quantities (voltages or currents) that are out of phase, by representing them as vectors in the complex plane. In a series RLC circuit, if the inductive and capacitive reactances are equal, their phasor contributions cancel each other, a condition known as resonance. This cancellation means that the total impedance is purely resistive, though individual elements may still develop voltage drops whose magnitudes can exceed the applied voltage when considered separately.

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