An alternating current (AC) series circuit contains a resistor, an inductor, and a capacit… — JAMB UTME

An alternating current (AC) series circuit contains a resistor, an inductor, and a capacitor. The root-mean-square (RMS) potential differences measured across the resistor, inductor, and capacitor are $80\text{ V}$ , $110\text{ V}$ , and $50\text{ V}$ , respectively. What is the total supply voltage across the circuit in volts?

Answer: 100 V

Answer

The total supply voltage across the series AC circuit is

100\text{ V}

In a series alternating current circuit, the voltages across the resistor, inductor, and capacitor are not in phase. The resistor voltage is in phase with the current, whereas inductor voltage leads by

90^\circ

and capacitor voltage lags by

90^\circ

. The total voltage is calculated using vector addition:

V = \sqrt{V_R^2 + (V_L - V_C)^2}

. Substituting the given values gives

V = \sqrt{80^2 + (110 - 50)^2} = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = 100\text{ V}

Step-by-Step Solution

Identify the RMS potential differences across each component.

V_R = 80\text{ V}

V_L = 110\text{ V}

, and

V_C = 50\text{ V}

In a series AC circuit, voltages across reactive components are out of phase with the resistor voltage.

Calculate the net reactive voltage difference between the inductor and capacitor.

V_L - V_C = 110\text{ V} - 50\text{ V} = 60\text{ V}

Inductive voltage leads current by

90^\circ

while capacitive voltage lags current by

90^\circ

, making them

180^\circ

out of phase with each other.

Determine total supply voltage using vector (phasor) addition.

V = \sqrt{V_R^2 + (V_L - V_C)^2} = \sqrt{80^2 + 60^2} = \sqrt{6400 + 3600} = \sqrt{10000} = 100\text{ V}

The resistive voltage and net reactive voltage are perpendicular (

90^\circ

phase angle difference).

Key Concept

Phasor Addition of Voltages in a Series AC Circuit

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