Series RC phasor sum — a resistor shows 6 V (rms) and a capacitor shows 8 V (rms) in the same series RC branch. Using vector (phasor) addition, what is the magnitude of the source voltage?

Introduction / Context:
In AC circuits containing reactance, voltage drops do not add arithmetically. Instead, they combine as vectors (phasors) because they are out of phase. This question tests your ability to use the right kind of addition for a series RC circuit where resistor and capacitor voltages are in quadrature (90 degrees apart).

Given Data / Assumptions:

• Series RC circuit under sinusoidal steady state.
• Measured magnitudes: V_R = 6 V (in phase with current), V_C = 8 V (lags current by 90 degrees).
• The source voltage equals the phasor sum of V_R and V_C.

Concept / Approach:
In a series RC circuit, current is common. The resistor voltage is V_R = I * R and is in phase with the current; the capacitor voltage is V_C = I * Xc, but since Xc introduces -90 degrees, V_C is 90 degrees behind the current. Therefore, V_R and V_C are orthogonal, making the source magnitude Vs the Pythagorean sum: Vs = sqrt(V_R^2 + V_C^2).

Step-by-Step Solution:

Verification / Alternative check:
If you convert to impedance and current, the same result emerges. For some I, V_R = I*R and V_C = I*Xc; geometry dictates a right triangle, yielding Vs = 10 V when the legs are 6 V and 8 V.

Why Other Options Are Wrong:

• Arithmetic sum (14 V) ignores the 90-degree phase difference.
• “2 V” is non-physical for these magnitudes; phasor subtraction is not appropriate here.
• Frequency is not needed once V_R and V_C are given; their vector relationship is fixed.

Common Pitfalls:
Adding AC drops like DC series voltages or forgetting that resistor and capacitor voltages are orthogonal.

Final Answer:
Correct — Vs = 10 V.