In a 20 V_ac RMS series RC circuit, the measured RMS voltages are 20 V across the resistor and 40 V across the capacitor. What is the applied source voltage (RMS)?

Introduction:
In series RC circuits, element voltages are out of phase by 90 degrees between the resistive and capacitive branches. The source voltage is the vector (phasor) sum, not the arithmetic sum, of the component voltages. This problem reinforces correct phasor addition.

Given Data / Assumptions:

• Series RC network, single frequency steady state.
• RMS voltages: VR = 20 V, VC = 40 V.
• Resistive voltage in phase with current, capacitive voltage lags current by 90 degrees.

Concept / Approach:
For orthogonal phasors, the magnitude of the source voltage is Vs = sqrt( VR^2 + VC^2 ). The series nature ensures the same current through both elements, but voltages add as vectors because of the phase difference.

Step-by-Step Solution:
1) Compute squares: VR^2 = 20^2 = 400; VC^2 = 40^2 = 1600.2) Add: 400 + 1600 = 2000.3) Take square root: Vs = sqrt(2000) ≈ 44.721 V.4) Round to nearest option: approximately 45 V.

Verification / Alternative check:
Check current I from VR = I * R and VC = I * Xc if desired; the 90 degree phase difference guarantees Vs slightly greater than the larger component voltage and much less than their arithmetic sum, which matches 45 V.

Why Other Options Are Wrong:
50 V, 55 V, 60 V: treat voltages as partially or fully in phase, overestimating Vs.
42 V: underestimates the vector sum; smaller than VR^2 + VC^2 square root.

Common Pitfalls:
Adding voltages arithmetically (20 + 40 = 60 V) or subtracting them directly. Always perform phasor addition for orthogonal components.

Final Answer:
45 V