In a parallel RC circuit excited by AC, which statement correctly describes the relationship between the branch currents and the total current?

Introduction / Context:
Parallel RC circuits are ubiquitous in filters and timing networks. Because the resistor branch current and capacitor branch current are out of phase, the way they combine is vectorial rather than simple arithmetic. Recognizing this prevents serious errors in current estimation and power factor analysis.

Given Data / Assumptions:

• Parallel connection of a resistor R and capacitor C across the same AC voltage.
• Steady-state sinusoidal excitation.
• Ideal components with negligible parasitics for conceptual clarity.

Concept / Approach:
In a parallel RC, branch currents are I_R = V / R (in phase with V) and I_C = V / X_C (leading V by 90 degree), where X_C = 1 / (2 * pi * f * C). The total current is the phasor sum: I_T = vector sum of I_R and I_C. Since the angle between I_R and I_C is 90 degree, the magnitude is I_T = sqrt( I_R^2 + I_C^2 ), which is strictly less than the arithmetic sum I_R + I_C whenever both are nonzero.

Step-by-Step Solution:
1) Compute I_R = V / R, aligned with voltage.2) Compute I_C = V / X_C, leading voltage by 90 degree.3) Combine as perpendicular vectors: I_T = sqrt( I_R^2 + I_C^2 ).4) Conclude I_T is less than I_R + I_C due to quadrature addition.

Verification / Alternative check:
Phasor diagram shows a right triangle with legs I_R and I_C and hypotenuse I_T. The hypotenuse is less than the sum of the legs, matching the inequality.

Why Other Options Are Wrong:

• Option A: Uses arithmetic sum, which ignores phase difference.
• Option B: Phases differ between resistor and capacitor branches.
• Option D: Greater than the sum is impossible for any triangle inequality.
• Option E: Incorrect because Option C is correct.

Common Pitfalls:
Adding AC branch currents arithmetically is a frequent mistake. Always combine out-of-phase currents vectorially using phasors or rectangular components.

Final Answer:
The total current is less than the sum of the currents for the resistance and capacitance.