Parallel RC branch currents: A parallel RC circuit has 100 mA through the resistive branch and 100 mA through the capacitive branch (rms). What is the total rms source current?
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A141 mA
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B100 mA
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C282 mA
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D200 mA
Answer
Correct Answer: 141 mA
Explanation
Introduction / Context:In parallel RC circuits, the resistive branch current is in phase with the voltage, while the capacitive branch current leads the voltage by 90°. Because these branch currents are orthogonal phasors, the total rms current is found using the Pythagorean sum, not simple arithmetic addition.
Given Data / Assumptions:
- I_R = 100 mA (in phase with V).
- I_C = 100 mA (leads V by 90°).
- Sinusoidal steady state; branch currents are orthogonal.
Concept / Approach:
Total current magnitude I_T = √(I_R^2 + I_C^2) because the phasors are at right angles. This is analogous to vector addition of perpendicular components on the complex plane.
Step-by-Step Solution:
Square and sum: I_R^2 + I_C^2 = (0.1)^2 + (0.1)^2 = 0.01 + 0.01 = 0.02 A^2.Take square root: I_T = √0.02 ≈ 0.1414 A.Convert to mA: 0.1414 A ≈ 141 mA.Verification / Alternative check:
Phasor diagram confirms orthogonality; if the two were in phase, the total would be 200 mA, but here orthogonal components yield 141 mA.
Why Other Options Are Wrong:
200 mA assumes in-phase addition. 100 mA ignores the other branch. 282 mA wrongly doubles the Pythagorean result.
Common Pitfalls:
Adding magnitudes linearly; forgetting phase relationships in reactive branches.
Final Answer:
141 mA