A resistor of 200 and a capacitor of 15.0 F are connected in series to a 220 V, 50 Hz ac source. (a) Calculate the current in the circuit; (b) Calculate the voltage (rms) across the resistor and the capacitor. Is the algebraic sum of these voltages more than the source voltage? If yes, resolve the paradox.
The current in the circuit is 0.755 A.
Voltage across resistor (V_R) = 151 V, Voltage across capacitor (V_C) = 160.3 V. The algebraic sum (311.3 V) exceeds the source voltage (220 V). This paradox is resolved by recognizing that V_R and V_C are out of phase by 90° and cannot be added algebraically. Using Pythagorean theorem: V = = = 220 V, which equals the source voltage.
Explanation
This example demonstrates the concept of phase difference in AC circuits. In a series RC circuit, the voltage across resistor (V_R) is in phase with current, while voltage across capacitor (V_C) lags current by 90°. Due to this phase difference, the voltages must be added vectorially, not algebraically. The impedance Z combines resistance and capacitive reactance using Pythagorean theorem, and similarly, the total voltage is the vector sum of individual voltages.