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Chapter 28: Problem 5

The same AC voltage appears across a capacitor and a resistor, and the same rms current flows in each. Is the power dissipation the same in each?

Short Answer

Expert verified
No, the power dissipation is not the same in each. The resistor dissipates power \(p_R = Vrms \times Irms\), but the capacitor dissipates no power as it is out of phase.

Step by step solution

01

Understanding RMS Current and AC Voltage

In AC circuits, power dissipation across resistive components (such as resistors) can be calculated using the RMS values of voltage and current because the current and voltage are in phase. This means ups and downsof voltage and current coincide. So here, power p = Vrms * Irms.
02

Power Dissipation Across Resistive Components

Apply the power formula for resistive loads, \(p = Vrms \times Irms\). As both the voltage and current are the same for the resistor and the capacitor, the power dissipation for the resistor is \(p_{R} = Vrms \times Irms\).
03

Power Dissipation in Capacitive Components

For power dissipation across capacitive components (such as capacitors), it's important to note that in an ideal capacitor in AC circuits, the current and voltage are out of phase by 90 degrees. Thus, power dissipated is zero because power is also defined by \(p = Vrms \times Irms \times cos(\Phi)\), where \(\Phi\) is the phase difference between current and voltage. For an ideal AC capacitor, this is 90 degrees and cos(90) is zero. Thus, the power dissipated by the capacitor is \(p_C = Vrms \times Irms \times cos(90) = 0\)
04

Compare Power Dissipation

As the power dissipated by the resistor is \(Vrms \times Irms\) and the power dissipated by the capacitor is 0, their power dissipations are not the same.

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