Final answer:
For a pure resistance R in an AC circuit, the average power absorbed is the product of the rms voltage and current, since voltage and current are in phase. For a pure inductance L or capacitance C, the average power absorbed is zero due to a 90° phase shift. At resonance in an RLC circuit, average power absorbed is maximized because impedance becomes purely resistive and voltage and current are in phase.
Step-by-step explanation:
If a voltage v(t) = Vmax cos(ω t) is applied across a pure resistance R, a pure inductance L, and a pure capacitance C, we need to consider the average power absorbed in each component separately. For a pure resistance R, the average power is given by Pave = Irms Vrms cos φ, where φ is the phase angle between voltage and current. Since a resistor does not introduce a phase shift between voltage and current (φ = 0°), cos φ is 1, and the average power is simply the product of the rms values of current and voltage divided by the resistance.
In a pure inductance L or pure capacitance C, the situation is different because they introduce a phase shift of φ = 90° between voltage and current. This means that cos φ is 0, and therefore, the average power absorbed by either the inductor or the capacitor is zero. Hence, in an AC circuit for a pure inductor or capacitor, no average power is dissipated as all the power is stored in magnetic or electric fields respectively and then returned to the circuit.
However, at resonance in an RLC circuit, the inductive reactance XL and capacitive reactance XC are equal but opposite in sign, canceling each other out. This means the impedance Z becomes purely resistive (Z=R), causing voltage and current to be in phase (φ = 0°). Consequently, at resonance, the average power (φ = 0°) absorbed is maximized, as indicated by the equation Pave = Vrms^2 / R, where Vrms is the rms voltage across the resistor.