Question

In an AC circuit, the current is $i=5 \sin(100t - \frac{\pi}{2})$ A and voltage is $e=200 \sin(100t)$ volt. Power consumption in the circuit is ($\cos 90^{\circ}=0$)

• A. 200 W
• B. 0 W
• C. 40 W
• D. 1000 W

Hint:

Logic Tip: When the phase difference between voltage and current is $90^{\circ}$, the current is called "wattless current" because no power is consumed.

Answer 1

The Correct Option is B

Concept: The average power consumed in an AC circuit is given by: \[ P = V_{\mathrm{rms I_{\mathrm{rms \cos \phi \] where $\phi$ is the phase difference between voltage and current.
Step 1: Identify voltage and current expressions Given: \[ v = V_0 \sin(100t), \qquad i = I_0 \sin\left(100t - \frac{\pi}{2}\right) \] Thus,

• Voltage phase = $100t$
• Current phase = $100t - \frac{\pi}{2}$

Step 2: Determine phase difference \[ \phi = 100t - \left(100t - \frac{\pi}{2}\right) = \frac{\pi}{2} \] \[ \phi = 90^\circ \]
Step 3: Calculate average power \[ P = V_{\mathrm{rms I_{\mathrm{rms \cos \phi \] \[ P = V_{\mathrm{rms I_{\mathrm{rms \cos 90^\circ \] \[ \cos 90^\circ = 0 \] \[ \Rightarrow P = 0 \]
Step 4: Physical interpretation Since the phase difference between voltage and current is $90^\circ$, the circuit behaves like a purely reactive circuit (either purely inductive or purely capacitive). In such circuits, energy is alternately stored and returned to the source, resulting in zero average power consumption. Final Answer: \[ \boxed{P = 0} \]