AI Tutor
If alternating current pushes electrons forward then pulls them backward dozens of times every second, does an electric toaster heater turn off and cool down during the negative half of the cycle?
Because electrical power in a resistor depends on the square of current, energy is released as heat regardless of which direction charges flow.

Instantaneous Power in a Resistor

Instantaneous power (pp) is the rate of energy transfer at one exact instant, found by multiplying the instantaneous voltage v(t)v(t) by the instantaneous current i(t)i(t).
In a purely resistive load, voltage and current are strictly in phase, meaning both reach their positive crests, zero crossings, and negative troughs at the exact same moments.
When both voltage and current drop below zero, multiplying two negative numbers gives a positive product, keeping power strictly positive (p(t)β‰₯0p(t) \ge 0).
Notice how the power curve oscillates twice as fast as the original voltage signal β€” what is the exact mathematical proof behind this double-frequency wave?

Deriving the Power-Time Equation

Let an AC supply provide voltage v(t)=V0sin⁑(Ο‰t)v(t) = V_0 \sin(\omega t) across resistance RR, driving current i(t)=I0sin⁑(Ο‰t)i(t) = I_0 \sin(\omega t), where V0V_0 and I0I_0 are peak values and Ο‰=2Ο€f\omega = 2\pi f is angular frequency.
Multiplying both gives p(t)=V0I0sin⁑2(Ο‰t)p(t) = V_0 I_0 \sin^2(\omega t).
Using the trigonometric identity sin⁑2(ΞΈ)=1βˆ’cos⁑(2ΞΈ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}, we rewrite instantaneous power as two separate terms:
p(t)=V0I02βˆ’V0I02cos⁑(2Ο‰t)p(t) = \frac{V_0 I_0}{2} - \frac{V_0 I_0}{2}\cos(2\omega t)