One of your equations is wrong but is corrected later in the video - it is not the square of the sum of voltages, it is the sum of the square of each voltage.
Current in a resistor is a start-stop motion of conduction band electrons due to their collision with the rocking lattice ions, and this causes a resistor with a sinusoidal voltage applied, to produce heat. The polarity reversals of an applied sinusoidal voltage (with the direction reversals of the applied electric field) do not affect electron collisions with the lattice ions. Electrons colliding with lattice ions from either direction will continue to produce heat and there is no cancellation of the heat developed! Mathematically, the average value of a sinusoid is zero, and so, the average value of the current will also be zero. The average values cannot therefore represent the heat developed in a resistor with a sinusoidal current. A resistor cannot develop heat due to a current in one half-cycle and then cool itself by a like amount of heat during the next half-cycle! It develops heat either way whether the current is positive or negative. The lattice ions vibrate from collisions irrespective of the directions in which the electrons collide with them. Therefore, since the average value is zero, it necessitates the use of the root-mean-square values of the voltage and current to compute the power, which is a statistical measure of the magnitude of a varying quantity and is the square root of the arithmetic mean of the square of the sinusoidal function. Electrostatics and circuits belong to one science not two. To learn the operation of circuits it is instructive to understand Current, the conduction process, resistors and Voltage at the fundamental level as in the following two videos: i. ua-cam.com/video/REsWdd76qxc/v-deo.html and ii. ua-cam.com/video/8BQM_xw2Rfo/v-deo.html It is not possible in this post to discuss in more detail average and rms values. The last frame References in video #1 lists textbook 4 which discusses in detail using a unified approach sinusoidal voltage, current, their average and root mean square values.
One of your equations is wrong but is corrected later in the video - it is not the square of the sum of voltages, it is the sum of the square of each voltage.
Current in a resistor is a start-stop motion of conduction band electrons due to their collision with the rocking lattice ions, and this causes a resistor with a sinusoidal voltage applied, to produce heat. The polarity reversals of an applied sinusoidal voltage (with the direction reversals of the applied electric field) do not affect electron collisions with the lattice ions. Electrons colliding with lattice ions from either direction will continue to produce heat and there is no cancellation of the heat developed!
Mathematically, the average value of a sinusoid is zero, and so, the average value of the current will also be zero. The average values cannot therefore represent the heat developed in a resistor with a sinusoidal current.
A resistor cannot develop heat due to a current in one half-cycle and then cool itself by a like amount of heat during the next half-cycle! It develops heat either way whether the current is positive or negative. The lattice ions vibrate from collisions irrespective of the directions in which the electrons collide with them. Therefore, since the average value is zero, it necessitates the use of the root-mean-square values of the voltage and current to compute the power, which is a statistical measure of the magnitude of a varying quantity and is the square root of the arithmetic mean of the square of the sinusoidal function.
Electrostatics and circuits belong to one science not two. To learn the operation of circuits it is instructive to understand Current, the conduction process, resistors and Voltage at the fundamental level as in the following two videos:
i. ua-cam.com/video/REsWdd76qxc/v-deo.html and
ii. ua-cam.com/video/8BQM_xw2Rfo/v-deo.html
It is not possible in this post to discuss in more detail average and rms values.
The last frame References in video #1 lists textbook 4 which discusses in detail using a unified approach sinusoidal voltage, current, their average and root mean square values.