Welcome, everyone! Today we are going to explore what happens when we apply alternating current, or AC voltage, to a resistor. Can anyone tell me what AC voltage looks like?

Exactly! AC voltage varies with time and can be represented by the equation v = v_m sin(ωt). Here, v_m is the peak voltage. Now, if we connect this to a resistor, we can find the current through it using Ohm's Law.

Great question! It's similar to DC circuits. We use $i = \frac{v_m}{R} \sin(ωt)$. This tells us that the current is directly proportional to the voltage divided by the resistance.

Absolutely! The current also varies sinusoidally, and both current and voltage are in phase in this case.

Correct! This points us to an important characteristic of resistors in AC circuits—both waveforms are synchronized.

To summarize, we learned that in AC circuits, the voltage and current through a resistor reach their maximum and minimum values simultaneously. This is a fundamental concept in electrical engineering.

Now, let’s delve into how we calculate power in these circuits. The average power dissipated in a resistor is given by the formula $P_{avg} = \frac{1}{2} i_m^2 R$. Can someone remind me what $i_m$ represents?

That's correct! To bridge it with DC circuits, we often use root mean square values. Can anyone explain what rms current is?

Spot on! We define it as $I = \frac{i_m}{\sqrt{2}}$. The same applies for voltage. What would the rms voltage be?

Exactly! The average power can then also be expressed as $P = IV$, which reflects the similarity between AC and DC circuits.

Absolutely! Now, to recap, we’ve covered how power is determined for AC circuits, and how rms values help us understand and calculate that power effectively.

Let’s now discuss a critical takeaway: the phase relationship in AC circuits. In resistors, both current and voltage are in phase. Why do you think that's significant?

Great observation! This allows us to apply similar techniques in different contexts. Also, remember that average power, in this case, is not zero even though the individual current flows oscillate.

Good question! Although instantaneous current oscillates, when squared and averaged over a cycle, the result is a positive value. Thus, Joule heating occurs—meaning energy is dissipated.

That's correct! The energy consumed translates to heat, which is why resistors are often rated for power dissipation.

In summary, we now understand that power in resistive circuits remains positive due to the consistent relationship between current and voltage sine waves.