Animated Solution for Physics - Electromagnetic Induction: An AC source rated 220 V, 50 Hz is connected to a resistor. The time taken by the current to change from its maximum to the rms value is

Imagine an AC circuit where the current is oscillating harmonically. The problem asks for the time taken for the current to change from its maximum value to its RMS (Root Mean Square) value.

To make our mathematical journey as elegant as possible, we should choose a function that naturally starts at its maximum. The cosine function is perfect for this! Let's define our instantaneous current as:

Here, $I_0$ is the peak (maximum) current, and $\omega$ is the angular frequency. At $t=0$, the current is $I_0\cos (0)=I_0$, which perfectly matches our starting condition.

We know from the fundamentals of alternating current that the RMS value is related to the peak value by a factor of $\sqrt{2}$. Specifically:

Our goal is to find the specific time $t$ when the instantaneous current $I(t)$ drops to this RMS value. So, we set up our master equation by equating the two:

The beauty of this setup is how quickly it simplifies. The peak current $I_0$ cancels out from both sides, leaving us with a pure trigonometric equation:

From our knowledge of standard angles, we know that the cosine of $\frac{\pi }{4}$ radians (or $45^{\circ }$) is $\frac{1}{\sqrt{2}}$. Therefore, the phase angle must be:

Now, we need to unpack the angular frequency $\omega$. We are given the frequency of the AC source as $f=50\text{ Hz}$. The relationship between angular frequency and standard frequency is $\omega =2\pi f$. Substituting the given value:

Finally, we substitute this back into our phase equation to isolate $t$:

To make this number more readable, we convert it into milliseconds by multiplying by $1000$:

And there we have it! It takes exactly $2.5\text{ ms}$ for a $50\text{ Hz}$ AC current to drop from its peak to its RMS value.