Animated Solution for Physics - Electromagnetic Induction: A series L-C-R circuit of $R=5\Omega$, $L=20\text{ mH}$ and $C=0.5\mu \text{F}$ is connected across an AC supply of 250 V, having variable frequency. The power dissipated at resonance condition is ...... $\times 10^2\text{ W}$.

Imagine you are in a lab, looking at a series L-C-R circuit. We have an inductor of $20\text{ mH}$, a capacitor of $0.5\mu \text{F}$, and a resistor of $5\Omega$. They are all hooked up to an alternating current supply of $250\text{ V}$. The frequency of this supply can be changed.

Our goal is to find the power dissipated when the circuit hits resonance. The key phrase here is resonance condition. What exactly happens at resonance? Well, the inductive reactance ($X_L$) and the capacitive reactance ($X_C$) become perfectly equal.

Because they are out of phase, they cancel each other out completely. This means the total impedance of the circuit, $Z$, simplifies to just the resistance, $R$. The circuit behaves as if it's purely resistive!

Now, how do we calculate the power dissipated? In any AC circuit, the average power is the square of the RMS current times the resistance:

Since our impedance is just the resistance, the RMS current is simply the RMS voltage divided by the resistance ($I_{\text{rms}}=\frac{V_{\text{rms}}}{R}$). Substituting this, we get a beautiful, simple formula:

Power equals the square of the RMS voltage divided by the resistance. Let's bring back the values we were given. We know the RMS voltage is $250\text{ V}$, and the resistance is $5\Omega$.

Let's carefully substitute these into our power equation. We get $250$ squared, divided by $5$. Notice how we don't even need the values of inductance and capacitance! They were just there to distract us.

Time for some quick arithmetic. Two hundred and fifty squared is $62500$. Now, divide that by $5$. It comes out to be exactly $12500\text{ W}$. That's a lot of power being dissipated as heat by our little resistor!

We are almost there. The question asks for the answer in a specific format: some number multiplied by $10^2$. So, we rewrite $12500$ as $125\times 10^2$.

Comparing this with the given format, our missing number is $125$. And that is our final answer! Before we wrap up, think about this: what if the frequency was slightly off resonance? The impedance would no longer be just the resistance; it would be higher. This means the current would drop, and the power dissipated would be less than this maximum value. This sharp peak in power at resonance is exactly how your radio tunes into a specific station!