Animated Solution for Physics - Electromagnetic Induction: The self-inductance of the motor of an electric fan is $10\text{ H}$. In order to impart maximum power at $50\text{ Hz}$, it should be connected to a capacitance of

Imagine you are looking inside an electric fan. What makes the blades spin? It is the motor, and at the heart of every electric motor are coils of copper wire. These coils act as electromagnets, but they also introduce a fundamental property into our circuit: self-inductance.

In this problem, our fan motor has a significant self-inductance of $L=10\text{ H}$ and operates on a standard AC supply frequency of $f=50\text{ Hz}$. Because of this inductance, the motor doesn't just consume real power; it also draws reactive power. The inductor causes the alternating current to lag behind the alternating voltage.

When the current and voltage are out of phase, the circuit cannot operate at its maximum efficiency. The power delivered to an AC circuit is given by $P=V_{\text{rms}}{I}_{\text{rms}}\cos \varphi$, where $\cos \varphi$ is the power factor.

To impart maximum power, we need the power factor to be exactly $1$. This means the phase difference $\varphi$ must be zero. How do we achieve this? We must bring the circuit into resonance. At resonance, the lagging effect of the inductor is perfectly cancelled out by the leading effect of a capacitor. Mathematically, this means the inductive reactance ($X_L$) must equal the capacitive reactance ($X_C$).

Let's set up our master equation for resonance:

We know the formulas for these reactances:

Rearranging this to solve for the unknown capacitance $C$, we get:

First, let's find the angular frequency $\omega$. Since the supply frequency is $50\text{ Hz}$:

Now, substitute $\omega$ and $L=10\text{ H}$ into our capacitance equation:

Here is where a classic physics approximation saves the day. We can approximate ${\pi }^2\approx 10$. Substituting this in:

Since $10^{-6}\text{ F}$ is exactly one microfarad, we arrive at our final answer:

By connecting a $1\mu \text{F}$ capacitor, we perfectly correct the power factor, allowing the fan motor to draw maximum real power from the $50\text{ Hz}$ supply!