Animated Solution for Physics - Electromagnetic Induction: In a series $L-C-R$ circuit, the inductive reactance ($X_L$) is $10\Omega$ and the capacitive reactance ($X_C$) is $4\Omega$. The resistance ($R$) in the circuit is $6\Omega$. The power factor of the circuit is

When you first learn about direct current (DC) circuits, calculating power is beautifully simple: you just multiply the voltage by the current ($P=VI$). But when you step into the world of Alternating Current (AC), things get a bit more complicated. In an AC circuit, the voltage and current are constantly oscillating like waves. If the circuit contains inductors or capacitors, these waves don't peak at the same time. They get out of sync.

This "out of sync" nature is described by the phase angle ($\varphi$). Because the voltage and current aren't pushing together perfectly, the actual real power the circuit consumes is less than the raw product of $V_{rms}$ and $I_{rms}$. To find the true power, we must multiply by a crucial term called the Power Factor, which is mathematically defined as $\cos \varphi$.

In our series $L-C-R$ circuit, we have three components battling it out: 1. The Resistor ($R=6\Omega$): The resistor is the only component that actually consumes real power (dissipating it as heat). It wants the voltage and current to be perfectly in phase. 2. The Inductor ($X_L=10\Omega$): The inductor opposes changes in current, causing the current to lag behind the voltage by $90^{\circ }$. 3. The Capacitor ($X_C=4\Omega$): The capacitor opposes changes in voltage, causing the current to lead the voltage by $90^{\circ }$.

Because the inductor and capacitor pull the phase in exactly opposite directions, they are locked in a $180^{\circ }$ tug-of-war. To find their combined effect, we simply subtract them. The net reactance is $X_L-X_C=10\Omega -4\Omega =6\Omega$.

The Impedance Triangle

We can't just add the resistance and the net reactance like normal numbers because they act at $90^{\circ }$ to each other. Instead, we use a geometric tool called the Impedance Triangle.

Imagine walking $6$ steps East (representing the resistance $R$) and then $6$ steps North (representing the net reactance $X_L-X_C$). Your direct distance from the starting point is the total opposition to the current, known as the Impedance ($Z$).

Using the Pythagorean theorem:

Let's plug in our values:

Now, look back at our impedance triangle. The angle between the resistance (base) and the impedance (hypotenuse) is our phase angle, $\varphi$.

By basic trigonometry, the cosine of this angle is the ratio of the adjacent side to the hypotenuse:

Substituting our calculated values:

The $6$ in the numerator and denominator elegantly cancel out, leaving us with our final answer:

This power factor corresponds to a phase angle of $45^{\circ }$. It tells us that the circuit is moderately efficient, converting a significant portion of the apparent power into real, usable work!