Animated Solution for Physics - Electromagnetic Induction: In an $L-C-R$ series AC circuit, the voltage across each of the components $L,C$ and $R$ is $50\text{ V}$. The voltage across the $L-C$ combination will be

When dealing with Alternating Current (AC) circuits, simple algebraic addition goes out the window. Why? Because voltages and currents are no longer just static numbers; they are oscillating waves. To make sense of these waves, we use a brilliant mathematical tool called Phasors—rotating vectors that capture both the magnitude and the phase of these oscillations.

Imagine you are looking at a series $L-C-R$ circuit. The most crucial thing to remember about a series circuit is that the current is the same everywhere. It flows through the resistor, the inductor, and the capacitor simultaneously. Because of this, we always take the current $I$ as our reference phasor, pointing horizontally to the right.

Now, let's see how the voltage across each component behaves relative to this current:

1. The Resistor ($V_R$): The resistor is the straightforward component. Its voltage is perfectly in sync with the current. When the current peaks, the voltage peaks. So, the phasor for $V_R$ points in the exact same direction as $I$.

2. The Inductor ($V_L$): Inductors are stubborn; they oppose changes in current. Because of this electrical inertia, the voltage across an inductor reaches its peak before the current does. Specifically, $V_L$ leads the current by exactly $90^{\circ }$. On our phasor diagram, $V_L$ points straight up.

3. The Capacitor ($V_C$): Capacitors are the opposite; they oppose changes in voltage. They need time to charge up, so the voltage across a capacitor reaches its peak after the current does. $V_C$ lags the current by exactly $90^{\circ }$. On the phasor diagram, $V_C$ points straight down.

Look at the phasor diagram now. You have $V_L$ pulling straight up with a force of $50\text{ V}$, and $V_C$ pulling straight down with a force of $50\text{ V}$. They are exactly $180^{\circ }$ out of phase.

What happens when two equal and opposite forces pull on the same point? They cancel each other out completely!

Mathematically, the net voltage across the $L-C$ combination is the vector sum of $V_L$ and $V_C$. Since they are anti-parallel, we simply subtract their magnitudes:

Substituting the given values:

This phenomenon, where the inductive and capacitive reactances perfectly cancel each other out, is the hallmark of electrical resonance. At this specific frequency, the $L-C$ combination acts like a perfect short circuit, and the total voltage of the entire circuit is simply equal to the voltage across the resistor, $V_R$.