Animated Solution for Physics - Current Electricity: An $R-C$ circuit as shown in the figure is driven by an AC source generating a square wave. The output wave pattern monitored by CRO would look close to

Imagine you are standing in front of an oscilloscope (CRO) hooked up to a simple $RC$ circuit. The circuit is being driven by a square wave generator. What exactly does a square wave do? Unlike a smooth sine wave, a square wave is abrupt. It acts like a switch that periodically flips between a constant positive voltage $V_0$ and zero.

This means the circuit is constantly alternating between two distinct states: a charging phase when the voltage is high, and a discharging phase when the voltage drops to zero. The CRO, connected directly across the capacitor, acts as our window into the capacitor's struggle to keep up with these sudden changes.

A Race to the Top

When the square wave hits its positive peak, the capacitor begins to charge. However, it doesn't charge instantly. The resistor in the circuit acts as a bottleneck, limiting the flow of current. The voltage across the capacitor during this phase is governed by the classic exponential growth equation:

If we analyze the mathematics of this curve, we see that the rate of charging is highest at the very beginning when the capacitor is empty. As it fills up, the repulsive force from the accumulated charge slows down the incoming current. Graphically, this rapid initial rise followed by a gradual leveling off creates a concave down curve.

The Gentle Fall

Suddenly, the square wave drops to zero. The capacitor, now acting like a temporary battery, begins to discharge its stored energy back through the resistor. The voltage across it decays according to the exponential decay equation:

Similar to the charging phase, the discharge is not linear. It drops very rapidly at first because the voltage (and thus the driving force) is at its maximum. As the voltage decreases, the discharge rate slows down, gently approaching zero. Graphically, this rapid initial drop followed by a gradual leveling out creates a concave up curve.

Putting it all together, the CRO will display a continuous, repeating pattern. During the positive half-cycle of the square wave, we will see a concave down curve as the capacitor charges. During the zero-voltage half-cycle, we will see a concave up curve as it discharges.

When we look at the given options, only one graph perfectly captures this alternating "concave down, then concave up" exponential dance. This elegant visual representation of differential equations in action is exactly what makes circuit analysis so fascinating!