For an AC oscillation at the instant when electrostatic PE is max. the rate of change of current at the instant is

To understand the relationship between electrostatic potential energy (PE) and current in an alternating current (AC) circuit, we need to delve into the concepts of energy storage in capacitors and the behavior of current in an AC system. When we talk about the instant when electrostatic potential energy is at its maximum, we are typically referring to a capacitor in an AC circuit.

Electrostatic Potential Energy in a Capacitor

In a capacitor, the electrostatic potential energy (PE) is given by the formula:

• PE = (1/2) C V²

Where:

• C is the capacitance of the capacitor.
• V is the voltage across the capacitor.

At the moment when the electrostatic potential energy is at its maximum, the voltage across the capacitor is also at its peak. This is a key point in the AC cycle, specifically at the point where the voltage waveform reaches its maximum amplitude.

Current in an AC Circuit

In an AC circuit, the current and voltage are sinusoidal and can be described mathematically as:

• V(t) = V₀ sin(ωt)
• I(t) = I₀ sin(ωt + φ)

Where:

• V₀ and I₀ are the peak voltage and current, respectively.
• ω is the angular frequency.
• φ is the phase difference between the voltage and current.

In a purely capacitive circuit, the current leads the voltage by 90 degrees (or π/2 radians). This means that when the voltage is at its maximum, the current is actually at zero. This is because the capacitor is fully charged at that moment, and there is no further flow of charge, which corresponds to the current being zero.

Rate of Change of Current

The rate of change of current can be described mathematically as the derivative of the current with respect to time:

• dI/dt

At the instant when the electrostatic potential energy is maximum (when the voltage is at its peak), the current is at its minimum, which is zero. Therefore, the rate of change of current at this instant is not zero; rather, it is at its maximum value. This is because the current is transitioning from a positive value to a negative value as the AC cycle continues.

Conclusion on Current Behavior

To summarize, at the instant when the electrostatic potential energy in a capacitor is at its maximum, the current is zero, but the rate of change of current (dI/dt) is at its peak. This reflects the dynamic nature of AC circuits, where energy is continuously exchanged between the electric field in the capacitor and the current flowing through the circuit.