Charging of a Capacitor in a Simple Circuit

Introduction:
In a simple circuit, a capacitor can be charged by connecting it in series with a resistance and a battery. When the circuit is closed, the capacitor gradually charges up, and a displacement current flows between the capacitor plates. Let's consider a circuit consisting of a 100 µF capacitor, a 10 kΩ resistance, and a 6-volt battery.

Explanation:

Step 1: Calculation of Charging Time Constant (RC):
The time constant (RC) of the circuit can be calculated by multiplying the resistance (R) and the capacitance (C). In this case, the time constant is given by:
RC = 10 kΩ * 100 µF
RC = 1 second

Step 2: Charging of the Capacitor:
When the circuit is closed, the capacitor starts to charge. Initially, the voltage across the capacitor is zero, and the current flowing through the circuit is at its maximum value. As time progresses, the voltage across the capacitor increases, and the current gradually decreases.

Step 3: Calculation of Charging Time:
The charging time (t) can be calculated using the formula:
t = 5 * RC
t = 5 seconds

Step 4: Calculation of Displacement Current:
After 500 milliseconds (0.5 seconds), the displacement current (id) between the capacitor plates can be determined. To calculate the displacement current, we need to consider the rate of change of charge on the capacitor over time.

The charge on the capacitor can be calculated using the formula:
Q = CV
where Q is the charge on the capacitor and V is the voltage across the capacitor.

At t = 0.5 seconds, the voltage across the capacitor (V) can be calculated using the formula:
V = V0 * (1 - e^(-t/RC))
where V0 is the battery voltage and t is the time.

Substituting the values, we get:
V = 6 * (1 - e^(-0.5/1))
V ≈ 5.27 volts

The charge on the capacitor (Q) can be calculated as:
Q = CV
Q = (100 µF) * (5.27 volts)
Q ≈ 0.527 milliCoulombs

The displacement current (id) between the capacitor plates can be calculated using the formula:
id = dQ/dt
where id is the displacement current and dQ/dt is the rate of change of charge.

At t = 0.5 seconds, the displacement current can be approximated as:
id ≈ (0.527 milliCoulombs - 0) / (0.5 seconds - 0)
id ≈ 1.054 milliAmpere

Conclusion:
After 500 milliseconds, the displacement current (id) between the capacitor plates is approximately 1.054 milliAmpere. It is important to note that the displacement current is not an actual flow of charge but rather a measure of the changing electric field within the capacitor.