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In an RC circuit, the charging of a capacitor is an important concept. When you connect a capacitor to a voltage source through a resistor, it starts charging. The voltage across the capacitor increases gradually.
This is because the resistor controls how fast the charge can flow to the capacitor. Let's consider an electromagnetic force (emf) source, a resistor, and a capacitor in series. The voltage builds up and approaches its maximum value slowly over time.
The charge, denoted as \(Q\), on the capacitor increases according to the formula:

• \(Q(t) = C \mathcal{E} (1 - e^{-\frac{t}{RC}})\)

This equation shows that initially, the charge on the capacitor is zero, and it reaches a maximum value \(C \mathcal{E}\) as time progresses toward infinity.

The behavior of the current through the resistor, \(I(t)\), during the charging process is crucial. Initially, the current is at its maximum value because the voltage difference is highest.
As the capacitor charges, the current decreases. This is due to the buildup of voltage across the capacitor which acts against the emf source. In our given problem, this current is found using the exponential decay formula:

• \(I(t) = \frac{\mathcal{E}}{R} e^{-\frac{t}{RC}}\)

This explains why, as time increases, \(I(t)\) approaches zero in a fully charged capacitor. At the moment given in the exercise, you see the current reduced from the maximum initial current.

Finding the charge on the capacitor plates involves understanding how the charge accumulates over time. The charge on the plates is directly proportional to the capacitance \(C\) and the applied voltage \(\mathcal{E}\).
At any given time \(t\), the charge \(Q\) can be calculated using the formula mentioned earlier:

• \(Q(t) = C \mathcal{E} (1 - e^{-\frac{t}{RC}})\)

Here, \(t\) is the precise moment when we need to know the charge. The term \(1 - e^{-\frac{t}{RC}}\) reflects how much of the total possible charge has been accumulated.
This is important because it shows the non-linear nature of capacitor charging, where the charge is gained most quickly at the beginning and slows down as \(t\) increases.

The time constant, denoted as \(\tau\), is defined as \(RC\) where \(R\) is resistance and \(C\) is capacitance. It is a measure of the speed at which the capacitor charges or discharges.
In simpler terms, the time constant gives us a sense of how fast the system reaches a certain charge level. After a time span equal to the time constant, the charge on the capacitor reaches approximately 63.2% of its maximum value.
Therefore, given a specific \(R\) and \(C\), the time constant indicates how quickly electrons are able to flow onto the capacitor’s plates, filling it up with charge.
This means the larger the resistor or capacitor, the slower the charge/discharge process, as more time is required to reach the full potential of the capacitor.