91Ó°ÊÓ

When a capacitor is connected to a power source, it starts charging up. Initially, a capacitor holds no charge, so the voltage across it is zero. This is because the electric field inside it has not yet been established.
As time progresses, the capacitor charges up, storing energy in the form of an electric field between its plates. The rate at which it charges depends on the resistance and capacitance in the circuit; these factors describe how quickly or slowly the capacitor can fill up with charge.
During this charging phase, the current flows from the power source, through the resistor, and into the capacitor. Eventually, the capacitor reaches a point where it is fully charged, meaning the voltage across its terminals equals the voltage from the source. At this stage, no more current flows in the circuit.

Ohm's Law is critical in analyzing circuits like the one described. It establishes a fundamental relationship between voltage (V), current (I), and resistance (R): \[I = \frac{V}{R} \]Where

• \(I\) is the current in amperes (A),
• \(V\) is the voltage in volts (V), and
• \(R\) is the resistance in ohms (Ω).

For a new circuit, such as one where the capacitor just starts to charge, the entire supplied voltage appears across the resistor. Since resistors are designed to limit current flow, Ohm's Law helps predict the rate of current flow when a given voltage is applied.
In our exercise, immediately after connection, all voltage from the source appears across the resistor as the capacitor has not charged up yet. This allows us to determine the initial current flowing through the circuit.

The time constant \(\tau\), a key concept in RC circuits, determines how fast the capacitor charges or discharges. It is defined as the product of the resistance (R) and the capacitance (C): \[\tau = R \cdot C \]

• For our circuit, \(R = 7.50 \text{k}\Omega\) and \(C = 4.60 \mu F\) (microfarads).
• The time constant gives us a measure: essentially, it is the time required for the voltage across the capacitor to reach approximately 63% of its maximum value, or in reverse, to drop to 37% during discharge.

After several time constants (about five times \(\tau\)), the capacitor is considered fully charged, and almost no current flows through the circuit anymore. This allows engineers to forecast and manage the behavior of the circuit over time.

The voltage across a resistor is significant in understanding the behavior of an RC circuit. Initially, just after the circuit is connected, the entire voltage from the power source is across the resistor since the capacitor has not started charging yet. Hence, we have: \[ V_r = \mathcal{E} \]Where

• \(V_r\) is the voltage drop across the resistor.
• \(\mathcal{E}\) is the electromotive force provided by the source.

As the capacitor charges over time, this voltage gradually decreases. Eventually, when the capacitor becomes fully charged, the voltage across the resistor drops to zero, because the capacitor assumes the full supply voltage. Understanding this transition helps in predicting how the circuit consumes energy and operates under steady-state conditions.