91Ó°ÊÓ

Ohm's Law is an essential principle in electronics, relating voltage (V), current (I), and resistance (R) in an electric circuit. According to this law, the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance.
The formula for Ohm's Law is given by:

• Current (\( I \)) = Voltage (\( V \)) / Resistance (\( R \))

In the context of an RC circuit—comprising a resistor and a capacitor in series—Ohm's Law helps determine the maximum current in the circuit right when the process starts.
At the initial moment when the capacitor is still uncharged, Ohm’s Law tells us the maximum current is achieved, calculated simply as the total voltage divided by the resistance.
In our exercise, with a total voltage of 6.0 V and a resistance of 3.0 MegaOhms, the initial maximum current is computed as:

• \( I_{max} = \frac{6.0 \,V}{3.0 \times 10^6 \, \Omega} = 2.0 \times 10^{-6} \, A \)

The time constant in an RC circuit is a critical parameter that determines how quickly the circuit responds to changes, such as charging and discharging of a capacitor.
It is represented by the Greek letter \( \tau \) (tau) and mathematically, it is the product of the resistance (R) and the capacitance (C).

• Time Constant (\( \tau \)) = Resistance (\( R \)) × Capacitance (\( C \))

For our circuit with a resistance of 3.0 MegaOhms and a capacitance of 0.28 microFarads, the time constant is:

• \( \tau = (3.0 \times 10^6) \times (0.28 \times 10^{-6}) = 0.84 \, s \)

The time constant indicates the time it takes for the charge on the capacitor to reach about 63.2% of its maximum value during charging.
This concept is crucial, providing a time frame for when major changes occur in the circuit’s behavior.

Capacitors are components that store electric charge and energy in an electric field. The amount of charge a capacitor can hold is determined by its capacitance and the voltage applied across it.
The charge on a capacitor at any point in time during charging or discharging can be modeled and predicted using exponential functions because, in RC circuits, these processes are not linear.
The maximum charge (\( Q_{max} \)) a capacitor can hold, when fully charged, can be calculated by multiplying the capacitance (\( C \)) by the voltage (\( V \)) across it.

• Maximum Charge (\( Q_{max} \)) = Capacitance (\( C \)) × Voltage (\( V \))
  • \( Q_{max} = 0.28 \times 10^{-6} \times 6.0 = 1.68 \times 10^{-6} \, C \)

This maximum charge occurs as time approaches infinity, illustrating that perfectly complete charging is theoretical and takes an indefinite period.

Capacitors store energy in an electric field, and the energy stored within can be calculated using the formula for the energy of a charged capacitor. The formula is given as:

• \( E = \frac{1}{2} C V^2 \)

Where:

• \( E \) is the energy stored
• \( C \) is the capacitance
• \( V \) is the voltage across the capacitor

After a certain period, specifically one time constant (\( \tau \)), the capacitor reaches about 63.2% of its total charge, allowing us to also determine the energy stored at this point.
For our specific circuit, using a voltage reduced to 63.2% of 6.0 V, the energy stored is calculated to demonstrate how capacitors accumulate energy over time, illustrating the gradual nature of charging:

• \( E \approx \frac{1}{2} \times 0.28 \times 10^{-6} \times (3.792)^2 \approx 2.01 \times 10^{-6} \, J \)