Problem 64 - College Physics

Question

A fully charged 6.0\(\mu\) F capacitor is connected in series with a \(1.5 \times 10^{5} \Omega\) resistor. What percentage of the original charge is left on the capacitor after 1.8 s of discharging?

Step-by-Step Solution

Answer

Approximately 13.5% of the original charge is left on the capacitor after 1.8 seconds.

Step 1: Identify the given parameters

We are provided with the following:- Capacitance of the capacitor, \( C = 6.0 \mu F = 6.0 \times 10^{-6} F \)- Resistance, \( R = 1.5 \times 10^{5} \Omega \)- Time, \( t = 1.8 \) secondsWe need to find the percentage of the original charge remaining after 1.8 seconds of discharging.

Step 2: Recall the formula for capacitive discharge

The charge \( Q \) on a discharging capacitor as a function of time is given by the equation:\[ Q(t) = Q_0 \cdot e^{-\frac{t}{RC}} \]where:- \( Q_0 \) is the initial charge,- \( R \) is the resistance,- \( C \) is the capacitance,- \( t \) is the time,- \( e \) is the base of the natural logarithm.

Step 3: Calculate the time constant \( \tau \)

The time constant \( \tau \) for an RC circuit is calculated as:\[ \tau = R \cdot C \]Substituting the given values:\[ \tau = 1.5 \times 10^{5} \Omega \times 6.0 \times 10^{-6} F = 0.9 \text{ seconds} \]

Step 4: Apply the discharge formula

Now, using the discharge formula:\[ Q(t) = Q_0 \cdot e^{-\frac{t}{\tau}} \]Substitute \( t = 1.8 \) seconds and \( \tau = 0.9 \) seconds:\[ Q(1.8) = Q_0 \cdot e^{-\frac{1.8}{0.9}} \]\[ Q(1.8) = Q_0 \cdot e^{-2} \]

Step 5: Calculate the remaining charge as a percentage

The charge remaining as a percentage of the initial charge can be expressed as:\[ \left( \frac{Q(1.8)}{Q_0} \right) \times 100\% \]Substitute the expression from Step 4:\[ \left( e^{-2} \right) \times 100\% \approx 13.5\% \]

Key Concepts

RC circuit

An RC circuit is a simple electrical circuit that includes a resistor (R) and a capacitor (C) connected in series. When you first connect the capacitor to a direct current (DC) power source, it begins to charge through the resistor. Once fully charged, the capacitor can release its stored energy through the resistor in a process known as capacitive discharge. This release of energy is precise and predictable, thanks to the mathematical nature of the circuit.

RC circuits are used in various applications such as timing circuits, audio equipment, and filters. They are fundamental in electronic circuits because they can manage time-based processes.

• The resistor controls how fast the capacitor charges or discharges.
• The capacitor stores electrical energy and determines the circuit's overall capacity.

Understanding how an RC circuit works is essential to grasping how capacitors behave when storing and releasing energy.

time constant

The time constant, often represented by the Greek letter \( \tau \) (tau), is a crucial concept when dealing with RC circuits. It is a measure of the time it takes for the capacitor to charge or discharge to about 63.2% of its full charge or discharge level. The time constant is calculated by multiplying the resistance \( R \) by the capacitance \( C \):

\[ \tau = R \cdot C \]

Here, \( R \) is in ohms (Ω) and \( C \) is in farads (F). The result, \( \tau \), is in seconds (s).

In our exercise, the time constant is \( 0.9 \) seconds, meaning this is the duration for the charge to fall to approximately 36.8% of its initial value.

• The time constant helps predict how quickly or slowly a circuit will respond.
• It allows engineers to design circuits that require specific timing characteristics.

This foundational understanding of the time constant in RC circuits offers insight into how electronic devices manage power and data signals.

capacitor

A capacitor is an essential component in electronic circuits that stores energy in the form of an electric field. It consists of two conductive plates separated by an insulating material known as the dielectric. When a voltage is applied across the plates, an electric field develops across the dielectric, allowing the capacitor to store electrical energy.

The amount of charge a capacitor can store is measured in farads (F). In the original exercise, the capacitor has a capacitance of \( 6.0 \mu F \) (microfarads), which determines how much charge it can store.

• Capacitors are used to smooth out voltage fluctuations in power supply circuits.
• They can also be used in timing applications and for filtering signals.

Understanding the role of capacitors helps in realizing how electronic circuits store and discharge energy efficiently.

resistor

A resistor is an electrical component that limits or regulates the flow of electrical current in a circuit. It essentially controls how much current flows through different parts of a circuit by presenting a resistance to the flow of electrons. The resistance is measured in ohms (Ω).

In an RC circuit, the resistor plays a critical role by influencing how quickly the capacitor charges or discharges. In our example, the resistor has a value of \( 1.5 \times 10^{5} \Omega \), regulating the discharge process of the capacitor.

• Resistors are vital for protecting sensitive components from excessive current.
• They are used in various applications, including voltage dividers and in setting time constants with capacitors in RC circuits.

Knowing how resistors function within an RC circuit allows one to understand and manipulate the timing and behavior of electronic circuits effectively.