Question

In the capacitor discharge formula \(q=q_{0} e^{-t / \tau}\) the symbol \(\tau\) represents : (1) the time, it takes for capacitor to loose \(\mathrm{q}_{0}\) /e charge. (2) the time it takes for capacitor to loose charge \(q_{0}\left(1-\frac{1}{e}\right)\) (3) the time it takes for capacitor to loose essentially all of its initial charge. (4) time in which charge of capacitor becomes half of initial value (5) none of the above.

Short answer

Option (1) is correct: \( \tau \) is the time it takes for the capacitor to lose \( \frac{q_{0}}{e} \) charge.

Step-by-step solution

Understand the given formula

The formula given is for the discharge of a capacitor: \( q = q_{0} e^{-t / \tau} \)where \( q \) is the charge remaining on the capacitor at time \( t \), \( q_{0} \) is the initial charge, and \( \tau \) (tau) is the time constant.

Definition of time constant \( \tau \)

The time constant \( \tau \) represents the time it takes for the charge on the capacitor to decrease to \( \frac{q_{0}}{e} \) (which is approximately 36.8% of the initial charge).

Analyze each option

1. Correct: This definition matches the definition of \( \tau \).2. Incorrect: This definition instead references the complementary fraction \( 1 - \frac{1}{e} \).3. Incorrect: The time it takes for a capacitor to lose all its charge is significantly longer than \( \tau \).4. Incorrect: The time it takes for the charge to be halved is known as the half-life, not \( \tau \).5. Incorrect: Since option 1 is correct, this option is incorrect.

Choose the correct option

Based on the analysis, the correct interpretation of \( \tau \) is given by option (1): \( \tau \) is the time it takes for the capacitor to lose \( \frac{q_{0}}{e} \) of its charge.

Key concepts

Time Constant

The time constant, represented by the symbol \( \tau \) (tau), is a fundamental concept in capacitor discharge. In the context of the given formula \( q = q_{0}e^{-t / \tau} \), \( \tau \) defines the time required for the charge on the capacitor to reduce to approximately 36.8% of its initial value. This is because 36.8% is \( \frac{1}{e} \), where \( e \) is the base of the natural logarithm (approximately equal to 2.718).

The time constant provides a measure of how quickly the capacitor discharges. When \( t = \tau \), the charge \( q \) drops to \( q_{0} / e \).

Applications of time constants include:

• Determining charge and discharge rates of capacitors in electronic circuits.
• Predicting the response times of RC (resistor-capacitor) circuits in timing applications.

It is essential to note that the time constant depends on the properties of both the resistor and the capacitor in the circuit, given by the product \( \tau = RC \).

Exponential Decay

Exponential decay describes the process by which the charge on a capacitor decreases over time. In the formula \( q = q_{0} e^{-t / \tau} \), we observe that the charge \( q \) decreases exponentially as time \( t \) increases.

This form of decay means that the quantity of charge diminishes faster initially but slows down as it progresses. The exponential nature is characterized by a constant percentage reduction over equal time increments.

Key points to remember about exponential decay in capacitors:

• It represents a continuous decrease in charge.
• The rate of decay is proportional to the amount of charge remaining.
• For each time interval equal to the time constant \( \tau \), the charge drops to about 36.8% of its previous amount.

Understanding this principle is crucial in electronics, where exponential decay is used to model behaviors such as discharging batteries and the response of filters.

Capacitor Charge

Capacitor charge, denoted as \( q \) in our formula, refers to the amount of electric charge stored on the capacitor's plates. Initially, a fully charged capacitor holds a charge of \( q_{0} \). As the capacitor discharges, this charge decreases over time following the exponential decay law.

Important aspects of capacitor charge include:

• Initial Charge: \( q_{0} \) is the charge when \( t = 0 \).
• Remaining Charge: \( q \) at any time \( t \), given by the equation \( q = q_{0} e^{-t / \tau} \).
• Lost Charge: The difference between the initial and remaining charge.

In practical scenarios, capacitors are used to store and release energy in circuits, smooth voltage fluctuations, and filter signals. Understanding how the charge changes over time with respect to the time constant is essential for designing and analyzing electronic circuits effectively.