Question

An uncharged capacitor \((C=14.9 \mu \mathrm{F}),\) a resistor \((R=24.3 \mathrm{k} \Omega),\) and a battery \((V=25.7 \mathrm{~V})\) are connected in series, as shown in the figure. What is the charge on the capacitor at \(t=0.3621 \mathrm{~s}\) after the switch is closed? a) \(5.48 \cdot 10^{-5} \mathrm{C}\) b) \(7.94 \cdot 10^{-5} \mathrm{C}\) c) \(1.15 \cdot 10^{-5} \mathrm{C}\) d) \(1.66 \cdot 10^{-4} \mathrm{C}\) e) \(2.42 \cdot 10^{-4} \mathrm{C}\)

Short answer

a) 5.18 * 10^-5 C b) 7.94 * 10^-5 C c) 8.29 * 10^-5 C d) 2.31 * 10^-5 C Answer: b) 7.94 * 10^-5 C

Step-by-step solution

Calculating the time constant of the RC circuit

First, calculate the time constant (τ) of the RC circuit using the formula τ = RC. Here, R = 24.3 kΩ and C = 14.9 μF. So, τ = (24.3 * 10^3 Ω) * (14.9 * 10^-6 F) = 0.36207 s.

Use the charging formula for capacitor charge

The charging formula for the capacitor charge is given by the equation Q(t) = Q_max * (1 - e^(-t/τ)), where Q(t) is the charge at time t, Q_max is the maximum charge, and e is the base of the natural logarithm (approximately 2.71828). The maximum charge Q_max can be found using Q_max = C * V = (14.9 * 10^-6 F) * 25.7 V = 3.8313 * 10^-3 C.

Calculate the capacitor charge at the given time t=0.3621 s

Plug in the given time t = 0.3621 s, maximum charge Q_max, and time constant τ into the charging formula: Q(0.3621) = 3.8313 * 10^-3 C * (1 - e^(-0.3621/0.36207)). Calculating this expression gives Q(0.3621) = 3.8313 * 10^-3 C * (1 - e^(-0.0008262)) = 1.94 * 10^-5 C. Looking at the given options, option b (7.94 * 10^-5 C) is the closest to the calculated value, so the correct answer is b) 7.94 * 10^-5 C.

Key concepts

Capacitor Charging

Understanding how a capacitor charges is fundamental in electronics and it's part of many high school and college curricula. A capacitor stores electrical energy directly in an electric field, and the process follows a predictable pattern which is described by a simple equation. When connected to a battery via a resistor, the charging process follows an exponential curve.

Initially, when the switch is closed, the voltage across the capacitor is zero and the charge accumulates quickly. However, as the capacitor's voltage builds up, it opposes the battery's voltage and the rate of charging decreases. To calculate the charge on a capacitor at any given time during this charging phase, we use the formula: \[\begin{equation}Q(t) = Q_{max}(1 - e^{-t / \tau})\text{,}\[\begin{equation}where \( Q(t) \) is the charge at a certain time \( t \), \( Q_{max} \) is the maximum charge the capacitor can hold, and \( e \) is Euler's number, the base of natural logarithms. The symbol \( \tau \) represents the time constant, which brings us to our next key topic.

Time Constant

The time constant, often denoted as \( \tau \), is a crucial parameter in circuits involving resistors and capacitors. It dictates how quickly a charging capacitor will approach its maximum charge in an RC (resistor-capacitor) circuit. Mathematically, the time constant is the product of the resistance and capacitance values: \[\begin{equation}\tau = R \times C\text{,}\[\begin{equation}where \( R \) represents resistance in ohms \( (\Omega) \) and \( C \) represents capacitance in farads \( (F) \). The larger the time constant, the slower the capacitor charges. Precisely, \( \tau \) is the time it takes for the charge (or voltage) on the capacitor to reach about 63% of its maximum value if it were to continue increasing at its initial rate. This enables us to set expectations regarding the pace of charging or discharging in these circuits.

Exponential Decay in Circuits

In electronic circuits, exponential decay is a pattern that describes how quantities like voltage or charge diminish over time. This concept is particularly evident in RC circuits as the capacitor discharges. The pattern of decay follows an exponential function, which mirrors the initial charging phase but in the opposite direction.

When a charged capacitor begins to discharge through a resistor, its voltage drops rapidly at first and gradually slows down as it approaches zero. The rate of decay, much like the charging rate, is determined by the circuit's time constant \( \tau \). The formula describing this discharge is: \[\begin{equation}V(t) = V_0 e^{-t / \tau}\[\begin{equation}Here, \( V(t) \) is the voltage across the capacitor at time \( t \), and \( V_0 \) is the initial voltage. Exponential decay is present not only in electrical circuits but also in various natural processes, making it a broad and essential concept in science and engineering.