Problem 93 - Physics for Scientists and Engineers with Modern Physics

Question

(II) An \(R C\) series circuit contains a resistor \(R=15 \mathrm{k} \Omega\), a capacitor \(C=0.30 \mu \mathrm{F},\) and a battery of emf \(\mathscr{E}=9.0 \mathrm{~V}\) Starting at \(t=0\), when the battery is connected, determine the charge \(Q\) on the capacitor and the current \(I\) in the circuit from \(t=0\) to \(t=10.0 \mathrm{~ms}\) (at 0.1-ms intervals). Make graphs showing how the charge \(Q\) and the current \(I\) change with time within this time interval. From the graphs find the time at which the charge attains \(63 \%\) of its final value, \(C^{\mathscr{C}},\) and the current drops to \(37 \%\) of its initial value, \(\mathscr{E} / R\).

Step-by-Step Solution

Answer

The charge attains 63% of its final value and current drops to 37% of its initial value at 4.5 ms.

Step 1: Understanding RC Circuit Charging

An RC circuit is a circuit where a resistor (R) and a capacitor (C) are connected in series. When a battery is connected, the charge on the capacitor and the current in the circuit change over time. The charge on the capacitor over time, as it charges, is described by the equation: \[ Q(t) = C \mathscr{E} (1 - e^{-t/RC}) \]where \( \mathscr{E} \) is the electromotive force of the battery and \( t \) is time. The initial current, \( I(0) \), is given by \( I(0) = \frac{\mathscr{E}}{R} \). The current at any time \( t \) is given by \[ I(t) = \frac{\mathscr{E}}{R} e^{-t/RC} \].

Step 2: Calculating Time Constants

The time constant \( \tau \) for an RC circuit is \( \tau = RC \).Using the given values:\[ R = 15 \times 10^3 \text{ ohms} \] \[ C = 0.30 \times 10^{-6} \text{ farads} \] \[ \tau = 15 \times 10^3 \times 0.30 \times 10^{-6} = 4.5 \text{ ms} \]

Step 3: Calculating Charge over Time

To find the charge \( Q(t) \) at intervals of 0.1 ms up to 10 ms:- For \( t = 0 \) ms:\[ Q(0) = 0 \]- For \( t = 0.1 \) ms, \( t = 0.2 \) ms,... up to \( t = 10 \) ms:Use\[ Q(t) = 0.30 \times 10^{-6} \times 9.0 \times (1 - e^{-t/4.5}) \].Calculate \( Q(t) \) for each interval.

Step 4: Calculating Current over Time

Using the current formula\[ I(t) = \frac{9.0}{15 \times 10^3} e^{-t/4.5} \],calculate \( I(t) \) for each time interval from 0 to 10 ms in steps of 0.1 ms.

Step 5: Plotting Graphs

Plot the graph for \( Q(t) \) versus time and \( I(t) \) versus time using the calculated values. Observe the exponential rise in charge towards its maximum \( C\mathscr{E} \) and the exponential decay in current towards zero.

Step 6: Determining Specific Percentages

- The charge \( Q \) reaches \( 63\% \) of its final value when \( t = \tau \), that is 4.5 ms.- The current \( I \) drops to \( 37\% \) of its initial value also at \( t = \tau \). Verify these points on the plotted graphs.

Key Concepts

Time Constant

The concept of a time constant, denoted as \( \tau \), is fundamental in understanding how RC circuits behave over time. In an RC circuit, which consists of a resistor and a capacitor in series, the time constant \( \tau \) is the product of the resistance \( R \) and the capacitance \( C \). Mathematically, this is represented as \( \tau = RC \). This value dictates how quickly the capacitor charges and discharges.

In the context of our given circuit, with \( R = 15 \times 10^3 \text{ ohms} \) and \( C = 0.30 \times 10^{-6} \text{ farads} \), the time constant is calculated to be \( \tau = 4.5 \text{ ms} \). The significance of \( \tau \) lies in its role as a measure of time over which the voltage across the capacitor reaches approximately 63% of its maximum value during charging. Similarly, it's the time over which the current through the circuit falls to about 37% of its initial value during discharging.

Understanding the time constant allows us to predict how quickly the RC circuit responds to changes, which is crucial for designing circuits that meet specific timing requirements.

Capacitor Charging

Charging a capacitor in an RC circuit involves the gradual accumulation of charge over time. When a voltage \( \mathscr{E} \) is applied, the capacitor starts with zero charge and gradually charges up to its maximum value of \( C\mathscr{E} \). The charging process is characterized by the equation:

• \( Q(t) = C \mathscr{E} (1 - e^{-t/RC}) \)

This formula shows that the charge \( Q \) on the capacitor at any time \( t \) is dependent on the time constant \( \tau = RC \).

At \( t = 0 \), the charge \( Q(0) = 0 \) since the capacitor is initially uncharged. As \( t \) approaches \( \tau \), the charge reaches approximately 63% of its maximum value. For long periods, specifically at \( t = 10 \times \tau \), the capacitor is considered to be fully charged as it reaches nearly 100% of \( C\mathscr{E} \).

In the practical scenario described in our exercise, understanding this charging mechanism helps to predict how the current and charge will evolve, which is crucial for applications where precise timing and charge levels are vital.

Exponential Growth and Decay

The behaviors of charging and discharging in an RC circuit are classic examples of exponential growth and decay. These processes are depicted in the mathematical expressions for the charge and current in the circuit over time.

- **Exponential Growth of Charge:** This occurs when the capacitor is charging. Initially, the charge grows rapidly and then slows as it approaches its maximum value. The formula \( Q(t) = C \mathscr{E} (1 - e^{-t/RC}) \) shows this exponential rise, where \( t/RC \) (or \( t/\tau \)) represents how far in time the process has reached compared to the time constant.- **Exponential Decay of Current:** Simultaneously, the current in the circuit decreases as the capacitor charges. Represented by \( I(t) = \frac{\mathscr{E}}{R} e^{-t/RC} \), the current starts at a maximum and declines exponentially to zero as time passes beyond one time constant \( \tau \).

These exponential functions illustrate how the rates of change initially affect the circuit's performance, leading to a rapid change at first, which tapers off as steady state is reached. This concept is pivotal in electronics, where timing and speed of responses are critical.