91Ó°ÊÓ

A potential difference of \(300 \mathrm{~V}\) is applied to a series connection of two capacitors of capacitances \(C_{1}=2.00 \mu \mathrm{F}\) and \(C_{2}=8.00 \mu \mathrm{F}\). What are (a) charge \(q_{1}\) and (b) potential difference \(V_{1}\) on capacitor 1 and (c) \(q_{2}\) and (d) \(V_{2}\) on capacitor \(2 ?\) The charged capacitors are then disconnected from each other and from the battery. Then the capacitors are reconnected with plates of the same signs wired together (the battery is not used). What now are (e) \(q_{1}\), (f) \(V_{1},\left(\right.\) g) \(q_{2}\), and (h) \(V_{2}\) ? Suppose, instead, the capacitors charged in part (a) are reconnected with plates of opposite signs wired together. What now are (i) \(q_{1},(\mathrm{j}) V_{1},(\mathrm{k}) q_{2}\), and (1) \(V_{2}\) ?

Step by step solution

01

Calculate Equivalent Capacitance

Since the capacitors are in series, the equivalent capacitance \( C_{eq} \) is given by: \[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \] Substitute \( C_1 = 2.00 \mu F \) and \( C_2 = 8.00 \mu F \):\[ \frac{1}{C_{eq}} = \frac{1}{2.00} + \frac{1}{8.00} \] Solving this gives \( C_{eq} = 1.60 \mu F \).

02

Calculate Total Charge

The total charge \( Q \) on the series capacitors can be found using \( Q = C_{eq} \, V \), where \( V = 300 \, V \). Then, \[ Q = 1.60 \times 10^{-6} \, F \times 300 \, V = 480 \times 10^{-6} \, C = 480 \, \mu C \].

03

Charge on Each Capacitor

In a series connection, the charge on each capacitor is the same. Thus, \( q_1 = q_2 = 480 \, \mu C \).

04

Calculate Voltage Across Capacitor 1

For capacitor 1, \( V_1 = \frac{q_1}{C_1} = \frac{480 \, \mu C}{2.00 \, \mu F} = 240 \, V \).

05

Calculate Voltage Across Capacitor 2

For capacitor 2, \( V_2 = \frac{q_2}{C_2} = \frac{480 \, \mu C}{8.00 \, \mu F} = 60 \, V \).

06

Reconnected with Plates of Same Signs

When the capacitors are reconnected with plates of the same sign, charge is conserved. The total charge is \( q_{1} = q_{2} = 480 \, \mu C \). Since they are in parallel now, \( V_{1} = V_{2} = \frac{480 \, \mu C}{10 \, \mu F} = 48 \, V \).

07

Reconnected with Plates of Opposite Signs

When reconnected with plates of opposite signs, the difference in charge is zero since charges cancel out. Thus, \( q_1 = 0 \) and \( q_2 = 0 \). The potential difference \( V_1 = V_2 = 0 \) V.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Capacitance

Capacitance is a fundamental property of capacitors, which relates to their ability to store charge. The capacitance, denoted as \( C \), is defined as the amount of charge \( Q \) a capacitor can store per unit voltage \( V \) across it, expressed by the equation \( C = \frac{Q}{V} \). Capacitors come in various capacitance values, often measured in microfarads (\( \mu F \)).

• The higher the capacitance, the more charge a capacitor can store at a given potential difference.
• In series connections, the equivalent capacitance is less than any individual capacitance. The combined capacitance \( C_{eq} \) of two capacitors in series is found using \( \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \).
• In parallel connections, capacitances add up linearly, \( C_{eq} = C_1 + C_2 \).

The exercise shows how calculations differ when combining capacitors in series versus parallel.

Potential Difference

The potential difference, also known as voltage, across a capacitor is crucial as it determines the energy stored. It is represented by the symbol \( V \). According to the formula \( V = \frac{Q}{C} \), a constant charge in a capacitor with lower capacitance results in a higher potential difference.

• In the given problem, a 300 V potential difference was initially applied to the series connection. This voltage drives the charge storage in each capacitor.
• For each capacitor in series, the potential difference is determined by the charge divided by its capacitance, as calculated for each capacitor separately.
• For example, the voltage on capacitor 1 is 240 V, while on capacitor 2 is 60 V, ensuring that their sum equals back to the total applied voltage.

Understanding potential difference helps predict energy distribution in capacitor networks.

Charge Conservation

Charge conservation is a key principle in capacitor circuits, meaning that the total charge remains constant in an isolated system. When capacitors in series are disconnected from their energy source and manipulated, the principle ensures that total charge remains constant even if configurations change.

• In the example, the entire charge of 480 \( \mu C \) is conserved when reconfiguring the capacitors in different setups.
• When capacitors are reconnected with plates of the same signs, the total charge is evenly distributed among them as they effectively move into a parallel configuration.
• When connected with plates of opposite signs, charges cancel each other out, simulating a state of neutral charge with potential differences dropping to zero.

Recognizing this conservation is essential for correctly analyzing capacitor reconfigurations.

Capacitor Circuits

Capacitor circuits encompass various configurations, such as series and parallel arrangements. Understanding these setups is vital for predicting electrical response and potential changes.

• Series circuits are characterized by having the same charge on all capacitors, but dividing the total voltage across each unit according to their capacitances.
• Parallel circuits, however, maintain the same voltage across all capacitors while allowing the total charge to sum across them.
• When modifying these circuits, like disconnecting and reconnecting with different plate orientations, the behavior and solutions vary significantly, impacting stored charge and voltage distribution.
• Practically, the manipulation of capacitor circuits is foundational in designing electronic devices, where specific energy storage or discharge levels must be achieved and managed efficiently.

Mastering capacitor circuits is crucial for optimizing energy handling in eletronic applications.