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Chapter 24: Problem 2

(1) How much charge flows from a \(12.0-\mathrm{V}\) battery when it is connected to a \(12.6-\mu \mathrm{F}\) capacitor?

Short Answer

Expert verified
The charge that flows is \(1.512 \times 10^{-4}\) Coulombs.

Step by step solution

01

Understand the Relationship

To find the charge (\(Q\)) that flows from the battery to the capacitor, we use the relationship given by \(Q = CV\), where \(C\) is the capacitance of the capacitor and \(V\) is the voltage of the battery.
02

Convert Units if Necessary

The capacitance is given as \(12.6\ \mu\mathrm{F}\), which must be converted to Farads before using the formula. Since \(1\ \mu\mathrm{F} = 10^{-6}\ \mathrm{F}\), the capacitance is \(12.6 \times 10^{-6}\ \mathrm{F} = 1.26 \times 10^{-5}\ \mathrm{F}\).
03

Apply the Formula

Substitute the values into the formula: \(Q = CV = (1.26 \times 10^{-5} \ \mathrm{F})(12.0 \ \mathrm{V})\).
04

Perform the Multiplication

Multiply the capacitance and voltage to find the charge, \(Q = 1.26 \times 10^{-5} \times 12.0 = 1.512 \times 10^{-4}\). This gives the charge in Coulombs.

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Key Concepts

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

Understanding Capacitance
Capacitance is a measure of a capacitor's ability to store charge. Think of it like a storage container for electric charge in the world of circuits. It is represented by the letter \( C \) and measured in units called Farads (\( F \)). A capacitor's ability to store charge depends on its physical characteristics, such as the area of its plates and the distance between them. Capacitance is typically given in microfarads (渭F) for practical purposes, since a Farad is a very large unit. One microfarad (\( \,\mu \mathrm{F} \)) is one millionth of a Farad, or \( 1 \ imes 10^{-6} \) Farads. This means that a large capacitance will allow more charge to be stored and released in a circuit.
The Role of Coulombs
Coulombs (\( Q \)) are the units used to measure electric charge in a circuit. When you think of electric charge, think of it like grains of sand that can be moved around and stored. The relationship between charge, capacitance, and voltage is linear and represented by the formula \( Q = CV \). In this formula, \( Q \) stands for the charge in Coulombs, while \( C \) represents capacitance and \( V \) the voltage. By knowing how much charge a capacitor can hold, you can understand how much energy is stored and how it can be used in an electrical circuit. One Coulomb is a large amount of charge, approximately equivalent to the charge of \( 6.242 \ imes 10^{18} \) electrons.
Understanding Voltage
Voltage (\( V \)) in a circuit represents the potential difference between two points and is often likened to the pressure that pushes electric charge through a circuit. It's measured in Volts (\( \,\mathrm{V} \)). Just like pressure can push water through a pipe, voltage pushes the electric charge through the wires. A high voltage means there's a strong push, driving the charge to flow more readily through the circuit. In the context of capacitors, voltage is crucial because it determines how much energy is stored when a charge is accumulated on the capacitor. This potential difference is what gets multiplied with capacitance to calculate how many Coulombs, or charge, will flow in a circuit.

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Most popular questions from this chapter

How much charge flows from a 12.0-V battery when it is connected to a \(12.6-\mu \mathrm{F}\) capacitor?

A 175 -pF capacitor is connected in series with an unknown capacitor, and as a series combination they are connected to a 25.0-V battery. If the 175-pF capacitor stores \(125 \mathrm{pC}\) of charge on its plates, what is the unknown capacitance?

What is the capacitance per unit length \((\mathrm{F} / \mathrm{m})\) of a coaxial cable whose inner conductor has a \(1.0-\mathrm{mm}\) diameter and the outer cylindrical sheath has a 5.0 -mm diameter? Assume the space between is filled with air.

In a dynamic random access memory } & \text { (DRAM) }\end{array}\( computer chip, each memory cell chiefly consists of a capacitor for charge storage. Each of these cells represents a single binary-bit value of 1 when its 35 -fF capacitor \)\left(1 \mathrm{fF}=10^{-15} \mathrm{~F}\right)\( is charged at \)1.5 \mathrm{~V},\( or 0 when uncharged at \)0 \mathrm{~V}\(. \)(a)\( When it is fully charged, how many excess electrons are on a cell capacitor's negative plate? (b) After charge has been placed on a cell capacitor's plate, it slowly "leaks" off (through a variety of mechanisms) at a constant rate of \)0.30 \mathrm{fC} / \mathrm{s} .\( How long does it take for the potential difference across this capacitor to decrease by \)1.0 \%$ from its fully charged value? (Because of this leakage effect, the charge on a DRAM capacitor is "refreshed" many times per second.)

(II) It takes 15 \(\mathrm{J}\) of energy to move a 0.20 \(\mathrm{-mC}\) charge from one plate of a \(15-\mu \mathrm{F}\) capacitor to the other. How much charge is on each plate?
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