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Chapter 11: Problem 32

Find the energy stored in a capacitor in terms of its capacitance and the voltage difference across it.(answer check available at lightandmatter.com)

Short Answer

Expert verified
The energy stored in a capacitor is \( E = \frac{1}{2} C V^2 \).

Step by step solution

01

Understand the Energy Formula

The energy stored in a capacitor is given by the formula \( E = \frac{1}{2} C V^2 \), where \( E \) is the energy in joules, \( C \) is the capacitance in farads, and \( V \) is the voltage in volts. This formula describes how much energy is stored based on the capacitance and voltage.
02

Put the Values into the Formula

To find the energy stored in the capacitor, you substitute the capacitance and voltage values into the formula \( E = \frac{1}{2} C V^2 \). For example, if the capacitance \( C \) is 2 farads and the voltage \( V \) is 5 volts: \[ E = \frac{1}{2} \cdot 2 \cdot 5^2 \].
03

Calculate the Result

Using the provided values in the example, compute \( E = \frac{1}{2} \cdot 2 \cdot 25 = 25 \text{ joules} \). This is the energy stored in the capacitor based on the given voltage and capacitance.

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Key Concepts

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

Capacitance
Capacitance is a fundamental property of capacitors. It describes a capacitor's ability to store electrical charge. The higher the capacitance, the more charge a capacitor can store at a given voltage.Capacitance is measured in farads (F), named after Michael Faraday, who contributed significantly to the study of electromagnetism. A farad is a large unit, and practical capacitors commonly have capacitance values in microfarads (\(\mu F\)), which is \(10^{-6}\) farads, or even smaller.
  • Formula for Capacitance: At a basic level, capacitance \(C\) can be calculated using the formula \( C = \frac{Q}{V} \), where \(Q\) is the charge in coulombs, and \(V\) is the voltage in volts.
  • Impact of Capacitance: The stored energy in a capacitor heavily relies on its capacitance, as shown in the energy formula \( E = \frac{1}{2} C V^2 \).
This makes understanding capacitance critical in determining how much energy the capacitor can hold when connected across a specific voltage.
Voltage Difference
The voltage difference, often referred to as potential difference, is a measure of the electric potential difference between two points. In the context of capacitors, it is the difference between the two plates of the capacitor.Voltage is measured in volts (V), a unit named after the physicist Alessandro Volta. The voltage across a capacitor is critical because it directly affects how much charge is stored.
  • Role in a Capacitor: The greater the voltage difference across the capacitor, the more energy it can store, provided the capacitance remains constant.
  • Energy Relationship: In the energy formula \( E = \frac{1}{2} C V^2 \), the square of the voltage term indicates that the energy stored increases quadratically with the voltage difference.
Thus, a small increase in voltage can significantly increase the energy stored in the capacitor.
Joules
Joules are the units of energy in the International System of Units (SI). Named after James Prescott Joule, a joule quantifies energy, work, or heat.In electrical systems, joules are used to describe the energy transferred or stored, such as in a capacitor. When you calculate the energy stored in a capacitor using the formula \( E = \frac{1}{2} C V^2 \), the result is expressed in joules.
  • Practical Example: In the example provided, using the formula \( E = \frac{1}{2} C V^2 \) for a capacitor with a capacitance of 2 farads and a voltage of 5 volts, the calculated energy is 25 joules. This energy is what the capacitor can release when discharged.
  • Importance of Joules: Understanding joules helps in assessing how much energy the capacitor can store and subsequently deliver. This is critical in designing and analyzing circuits.
So, the energy expressed in joules provides a tangible measure of what a capacitor can "do" in terms of energy storage and release.

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Most popular questions from this chapter

Find the capacitance of the surface of the earth, assuming there is an outer spherical "plate鈥� at infinity. (In reality, this outer plate would just represent some distant part of the universe to which we carried away some of the earth's charge in order to charge up the earth.)(answer check available at lightandmatter.com)

(a) At time \(t=0\), a positively charged particle is placed, at rest, in a vacuum, in which there is a uniform electric field of magnitude \(E\). Write an equation giving the particle's speed, \(v\), in terms of \(t, E\), and its mass and charge \(m\) and \(q\). (answer check available at lightandmatter.com) (b) If this is done with two different objects and they are observed to have the same motion, what can you conclude about their masses and charges? (For instance, when radioactivity was discovered, it was found that one form of it had the same motion as an electron in this type of experiment.)

(a) Find the capacitance of two identical capacitors in series. (b) Based on this, how would you expect the capacitance of a parallel-plate capacitor to depend on the distance between the plates?

A solid cylinder of radius \(b\) and length \(\ell\) is uniformly charged with a total charge \(Q .\) Find the electric field at a point at the center of one of the flat ends.

(a) Use complex number techniques to rewrite the function \(f(t)=4 \sin \omega t+3 \cos \omega t\) in the form \(A \sin (\omega t+\delta)\).(answer check available at lightandmatter.com) (b) Verify the result using the trigonometric identity \(\sin (\alpha+\beta)=\sin \alpha \cos \beta+\sin \beta \cos \alpha\).
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