8.3 Energy Stored in a Capacitor
3 min read•june 24, 2024
Capacitors store electrical energy in their electric fields. This energy is proportional to the square of the and inversely proportional to . Understanding how capacitors store energy is crucial for many electrical applications.
The energy stored in a relates to the done to charge it against the electric field. This concept connects to broader ideas of electric potential energy and fields, which are fundamental in electrostatics and circuit theory.
Energy Stored in a Capacitor
Energy storage in capacitors
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- Energy stored in a directly proportional to the square of the charge on the capacitor (doubling charge quadruples energy)
- Energy stored in a capacitor inversely proportional to the (halving capacitance doubles energy)
- Formula for energy stored in a capacitor:
- = energy stored in joules (J)
- = charge in coulombs (C)
- = capacitance in farads (F)
- Alternative formula using voltage across the capacitor:
- = voltage in volts (V)
- The energy stored is a form of
Capacitor energy and electric fields
- Energy stored in a capacitor equals done to charge it
- External source (battery) moves charges from one plate to the other against the electric field
- Electric field between capacitor plates is uniform and perpendicular
- Electric field magnitude: ( = distance between plates)
- Electric field (energy per unit volume):
- = ( F/m)
- Total capacitor energy is times volume between plates:
- = area of each plate
- The between the plates determines the amount of energy stored
Applications of capacitor energy
- combination:
- Total capacitance is sum of individual capacitances:
- Voltage across each capacitor same and equal to source voltage
- Total energy stored is sum of energies in each capacitor:
- combination:
- Reciprocal of total capacitance is sum of reciprocals of individual capacitances:
- Charge on each capacitor same and equal to total charge
- Total energy stored is sum of energies in each capacitor:
- Defibrillators deliver controlled electric shock to heart to restore normal rhythm
- Use capacitors to store large energy (100-400 J) and release quickly
- Energy delivered: ( = capacitance, = voltage charged to)
Dielectrics and Capacitor Energy
- Dielectrics are insulating materials placed between capacitor plates
- Dielectrics increase the capacitance of a capacitor
- The work required to charge a capacitor with a is less than without
- Dielectrics affect the potential difference between the plates
- The energy stored in a capacitor with a is influenced by its capacitance
Key Terms to Review (32)
$\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$: This key term represents the formula for calculating the total capacitance of a set of capacitors connected in parallel. It states that the reciprocal of the total capacitance is equal to the sum of the reciprocals of the individual capacitances.
$arepsilon_0$: $arepsilon_0$, also known as the permittivity of free space, is a fundamental physical constant that represents the capability of free space to support an electric field. It is a measure of the amount of electrical charge that can be stored per unit volume of free space under the influence of an electric field. $arepsilon_0$ is a crucial parameter in the study of electromagnetism and the behavior of capacitors.
$C_{total} = C_1 + C_2 + ... + C_n$: The total capacitance of a circuit is equal to the sum of the individual capacitances connected in parallel. This formula represents the relationship between the total capacitance and the individual capacitances in a parallel capacitor network.
$E = \frac{V}{d}$: $E = \frac{V}{d}$ is an equation that describes the relationship between the electric field strength (E), the potential difference (V), and the distance (d) between two points in an electric field. This equation is particularly relevant in the context of energy stored in a capacitor, as it helps quantify the electric field within the capacitor and the energy stored due to the potential difference across its plates.
$U = \frac{1}{2} \frac{Q^2}{C}$: $U = \frac{1}{2} \frac{Q^2}{C}$ is the formula that describes the energy stored in a capacitor. It represents the potential energy stored in the electric field of a capacitor, where $U$ is the energy stored, $Q$ is the charge stored on the capacitor, and $C$ is the capacitance of the capacitor.
$u = \frac{1}{2} \varepsilon_0 E^2$: $u = \frac{1}{2} \varepsilon_0 E^2$ is the equation that describes the energy density, or the energy per unit volume, stored in an electric field. It represents the amount of energy stored in a capacitor per unit volume of the electric field between the capacitor plates.
$U = \frac{1}{2} CV^2$: $U = \frac{1}{2} CV^2$ is the formula used to calculate the energy stored in a capacitor. It represents the potential energy that is stored in the electric field of a capacitor when it is charged to a voltage $V$, where $C$ is the capacitance of the capacitor. This formula is a fundamental concept in understanding the behavior and applications of capacitors in electrical circuits and systems.
Capacitance: Capacitance is the ability of a system to store charge per unit voltage. It is measured in farads (F).
Capacitance: Capacitance is a measure of the ability of a capacitor to store electric charge. It is a fundamental quantity in the study of electricity and electronics, and it plays a crucial role in various topics related to electrostatic equilibrium, electric potential, and energy storage.
Capacitor: A capacitor is an electrical component that stores energy in the form of an electric field, created between two conductive plates separated by an insulating material. It is used to temporarily hold charge and release it when needed.
Capacitor: A capacitor is a passive electronic component that is used to store electrical energy in an electric field. It consists of two conductors separated by an insulator, and it is a fundamental component in many electrical and electronic circuits.
Charge: Charge is a fundamental property of matter that is the source of all electrical phenomena. It is a scalar quantity that can be positive or negative and is the basis for the attractive and repulsive forces between objects, as well as the flow of electric current.
Coulomb: A coulomb (C) is the SI unit of electric charge, representing the amount of charge transported by a constant current of one ampere in one second. One coulomb is equivalent to approximately $6.242 \times 10^{18}$ elementary charges.
Coulomb: The coulomb (symbol: C) is the SI unit of electric charge, named after the French physicist Charles-Augustin de Coulomb. It is a fundamental quantity that describes the amount of electric charge and is used extensively in the study of electric phenomena across various physics topics.
Defibrillator: A defibrillator is a medical device that delivers a dose of electric current to the heart to treat life-threatening cardiac dysrhythmias. It utilizes capacitors to store and release electrical energy quickly.
Defibrillator: A defibrillator is a device used to apply an electric shock to the heart of a person experiencing cardiac arrest. This electric shock is intended to stop the abnormal heart rhythm and allow the heart to resume a normal, effective pumping rhythm.
Dielectric: A dielectric is a non-conductive material that, when placed between the plates of a capacitor, increases its capacitance by reducing the electric field strength. This effect occurs due to polarization within the dielectric material.
Dielectric: A dielectric is an insulating material that can be polarized by an electric field. It is a material that does not conduct electricity but can support an electrostatic field by storing energy in the form of an electric field. Dielectrics are essential in the operation of capacitors and other electronic devices, and they play a crucial role in understanding the behavior of electric charges, conductors, insulators, and the applications of electrostatics.
Electric potential difference: Electric potential difference is the work done to move a unit charge between two points in an electric field. It is measured in volts (V).
Electrostatic Potential Energy: Electrostatic potential energy is the potential energy possessed by an electric charge due to its position within an electric field. It is the work done by an external force in bringing a charge from infinity to a specific location in the electric field.
Energy density: Energy density is the amount of energy stored in a given system or region of space per unit volume. It is often used to describe the energy stored in capacitors and electromagnetic fields.
Energy Density: Energy density is a measure of the amount of energy stored or carried per unit volume or mass of a material or system. It is a crucial concept in understanding the storage and transmission of energy in various physical contexts.
Farad: A farad (F) is the SI unit of capacitance, defined as one coulomb of electric charge stored per one volt of potential difference. It quantifies a capacitor's ability to store electrical energy.
Farad: The farad (symbol: F) is the SI unit of electrical capacitance, which is the ability of a body or system to store an electrical charge. It is a fundamental unit that is essential in understanding the behavior of capacitors, which are key components in electrical circuits and devices.
Joule: The joule (J) is the fundamental unit of energy in the International System of Units (SI). It represents the amount of work done or energy expended when a force of one newton acts through a distance of one meter. The joule is a versatile unit that can be used to quantify various forms of energy, including thermal, electrical, and mechanical energy.
Parallel Capacitor: A parallel capacitor is an arrangement of two or more capacitors connected in parallel, where the positive terminal of one capacitor is connected to the positive terminal of the other capacitor(s), and the negative terminal of one capacitor is connected to the negative terminal of the other(s). This configuration allows the capacitors to share the same voltage, while their individual capacitances add up to create a total capacitance for the circuit.
Permittivity of Free Space: Permittivity of free space is a fundamental physical constant that measures the ability of a vacuum to permit electric field lines. It plays a crucial role in electrostatics, affecting the strength of electric fields and the behavior of charge distributions in free space.
Potential Difference: Potential difference, also known as voltage, is the measure of the work done per unit charge in moving an electric charge between two points in an electric field. It represents the potential energy difference between two locations and is a fundamental concept in the study of electric circuits and the behavior of charged particles.
Series Capacitor: A series capacitor is an electrical component that is connected in series with other components in a circuit. It is used to store and release electrical energy, and its behavior is governed by the principles of capacitance and energy stored in a capacitor.
Volt: The volt is the unit of electric potential and electromotive force in the International System of Units (SI). It represents the potential difference across a conductor when a current of one ampere dissipates one watt of power. The volt is a fundamental unit that is essential in understanding and quantifying various electrical phenomena, from the storage of energy in capacitors to the generation of alternating current in household wiring.
Work: Work is the energy transferred to or from an object via a force acting upon it over a displacement. In physics, work is mathematically expressed as $W = F \cdot d \cdot \cos(\theta)$, where $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between them.
Work: Work is a fundamental concept in physics that describes the transfer of energy due to the application of a force over a distance. It is a measure of the energy expended or transferred during a physical process and is a crucial factor in understanding the behavior of thermodynamic systems, electric potential, and the storage of energy in capacitors.