This document discusses conductors and dielectrics. It defines conductors as materials that allow free movement of charges, like metals. The key properties of conductors are that the electric field inside is zero, the charge density inside is zero, and free charges exist only on the surface. These properties influence how conductors behave in external electric fields, inducing opposite charges on surfaces. The document also discusses equipotential surfaces, Poisson's and Laplace's equations, and provides examples of calculating electric fields and charge distributions for various conductor configurations.
This document provides an overview of electrostatics and electric fields. It discusses frictional electricity, properties of electric charges, Coulomb's law, units of charge, and continuous charge distributions. It also covers electric fields, electric field intensity due to point charges, the superposition principle, electric field lines, electric dipoles, and properties of electric field lines. The key topics covered in 3 sentences or less are: Electrostatic forces arise from the transfer of electrons when two materials are rubbed together. Coulomb's law describes the electrostatic force between point charges, which depends on the product of the charges and inversely on the square of the distance between them. Electric field lines represent the direction and strength of the electric field and eman
A Josephson junction is an electronic circuit capable of switching at very high speeds near absolute zero. It consists of two superconductors separated by a thin insulating barrier. Josephson showed that Cooper pairs can tunnel through this barrier coherently, producing effects like a DC supercurrent not requiring a voltage (DC Josephson effect) and an AC supercurrent oscillating at a frequency proportional to an applied voltage (AC Josephson effect). Josephson junctions have applications in areas like SQUIDs, qubits, digital electronics, voltage standards, and sensors.
1. The Hall Effect describes how a magnetic field exerts a force on moving charge carriers in a conductor, causing them to accumulate on one side and creating a measurable Hall voltage.
2. Key parameters like charge carrier density and type can be determined by measuring the Hall voltage under different magnetic field orientations and current directions.
3. The procedure involves applying known currents and magnetic fields to a sample and measuring the resulting Hall and series resistor voltages to determine the Hall coefficient and eliminate contributions from thermoelectric and magnetoresistance effects.
The document discusses the wave properties of particles. Some key points:
1) Louis de Broglie hypothesized in 1924 that matter has an associated wave-like nature with a wavelength given by Planck's constant divided by momentum.
2) A particle can be represented as a localized "wave packet" resulting from the interference and superposition of multiple waves with slightly different wavelengths and frequencies.
3) Davisson and Germer's electron diffraction experiment in 1927 provided direct evidence of the wave nature of electrons and supported de Broglie's hypothesis by measuring electron wavelengths matching those expected.
There are 4 pillars that make up the foundation of Electricity & Magnetism:
1) Gauss' Law (Electricity), which states that the electric field through a closed surface is proportional to the enclosed charge.
2) Gauss' Law (Magnetism), 3) Faraday's Law of Induction, and 4) Ampere's Law. Gauss' Law for electricity, proposed by Carl Friedrich Gauss, relates the total electric flux through a closed surface to the electric charge enclosed by the surface.
The document discusses Maxwell's equations, which describe the fundamental interactions between electricity and magnetism. It provides an overview of each of Maxwell's equations, including Gauss's law for electric and magnetic fields, Faraday's law of induction, and the Ampere-Maxwell law. For each equation, it presents both the integral and differential forms, and provides explanatory notes about the meaning and implications of the equations.
Superconductivity is characterized by zero electrical resistance and the Meissner effect, where magnetic fields are expelled. There are two types of superconductors - Type I, which have an abrupt transition to the normal state, and Type II, which have a more gradual transition. The BCS theory explains superconductivity as electrons pairing up into Cooper pairs at low temperatures, acting as bosons that condense into the same quantum state. Superconductors have applications in medical imaging, maglev trains, and power transmission due to their ability to carry high currents and create strong magnetic fields.
1. The document discusses electrostatic force and electric charge. It explains that when certain materials are rubbed together, they acquire a property called electricity that allows them to attract small pieces of paper.
2. Electric charge is a property of objects that causes electrical and electromagnetic effects. Charge can be positive, negative, or neutral. Charge is quantized and can only occur in discrete integer multiples of the elementary charge.
3. An electric field is the region of space around a charged object where other charged objects will feel an electrostatic force. The electric field strength is defined as the force on a small test charge placed in the field.
This document provides an overview of electrostatics and electric fields. It discusses frictional electricity, properties of electric charges, Coulomb's law, units of charge, and continuous charge distributions. It also covers electric fields, electric field intensity due to point charges, the superposition principle, electric field lines, electric dipoles, and properties of electric field lines. The key topics covered in 3 sentences or less are: Electrostatic forces arise from the transfer of electrons when two materials are rubbed together. Coulomb's law describes the electrostatic force between point charges, which depends on the product of the charges and inversely on the square of the distance between them. Electric field lines represent the direction and strength of the electric field and eman
A Josephson junction is an electronic circuit capable of switching at very high speeds near absolute zero. It consists of two superconductors separated by a thin insulating barrier. Josephson showed that Cooper pairs can tunnel through this barrier coherently, producing effects like a DC supercurrent not requiring a voltage (DC Josephson effect) and an AC supercurrent oscillating at a frequency proportional to an applied voltage (AC Josephson effect). Josephson junctions have applications in areas like SQUIDs, qubits, digital electronics, voltage standards, and sensors.
1. The Hall Effect describes how a magnetic field exerts a force on moving charge carriers in a conductor, causing them to accumulate on one side and creating a measurable Hall voltage.
2. Key parameters like charge carrier density and type can be determined by measuring the Hall voltage under different magnetic field orientations and current directions.
3. The procedure involves applying known currents and magnetic fields to a sample and measuring the resulting Hall and series resistor voltages to determine the Hall coefficient and eliminate contributions from thermoelectric and magnetoresistance effects.
The document discusses the wave properties of particles. Some key points:
1) Louis de Broglie hypothesized in 1924 that matter has an associated wave-like nature with a wavelength given by Planck's constant divided by momentum.
2) A particle can be represented as a localized "wave packet" resulting from the interference and superposition of multiple waves with slightly different wavelengths and frequencies.
3) Davisson and Germer's electron diffraction experiment in 1927 provided direct evidence of the wave nature of electrons and supported de Broglie's hypothesis by measuring electron wavelengths matching those expected.
There are 4 pillars that make up the foundation of Electricity & Magnetism:
1) Gauss' Law (Electricity), which states that the electric field through a closed surface is proportional to the enclosed charge.
2) Gauss' Law (Magnetism), 3) Faraday's Law of Induction, and 4) Ampere's Law. Gauss' Law for electricity, proposed by Carl Friedrich Gauss, relates the total electric flux through a closed surface to the electric charge enclosed by the surface.
The document discusses Maxwell's equations, which describe the fundamental interactions between electricity and magnetism. It provides an overview of each of Maxwell's equations, including Gauss's law for electric and magnetic fields, Faraday's law of induction, and the Ampere-Maxwell law. For each equation, it presents both the integral and differential forms, and provides explanatory notes about the meaning and implications of the equations.
Superconductivity is characterized by zero electrical resistance and the Meissner effect, where magnetic fields are expelled. There are two types of superconductors - Type I, which have an abrupt transition to the normal state, and Type II, which have a more gradual transition. The BCS theory explains superconductivity as electrons pairing up into Cooper pairs at low temperatures, acting as bosons that condense into the same quantum state. Superconductors have applications in medical imaging, maglev trains, and power transmission due to their ability to carry high currents and create strong magnetic fields.
1. The document discusses electrostatic force and electric charge. It explains that when certain materials are rubbed together, they acquire a property called electricity that allows them to attract small pieces of paper.
2. Electric charge is a property of objects that causes electrical and electromagnetic effects. Charge can be positive, negative, or neutral. Charge is quantized and can only occur in discrete integer multiples of the elementary charge.
3. An electric field is the region of space around a charged object where other charged objects will feel an electrostatic force. The electric field strength is defined as the force on a small test charge placed in the field.
1. The document discusses electric flux and Gauss's law. Electric flux is defined as the product of the electric field and the perpendicular surface area. Gauss's law states that the total electric flux through a closed surface is proportional to the enclosed charge.
2. Examples are presented for applying Gauss's law to calculate electric fields produced by spherically symmetric charge distributions like a point charge or thin spherical shell, as well as a cylindrically symmetric charged rod. The calculations involve setting up Gaussian surfaces and relating the flux to the enclosed charge.
3. Key results are that the electric field of a point charge follows an inverse square law, a thin spherical shell produces no field inside but an inverse square field outside, and the
1) Los capacitores almacenan energía eléctrica mediante la transferencia de cargas entre dos conductores aislados. La capacitancia de un capacitor depende de las dimensiones y formas de los conductores y del material aislante entre ellos.
2) Los capacitores tienen muchas aplicaciones prácticas como unidades flash, láseres de pulso y sensores de bolsas de aire. La energía almacenada en un capacitor guarda relación con el campo eléctrico entre los conductores.
3) Existen fórmulas para calcular la capacitancia de
This document discusses Laplace's equation, Poisson's equation, and the uniqueness theorem. It begins by introducing Laplace's equation and Poisson's equation, which are derived from Gauss's law. Poisson's equation applies to problems with a non-zero charge density, while Laplace's equation applies when the charge density is zero. The uniqueness theorem states that for the potential solution to be unique, it must satisfy Laplace's equation and the known boundary conditions. Several examples are then provided to demonstrate solving Laplace's and Poisson's equations for different boundary value problems.
Consider a sample of hydrogen gas in the glass discharge tube. The electric current is passed through the hydrogen gas present in the discharge tube under low pressure. When the hydrogen atoms absorb energy from the electric discharge, they get excited to higher energy states. And the unsettled electron in the excited state then returns to its initial position with the emission of photons of suitable wavelengths.
Now, the hydrogen gas in the discharge tube glows red indicating, the electron transition between the two different energy levels. And the emitted light radiation is passed through the slit and made to fall on the glass prism that separates the light radiation into constituent wavelengths. Finally, the photographic plate placed over there records the line emission spectrum of hydrogen.
The spectrum contains a set of lines in the ultraviolet, visible, and infrared regions. And the wavelength of lines obtained below 400 nm falls in the ultraviolet part of the electromagnetic spectrum. Similarly, wavelengths of lines obtained above 700 nm are in the infrared zone. The spectral lines in the visible region have wavelengths between 400-700 nm. The different wavelengths of light energy produced by hydrogen atoms are also known as the hydrogen light spectrum.
POLARIZATION
Polarization is a property of waves that can oscillate with more than one orientation.
Electromagnetic waves such as light exhibit polarization, as do some other types of wave, such as gravitational waves.
Sound waves in a gas or liquid do not exhibit polarization, since the oscillation is always in the direction the wave travels.
El documento trata sobre conceptos de potencial eléctrico, diferencia de potencial, campo eléctrico y energía potencial. Incluye varios problemas de cálculo relacionados con estas cantidades en diferentes configuraciones de cargas eléctricas puntuales y distribuciones de carga superficial uniforme.
Plane waves have parallel planar wavefronts that propagate indefinitely, while a laser beam has wavefronts that diverge from an initial waist, transitioning from planar to spherical. A laser produces a coherent, monochromatic beam through stimulated emission within an optical cavity. The Gaussian beam equation describes the amplitude, phase, and beam width of a laser beam as it propagates, accounting for its localized amplitude profile and changing wavefront curvature compared to plane waves.
Laser Action
The combination of spontaneous emission first, and then stimulated emission, causes the laser to "lase," which means it generates a coherent beam of light at a single frequency.
This document discusses boundary value problems in electrostatics. It provides the Poisson and Laplace equations, which can be used to solve for electric potential or field when the charge or potential is only known at boundaries. The general procedure is to solve the equations to find the potential, then take derivatives to find the electric field and displacement field. Examples are given to demonstrate applying boundary conditions to determine constants and unique solutions for potential and field.
This document discusses the application of Gauss's law to calculate electric field intensity in two situations:
1) Inside a hollow metallic sphere with charge distributed over its surface. Using Gauss's law, the electric field inside the sphere is found to be zero, showing how electric force can be shielded.
2) Near an infinite sheet with uniform surface charge density. A Gaussian cylinder is used to calculate the flux, yielding an electric field intensity of σ/(2ε0) perpendicular to the sheet. Gauss's law allows calculating electric fields for complex charge distributions.
This document discusses the methodology of thermodynamics and statistical mechanics. It explains that thermodynamics studies the relationships between macroscopic properties like volume and pressure, while statistical mechanics links these macroscopic properties to the microscopic properties of individual molecules through analysis of their positions and momenta. It introduces the key ensembles used in statistical mechanics - the canonical ensemble, which models systems in thermal equilibrium with a heat bath at fixed temperature; the grand canonical ensemble, which models open systems that can exchange both energy and particles with a reservoir; and the microcanonical ensemble, which models isolated systems with a fixed total energy.
1. The document discusses semiconductors and their energy band structure. Semiconductors have a small forbidden band gap that allows electrons to move between the valence and conduction bands with small amounts of energy.
2. In semiconductors, the conduction band is partially filled or overlaps with the valence band, allowing electrons to move freely. This gives semiconductors the ability to be insulators or conductors depending on conditions.
3. The Fermi level lies between the valence and conduction bands and represents the highest energy level electrons can occupy at absolute zero temperature. Charge carriers in semiconductors include both holes in the valence band and electrons in the conduction band.
This document discusses the key principles of quantum physics including:
(1) The wave-particle duality of microparticles like electrons described by de Broglie's equation.
(2) Energy quantization described by Planck's equation.
(3) Heisenberg's uncertainty principle.
It describes how Schrodinger's equation is used to model the wave-like behavior of electrons in solids. The energy and behavior of electrons is quantized based on solutions to Schrodinger's equation under different boundary conditions, such as electrons confined in a potential well or interacting with a potential barrier. Quantum theory was needed to fully explain properties of electrons in solids and failures of classical free electron theory
Electrostatic induction occurs when a charged object is brought near an uncharged conductor. This causes the redistribution of the conductor's electrons, leaving some regions positively charged and others negatively charged. If the conductor is grounded, electrons will flow to the ground, leaving it positively charged. When the charged object is removed, the conductor's charges spread uniformly again. Dielectrics, made of polar or nonpolar molecules, can be polarized by an external electric field through the alignment of their molecules' dipole moments, resulting in an internal electric field that opposes the external field. Dielectrics break down if the external field exceeds their dielectric strength.
This document provides an overview of electrostatics and defines key terms:
- Electrostatics is the study of charges at rest and there are two types of electric charges: positive and negative.
- The fundamental law states that like charges repel and unlike charges attract.
- Bodies can become charged through rubbing, induction, or conduction. Devices like electroscopes and Van de Graff generators are used to study electric charges.
- Coulomb's law describes the electric force between two point charges. The electric field is defined as the force experienced by a test charge per unit of charge. Electric field lines indicate the direction of the electric force.
Los documentos describen experimentos sobre partículas cargadas que se mueven en campos magnéticos uniformes. En particular, se analizan las trayectorias circulares que describen al entrar perpendicularmente al campo, y cómo depende el radio de la órbita de parámetros como la carga, masa, velocidad y intensidad del campo magnético. También se discuten algunas propiedades generales de la fuerza magnética sobre partículas cargadas.
1. A field is defined as a property of space in which a material object experiences a force. For example, above the earth there is a gravitational field where a mass experiences a downward force.
2. The electric field is a vector field that exists around an electric charge even when no other charges are present. The direction of the electric field points away from a positive charge and towards a negative charge.
3. The electric field strength is defined as the force experienced by a hypothetical positive test charge per unit of charge, located at that point in space.
The document discusses the Hall effect, which is when a conductor carrying an electric current is placed perpendicular to a magnetic field. This causes the charges in the conductor to experience a force perpendicular to both the current and the magnetic field. This displacement of charges establishes a voltage difference known as the Hall voltage across the conductor. The Hall effect can be used to determine various properties of materials like charge carrier types and densities. Precise measurement techniques like Van der Pauw and Hall coefficient calculations are used to characterize semiconductor samples.
This document discusses Poisson's and Laplace's equations which relate electric potential to charge density. Poisson's equation applies to regions with charge density, while Laplace's equation applies to charge-free regions. The equations are derived and their applications are demonstrated, including calculating electric fields and potentials. Examples are provided of solving Laplace's equation for different boundary conditions. The document also covers capacitance of parallel plate capacitors, with and without a dielectric, as well as resistance and combinations of resistors in series and parallel.
1) The document discusses key concepts related to electric forces, fields, and circuits including that electric fields point in the direction of force on a positive test charge and capacitance depends on surface area, plate separation, and dielectric material.
2) Important vocabulary is defined like conductors, electric fields, capacitors, and dielectrics.
3) Formulas are presented for calculating values like charge, Coulomb's law, electric field, potential, capacitance, current, and resistance.
This document discusses the applications of dielectric materials. It begins by defining dielectric materials as insulating materials used for charge storage, as opposed to simple insulation. Common dielectric materials include ceramics, mica, glass, and plastics. The document then discusses two major applications of dielectric materials: capacitors and transformers. In capacitors, dielectrics allow greater charge storage at a given voltage. Transformers also rely on dielectrics to insulate coils and prevent electrical conduction while allowing magnetic energy transfer between coils. Common dielectric materials used in each application are also outlined.
1. The document discusses electric flux and Gauss's law. Electric flux is defined as the product of the electric field and the perpendicular surface area. Gauss's law states that the total electric flux through a closed surface is proportional to the enclosed charge.
2. Examples are presented for applying Gauss's law to calculate electric fields produced by spherically symmetric charge distributions like a point charge or thin spherical shell, as well as a cylindrically symmetric charged rod. The calculations involve setting up Gaussian surfaces and relating the flux to the enclosed charge.
3. Key results are that the electric field of a point charge follows an inverse square law, a thin spherical shell produces no field inside but an inverse square field outside, and the
1) Los capacitores almacenan energía eléctrica mediante la transferencia de cargas entre dos conductores aislados. La capacitancia de un capacitor depende de las dimensiones y formas de los conductores y del material aislante entre ellos.
2) Los capacitores tienen muchas aplicaciones prácticas como unidades flash, láseres de pulso y sensores de bolsas de aire. La energía almacenada en un capacitor guarda relación con el campo eléctrico entre los conductores.
3) Existen fórmulas para calcular la capacitancia de
This document discusses Laplace's equation, Poisson's equation, and the uniqueness theorem. It begins by introducing Laplace's equation and Poisson's equation, which are derived from Gauss's law. Poisson's equation applies to problems with a non-zero charge density, while Laplace's equation applies when the charge density is zero. The uniqueness theorem states that for the potential solution to be unique, it must satisfy Laplace's equation and the known boundary conditions. Several examples are then provided to demonstrate solving Laplace's and Poisson's equations for different boundary value problems.
Consider a sample of hydrogen gas in the glass discharge tube. The electric current is passed through the hydrogen gas present in the discharge tube under low pressure. When the hydrogen atoms absorb energy from the electric discharge, they get excited to higher energy states. And the unsettled electron in the excited state then returns to its initial position with the emission of photons of suitable wavelengths.
Now, the hydrogen gas in the discharge tube glows red indicating, the electron transition between the two different energy levels. And the emitted light radiation is passed through the slit and made to fall on the glass prism that separates the light radiation into constituent wavelengths. Finally, the photographic plate placed over there records the line emission spectrum of hydrogen.
The spectrum contains a set of lines in the ultraviolet, visible, and infrared regions. And the wavelength of lines obtained below 400 nm falls in the ultraviolet part of the electromagnetic spectrum. Similarly, wavelengths of lines obtained above 700 nm are in the infrared zone. The spectral lines in the visible region have wavelengths between 400-700 nm. The different wavelengths of light energy produced by hydrogen atoms are also known as the hydrogen light spectrum.
POLARIZATION
Polarization is a property of waves that can oscillate with more than one orientation.
Electromagnetic waves such as light exhibit polarization, as do some other types of wave, such as gravitational waves.
Sound waves in a gas or liquid do not exhibit polarization, since the oscillation is always in the direction the wave travels.
El documento trata sobre conceptos de potencial eléctrico, diferencia de potencial, campo eléctrico y energía potencial. Incluye varios problemas de cálculo relacionados con estas cantidades en diferentes configuraciones de cargas eléctricas puntuales y distribuciones de carga superficial uniforme.
Plane waves have parallel planar wavefronts that propagate indefinitely, while a laser beam has wavefronts that diverge from an initial waist, transitioning from planar to spherical. A laser produces a coherent, monochromatic beam through stimulated emission within an optical cavity. The Gaussian beam equation describes the amplitude, phase, and beam width of a laser beam as it propagates, accounting for its localized amplitude profile and changing wavefront curvature compared to plane waves.
Laser Action
The combination of spontaneous emission first, and then stimulated emission, causes the laser to "lase," which means it generates a coherent beam of light at a single frequency.
This document discusses boundary value problems in electrostatics. It provides the Poisson and Laplace equations, which can be used to solve for electric potential or field when the charge or potential is only known at boundaries. The general procedure is to solve the equations to find the potential, then take derivatives to find the electric field and displacement field. Examples are given to demonstrate applying boundary conditions to determine constants and unique solutions for potential and field.
This document discusses the application of Gauss's law to calculate electric field intensity in two situations:
1) Inside a hollow metallic sphere with charge distributed over its surface. Using Gauss's law, the electric field inside the sphere is found to be zero, showing how electric force can be shielded.
2) Near an infinite sheet with uniform surface charge density. A Gaussian cylinder is used to calculate the flux, yielding an electric field intensity of σ/(2ε0) perpendicular to the sheet. Gauss's law allows calculating electric fields for complex charge distributions.
This document discusses the methodology of thermodynamics and statistical mechanics. It explains that thermodynamics studies the relationships between macroscopic properties like volume and pressure, while statistical mechanics links these macroscopic properties to the microscopic properties of individual molecules through analysis of their positions and momenta. It introduces the key ensembles used in statistical mechanics - the canonical ensemble, which models systems in thermal equilibrium with a heat bath at fixed temperature; the grand canonical ensemble, which models open systems that can exchange both energy and particles with a reservoir; and the microcanonical ensemble, which models isolated systems with a fixed total energy.
1. The document discusses semiconductors and their energy band structure. Semiconductors have a small forbidden band gap that allows electrons to move between the valence and conduction bands with small amounts of energy.
2. In semiconductors, the conduction band is partially filled or overlaps with the valence band, allowing electrons to move freely. This gives semiconductors the ability to be insulators or conductors depending on conditions.
3. The Fermi level lies between the valence and conduction bands and represents the highest energy level electrons can occupy at absolute zero temperature. Charge carriers in semiconductors include both holes in the valence band and electrons in the conduction band.
This document discusses the key principles of quantum physics including:
(1) The wave-particle duality of microparticles like electrons described by de Broglie's equation.
(2) Energy quantization described by Planck's equation.
(3) Heisenberg's uncertainty principle.
It describes how Schrodinger's equation is used to model the wave-like behavior of electrons in solids. The energy and behavior of electrons is quantized based on solutions to Schrodinger's equation under different boundary conditions, such as electrons confined in a potential well or interacting with a potential barrier. Quantum theory was needed to fully explain properties of electrons in solids and failures of classical free electron theory
Electrostatic induction occurs when a charged object is brought near an uncharged conductor. This causes the redistribution of the conductor's electrons, leaving some regions positively charged and others negatively charged. If the conductor is grounded, electrons will flow to the ground, leaving it positively charged. When the charged object is removed, the conductor's charges spread uniformly again. Dielectrics, made of polar or nonpolar molecules, can be polarized by an external electric field through the alignment of their molecules' dipole moments, resulting in an internal electric field that opposes the external field. Dielectrics break down if the external field exceeds their dielectric strength.
This document provides an overview of electrostatics and defines key terms:
- Electrostatics is the study of charges at rest and there are two types of electric charges: positive and negative.
- The fundamental law states that like charges repel and unlike charges attract.
- Bodies can become charged through rubbing, induction, or conduction. Devices like electroscopes and Van de Graff generators are used to study electric charges.
- Coulomb's law describes the electric force between two point charges. The electric field is defined as the force experienced by a test charge per unit of charge. Electric field lines indicate the direction of the electric force.
Los documentos describen experimentos sobre partículas cargadas que se mueven en campos magnéticos uniformes. En particular, se analizan las trayectorias circulares que describen al entrar perpendicularmente al campo, y cómo depende el radio de la órbita de parámetros como la carga, masa, velocidad y intensidad del campo magnético. También se discuten algunas propiedades generales de la fuerza magnética sobre partículas cargadas.
1. A field is defined as a property of space in which a material object experiences a force. For example, above the earth there is a gravitational field where a mass experiences a downward force.
2. The electric field is a vector field that exists around an electric charge even when no other charges are present. The direction of the electric field points away from a positive charge and towards a negative charge.
3. The electric field strength is defined as the force experienced by a hypothetical positive test charge per unit of charge, located at that point in space.
The document discusses the Hall effect, which is when a conductor carrying an electric current is placed perpendicular to a magnetic field. This causes the charges in the conductor to experience a force perpendicular to both the current and the magnetic field. This displacement of charges establishes a voltage difference known as the Hall voltage across the conductor. The Hall effect can be used to determine various properties of materials like charge carrier types and densities. Precise measurement techniques like Van der Pauw and Hall coefficient calculations are used to characterize semiconductor samples.
This document discusses Poisson's and Laplace's equations which relate electric potential to charge density. Poisson's equation applies to regions with charge density, while Laplace's equation applies to charge-free regions. The equations are derived and their applications are demonstrated, including calculating electric fields and potentials. Examples are provided of solving Laplace's equation for different boundary conditions. The document also covers capacitance of parallel plate capacitors, with and without a dielectric, as well as resistance and combinations of resistors in series and parallel.
1) The document discusses key concepts related to electric forces, fields, and circuits including that electric fields point in the direction of force on a positive test charge and capacitance depends on surface area, plate separation, and dielectric material.
2) Important vocabulary is defined like conductors, electric fields, capacitors, and dielectrics.
3) Formulas are presented for calculating values like charge, Coulomb's law, electric field, potential, capacitance, current, and resistance.
This document discusses the applications of dielectric materials. It begins by defining dielectric materials as insulating materials used for charge storage, as opposed to simple insulation. Common dielectric materials include ceramics, mica, glass, and plastics. The document then discusses two major applications of dielectric materials: capacitors and transformers. In capacitors, dielectrics allow greater charge storage at a given voltage. Transformers also rely on dielectrics to insulate coils and prevent electrical conduction while allowing magnetic energy transfer between coils. Common dielectric materials used in each application are also outlined.
Karl Gauss developed Gauss's law, which relates the electric flux through a closed surface to the electric charge enclosed by the surface. Gauss's law can be used to calculate electric fields produced by charge distributions or to determine charge distributions that produce known electric fields. When applying Gauss's law, it is important to choose a Gaussian surface where the electric field is constant, zero, or parallel to the surface area elements to simplify the flux calculations. Conductors in electrostatic equilibrium have no electric field inside their volume, any excess charge resides on the surface, and the electric field just outside is perpendicular to the surface with a magnitude proportional to the surface charge density.
This document discusses the topic of superconductivity. It begins by introducing superconductivity as a phenomenon where certain materials conduct electricity without resistance below a critical temperature. It then describes the general properties of superconductors such as critical temperature, magnetic field effect, and persistent current. The document goes on to classify superconductors into two types and discusses their different behaviors in magnetic fields. It concludes by outlining several applications that utilize the unique properties of superconductors, such as Maglev trains, SQUIDs, and efficient power transmission.
Metals are opaque and highly reflective. They reflect most visible light due to their continuous empty electron states that allow electrons to absorb and re-emit light of the same wavelength. Silver reflects almost all light wavelengths, while gold's color is due to some light photons not being reemitted visibly. Nonmetals may absorb, reflect, refract, or transmit light depending on their electron band structure and properties like index of refraction.
This document provides information on various topics related to magnetism and magnetic materials:
1. It discusses different types of magnetic behavior such as diamagnetism, paramagnetism, and ferromagnetism. It also discusses the properties of hard and soft magnetic materials.
2. Key magnetic parameters are defined, including magnetic permeability, susceptibility, intensity of magnetization, Curie temperature, magnetic dipole moment, magnetic flux, and relative permeability.
3. The differences between diamagnetic, paramagnetic, and ferromagnetic materials are summarized in a table comparing their behaviors and properties.
4. The document also explains hysteresis loops, hard and soft magnets, and fer
1. Magnetism arises from the magnetic moments of electrons, both from their orbital motion and spin.
2. Magnetic materials can be classified as diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, or ferrimagnetic based on their magnetic properties.
3. Ferromagnetic materials exhibit hysteresis, where magnetization lags behind an applied magnetic field, leading to a hysteresis loop. Hard magnetic materials have large hysteresis loops while soft magnetic materials have small loops.
Dielectrics are materials that have permanent electric dipole moments. All dielectrics are electrical insulators and are mainly used to store electrical energy by utilizing bound electric charges and dipoles within their molecular structure. Important properties of dielectrics include their electric intensity or field strength, electric flux density, dielectric parameters such as dielectric constant and electric dipole moment, and polarization processes including electronic, ionic, and orientation polarization. Dielectrics are characterized by their complex permittivity, which relates to their ability to transmit electric fields and is dependent on factors like frequency, temperature, and humidity that can influence dielectric losses.
Dielectrics are materials that contain permanently aligned electric dipoles. When an electric field is applied, the dipoles in dielectric materials can undergo several types of polarization, including electronic, ionic, orientational, and space charge polarization. This polarization leads to an increase in the electric flux density and dielectric constant within the material. The dielectric constant is the ratio of the material's permeability to the permeability of free space and determines the material's behavior in electric fields.
Basic Information regarding superconductors.
Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a characteristic critical temperature.
This power-point presentation include
1. Introduction to Superconductors
2. Discovery
3. Properties
4. Important factors
5. Types
6. High Tc Superconductors
7. Magnetic Levitation and its application
8. Josephson effect
9. Application of superconductors
#Tip- You can further add videos which are available in vast amount on YouTube regarding superconductivity(specially magnetic levitation)
P.S.Does not contain information about Cooper pairs and BCS theory
Superconductivity is a phenomenon where certain materials have zero electrical resistance and expel magnetic fields when cooled below a critical temperature. It was discovered in 1911 by Heike Kamerlingh Onnes. The Meissner effect describes how superconducting materials actively push magnetic fields out of their interior when transitioning to the superconducting state. The BCS theory explains superconductivity as electrons forming Cooper pairs that pass through the material unimpeded. Superconducting materials include various metals, metal alloys, iron-based compounds, cuprates, and organic materials. Applications include maglev trains, medical imaging, and more efficient power transmission.
This document provides 50 common interview questions and suggested answers. It advises preparing responses that highlight relevant experience, focus on benefits to the organization, and remain positive. Some key questions covered include telling about yourself, weaknesses of past employers, how to handle salary discussions, and being able to discuss accomplishments and what motivates you. The document concludes by reminding the reader to have their own questions prepared to ask the interviewer.
The document summarizes optical properties of nanomaterials. It discusses topics like optics, optical properties of materials, thin film interference, luminescence, photonic crystals, photoconductivity, solar cells, and optical properties of quantum wells and quantum dots. In particular, it explains how the size-dependent band gap of quantum dots leads to size-tunable fluorescence colors, making quantum dots useful for applications like biological imaging and white LEDs.
This document provides materials to help prepare for an electrical engineering interview, including 92 electrical interview questions and answers and tips for answering common questions. Some key questions covered include explaining differential amplifiers and CMRR, discussing the advantages of AC over DC systems, describing experience and goals. The document also provides links to additional interview preparation resources.
1. The document discusses several key concepts in electrostatics including:
2. Line integral of electric field is equal to the negative of the potential difference between two points. Work done by an external force in moving a test charge between two points is equal to the potential difference.
3. Electrostatic force is a conservative force since the work done in moving a test charge along a closed path is zero, meaning the work is independent of the path taken.
4. Conductors allow the flow of electric charge through them, while insulators do not. Dielectrics can transmit electric effects when placed in an electric field through the induction of surface charges.
1. Gauss's law relates the electric flux through a closed surface to the net electric charge enclosed. The net electric flux is directly proportional to the enclosed charge.
2. For a point charge inside a spherical Gaussian surface, the electric flux is independent of the sphere's radius and depends only on the enclosed charge.
3. According to Gauss's law, the electric field is zero inside a conductor with excess surface charge because there is no enclosed charge, and excess charge resides entirely on the conductor's surface.
1) The document discusses electric potential, electric field, and capacitors. It provides definitions and equations for electric potential due to point charges, conducting spheres, rings, and other configurations.
2) Examples are given for calculating electric potential and electric field due to various charge distributions, including point charges, conducting spheres, disks, rings, and charged rods. Integral techniques are used for non-uniform charge distributions.
3) Boundary conditions for electric potential and fields are explained. The relationship between electric potential and electric field is emphasized.
- A capacitor is a device that stores electric charge and consists of two conductors carrying equal but opposite charges. There is a potential difference between the conductors.
- The amount of charge Q stored in a capacitor is proportional to the potential difference ΔV between the conductors. The constant of proportionality is called the capacitance C.
- Inserting a dielectric material between the capacitor plates increases the capacitance by a factor called the dielectric constant. Within the dielectric, the electric field is reduced even though the surface charges remain the same.
1) The document discusses various concepts related to electrostatics including capacitance, capacitors, and dielectrics. It defines capacitance as the ability of a conductor to store charge and explains the factors that affect capacitance such as area and distance of separation for a parallel plate capacitor.
2) Energy storage in capacitors is also covered, explaining that work must be done to charge a capacitor and this work is stored as electrostatic potential energy. Expressions are given for calculating energy stored in series and parallel combinations of capacitors.
3) The document introduces the concept of dielectrics, noting that polar molecules can have their dipole moments aligned by an electric field to induce polarization while non-polar
1. The document discusses various topics in electrostatics including line integrals of electric fields, electric potential and potential differences, Gauss's theorem, and applications of Gauss's theorem.
2. Key concepts covered are the definitions of electric potential and potential difference, the relationship between electric field and potential via line integrals, and Gauss's theorem that the electric flux through any closed surface is equal to the enclosed charge divided by the permittivity of free space.
3. Examples are given of using Gauss's theorem to calculate electric fields, such as for an infinite line charge, planar sheet of charge, and spherical shell of charge.
1. The document discusses concepts in electrostatics including the line integral of electric field, electric potential, Gauss's theorem, and applications.
2. Gauss's theorem states that the electric flux through any closed surface is equal to the net charge enclosed divided by the permittivity of free space.
3. The theorem can be used to derive Coulomb's law and calculate electric field intensities due to various charge distributions like line charges and spherical shells.
This document provides an overview of electrostatics and electric current concepts. It defines electrostatics as electricity from the Greek word for amber, where static electricity is generated by rubbing materials together. The key concepts covered include:
- Coulomb's law which describes the force between electric charges.
- The properties of electric fields and field intensity.
- How capacitors store electric charge and the differences between capacitors connected in parallel versus series.
- Definitions of electric current, resistance, voltage, and potential drop in circuits.
This document discusses the concepts of capacitance and capacitors. It defines capacitance as the ability of a conductor to store charges, with the capacitance of a conductor being equal to the amount of charge needed to change its potential by 1 volt. A capacitor is formed when two conductors are separated by an insulator. When charged, one plate gains electrons and becomes negatively charged while the other loses electrons and becomes positively charged. The capacitance of a capacitor depends on the size, shape, and distance between its conducting plates as well as the dielectric material between them. Common units of capacitance include the farad and its submultiples like microfarads and picofarads.
The document discusses several topics in electrostatics including electric potential, potential difference, equipotential surfaces, Gauss's law, and applications of Gauss's law. Gauss's law states that the electric flux through any closed surface is equal to the enclosed charge divided by the permittivity of free space. This relationship can be used to derive Coulomb's law and calculate electric fields due to various charge distributions like line charges, plane sheets of charge, and spherical shells.
This document discusses the principles and components of a Van de Graaff generator, which is used to generate very high voltages. It consists of a large metal sphere and two combs with sharp points that are attached to moving belts. When the belts transfer charge to the sphere via the combs, the potential of the sphere increases greatly. The sharp points on the combs ionize the surrounding air through corona discharge, spraying charges onto the belts. This allows the generator to continually build up charge on the sphere over time.
1) The document provides one mark, two mark and three mark questions from the chapter on Electric Charges and Fields.
2) It includes questions testing definitions of key terms like electric charge, electric field, electric dipole moment, Gauss's law.
3) It also has questions requiring diagrams of electric field patterns and derivations of expressions for force between charges and electric field.
1) The document provides one mark, two mark and three mark questions from the chapter on Electric Charges and Fields.
2) It includes questions testing definitions of key terms like electric charge, electric field, electric dipole moment, Gauss's law.
3) It also has questions requiring diagrams of electric field patterns and derivations of expressions for force between charges and electric field.
This document contains 29 multi-part physics problems related to electric fields, electric potential, and capacitance. The problems cover a range of concepts including Gauss's law, electric fields due to various charge distributions, capacitors in series and parallel, energy stored in capacitors, and more. Detailed calculations and explanations are required to fully solve each problem.
This document contains conceptual problems and their solutions related to electric fields and Gauss's law.
Problem 29 asks about an electric field given by a formula and calculates (a) the electric flux through each end of a cylinder in that field, (b) the flux through the curved surface, and (c) the total flux through the closed cylindrical surface. It then (d) uses Gauss's law to find the net charge inside the cylinder.
Problem 33 gives the electric flux out of one side of an imaginary cube and asks the reader to use Gauss's law to determine the amount of charge at the center of the cube.
The document provides an overview of electromagnetism and various concepts in physics such as electric charge, electric fields, electric potential, capacitors, and dielectrics. It discusses key historical discoveries, fundamental laws and equations, and examples of calculating values for different concepts through problem solving.
Electricity and magnetism are fundamentally linked phenomena that underpin modern technology. Electric forces exist between charged objects and atoms due to the interactions of positive protons in atomic nuclei and negative electrons orbiting them. Electricity controls many biological processes in living things. While objects are electrically neutral overall, applying forces can cause separations of charge that result in objects or parts of objects becoming electrically charged.
MAHARASHTRA STATE BOARD
CLASS XI AND XII
PHYSICS
CHAPTER 8
ELECTROSTATICS
Introduction.
Coulomb's law
Calculating the value of an electric field
Superposition principle
Electric potential
Deriving electric field from potential
Capacitance
Principle of the capacitor
Dielectrics
Polarization, and electric dipole moment
Applications of capacitors.
This document provides an overview of Gauss's law and its applications. It begins with definitions of electric flux and how to calculate flux through surfaces. It then introduces Gauss's law, which relates the electric flux through a closed surface to the enclosed charge. Examples are provided to demonstrate how to use Gauss's law to determine electric fields for symmetric charge distributions. The document also discusses how the electric field is zero inside conductors in electrostatic equilibrium and how Gauss's law can be used to show this. Worked examples further illustrate applying Gauss's law.
This document discusses methods of generating frequency-modulated (FM) signals. There are two basic methods: direct FM, where the carrier frequency is directly varied by the information signal using a voltage-controlled oscillator; and indirect FM, where the information signal modulates the phase of the carrier signal. FM generation can be achieved easily using a linear voltage-to-frequency converter like the LM566C integrated circuit, which generates a square or triangular FM wave that can be filtered into a sine wave. The LM566C is capable of FM frequencies up to 1 MHz. Noise has little effect on FM signals because FM receivers use limiter circuits that clip amplitude variations while preserving the frequency modulation carrying the information.
This document discusses interfacing a 7-segment display with an AVR microcontroller. It begins by introducing 7-segment displays and their use in common devices. It then explains the fundamentals of how a 7-segment display works, showing the individual segments that combine to display numbers. The document outlines the pin configurations for common anode and cathode displays and shows a block diagram of interfacing the display with a microcontroller port. It includes a table mapping hexadecimal values to the on/off states of the 7 segments needed to display each number and letter. Programming details are provided for initializing the controller and enabling the display output at a set brightness level.
This document describes interfacing an LCD display and keyboard with an AVR microcontroller board. It includes introductions to the AVR board, LCD, and keyboard. It describes connecting these components and provides the code for initializing the LCD, reading the keyboard matrix, and displaying pressed keys on the LCD. The main program continuously scans the keyboard matrix and uses a check function to display the pressed key on the LCD.
This document provides a history of computer crime, beginning with traditional problems of anonymity and ineffective law enforcement. It discusses early cases of computer sabotage in the 19th century. Three major incidents in the 1980s shook complacency: the compromise of Milnet which stole military data, the Morris Worm which caused millions in damages, and the crash of AT&T's network. The document categorizes computer crimes as those using computers as a means, targeting computers, or being incidental to computers. It provides examples such as phreaking, web-cramming, spoofing, and cybersquatting.
This document discusses CURL and its applications in vector fields. CURL describes the tendency of a fluid to cause rotation and was introduced by physicist James Clerk Maxwell. The CURL of a velocity field F measures how a fluid would turn an inserted paddle device. A flow with zero CURL is considered irrotational without vortices. DIVERGENCE describes the flux of a vector field through a surface, indicating fluid sources and sinks. For incompressible fluids, the DIVERGENCE is zero with no change in density. CURL and DIVERGENCE provide physical insight into fluid motion and flows.
Coulomb's law describes the electrostatic force between two point charges. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The law leads to the concept of electric field as a vector field that describes the electromagnetic force exerted on charged objects. Electric field lines are a graphical representation of the electric field, with properties like direction, strength, and superposition. Coulomb's law and electric fields can be applied to understand phenomena like dipoles, parallel plates in a capacitor, and spherically symmetric charge distributions.
Wireshark is a free and open-source packet analyzer that allows users to examine network traffic and protocol data in real-time. It can be used by network administrators to troubleshoot issues, security engineers to examine security problems, and developers to debug protocol implementations. Wireshark captures packets in real-time and displays them in an easy-to-read format with filters, color-coding and other features to analyze individual packets and network traffic.
This document discusses noise and its effects on systems. It defines noise as any unwanted input that limits a system's ability to process weak signals. Noise can come from various sources like resistors, transistors, mixers, power supplies, and the environment. The signal-to-noise ratio is used to measure the "noisiness" of a signal. Noise factor and noise figure are introduced as measures of a receiver's performance in the presence of noise. The concepts of noise temperature and its relationship to noise factor are also explained. Cascaded amplifiers and lossy networks are discussed in the context of calculating overall noise factor.
This document discusses interfacing an ultrasonic rangefinder module with an AVR microcontroller. It begins by describing the limitations of basic infrared obstacle sensors, such as not being able to measure accurate distances. Ultrasonic rangefinder modules are introduced as a better solution, being able to measure distances from 1cm to 400cm with 1cm accuracy. The document then discusses the characteristics and advantages of ultrasonic sensors, how they interface with microcontrollers, and provides an overview of the hardware that will be used to build a test circuit around an ATmega32 microcontroller and LCD display.
Mixers are electronic devices used to combine audio signals by routing and changing their level, tone, and dynamics. They allow adjustment of levels, equalization, effects, monitoring, and recording. Mixers come in various sizes from small portable units to large studio consoles. While intimidating for beginners due to many controls, mixers essentially have duplicated channel strips that make them easier to understand once you know how each channel works. Each channel strip contains gain, EQ, auxiliary sends, panning, and a level fader to control the signal flow and mix.
An auditorium is a room designed for audiences to watch performances. It contains seating areas like stalls at the same level as the stage, balconies on raised platforms towards the back, and private boxes near the front. Balconies may be stacked vertically and the highest level is sometimes called the "gods". Auditoriums are found in venues like theaters, opera houses, amphitheaters, concert halls, and stadiums. From an audio perspective, improving acoustics involves absorbing, blocking, and covering up sounds using materials like drapes, walls, and sound masking. Building services noise control focuses on reducing noise from HVAC, elevators, and generators through vibration isolation and sound traps.
The standard dynamic loudspeaker was developed in the 1920s and uses a magnetic field to move a coil connected to a diaphragm. There are several types of speakers that work differently, such as horns, electrodynamic, flat panel, plasma arc, and piezoelectric speakers. Electrodynamic speakers are the most common today and work by using a magnet to move a copper coil attached to a diaphragm, which vibrates the air to produce sound. Engineers have developed different types of drivers like tweeters, mid-ranges, and woofers to effectively reproduce specific frequency ranges.
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2. Conductors
A conductor (typically, a metal like Cu, Ag etc. or ionic conductors like HCl or
NaCl dissolved in water) allows free movement of charges. They have low resis-
tivity 10−8Ωm as compared to typical insulators like quartz, glass etc. which have
resistivity of the order of 1017Ωm. However, the property that really distinguishes
a metal from insulators or semi-conductors is the fact their temperature coefficient
of resistivity is positive while that of semi-conductors is negative.
• The electric field inside a conductor is zero. In an equilibrium situa- tion,
there cannot be an electric field inside a conductor as this would cause
charges (electrons or ions) to move around. In the presence of an external
field, there is charge separation inside a conductors with opposite charges
accumulating on the surface. This creates an internal electric field which
cancels the effect of the external field in such a way that the net electric
field inside the conductor volume is zero.
+
+
+
+
+
+
+
+
2
− E
−
−
−
−
−
−
−
int
Eext
• Charge density inside a conductor is zero. This follows from Gauss’slaw
∇ ·E = ρ/s0
As E˙ = 0,the charge density ρ = 0.
3. (This does not suggest that there is no charge inside, only that the positive
and negative charges cancel inside a conductor.)
• Free charges exist only on the surface of a conductor. Since there is no
net charge inside, free charges, if any, have to be on the surface.
• At the surface of a conductor, the electric field is normal to the surface.
If this were not so, the charges on the surface would move along the surface
because of the tangential component of the field, disturbing equilibrium.
E=0
Induced Charges in a conductor:
The above properties of a conductor influence the behaviour of a conductor placed
in an electric field. Consider, for instance, what happens when a charge +q is
brought near an uncharged conductor. The conductor is placed in the electric field
of the point charge. The field inside the conductor should, however, be zero. his
is achieved by a charge separation within the conductor which creates its own
electric field which will exactly compensate the field due to the charge +q. The
separated charges must necessarily reside on the surface.
Another way of looking at what is
happening is to think of the free
charges in the conductor being at-
tracted towards (or repelled from)
the external charge. Thus thesurface
nal charge is oppositely charged. To
keep the charge neutrality, the sur-
face away from the external charge
is similarly charged.
+q
++
+
+
+
of the conductor towards the exter- +
+
++
−
− −
−
−
−
−
−−
3
4. Example 1 :
A charge Qis located in the cavity inside a conducting shell. In addition, a charge
2Q is distributed in the conducting shell. Find the distribution of charge in the
shell. What is the electric field in the region outside the shell.
+Q
4
+
+
+
+
+
+
+
+
+
+
+ +
+
+
+
+
+
+
++
Take a gaussian surface entirely within the conducting shell, completely enclosing
the cavity. Everywhere on the gaussian surface E˙= 0. The flux and therefore, the
charge enclosed is zero within the gaussian surface. As the cavity contains a
charge Q, the surface of the cavity must have charge −Q. As the conductor has
distributed charge 2Q, the charge on the outside surface is 3Q.
The principle illustrated in the above problem is known as Faraday’s Cage. If a
hollow conducting box is kept in an electric field, the charges in the cavity are
redistributed in such a way that the electric field inside the cavity is zero. This is
used to provide an enclosure for sensitive electronic equpment which must be
kept free of external electronic disturbance.
Example 2 :
Calculate the electric field outside a conductor carrying a surface charge density
σ.
5. ++ +
+ +
dSE
+
+ ++
+ +
r +
L
+ +
+ +
+
+
+
+
+
+
+
+
+ ++
+ +++
Take a gaussian pillbox in the shape of a cylinder of height h with h/2 inside
and h/2 outside the conductor. Lat the cross sectional area be dS normal to the
surface. The electric field is normal to the surface. As the field inside is zer and
there is no tangential component of the field at the surface, the flux goes out only
through the outer cap of the cylider. The charge enclosed is σdS and the flux is
EdS. The electric field is normal to the surface. applyinng Gauss’s law
σ
s0
E = nˆ
.
Exercise :
Two parallel, infinite plates made of material of perfect conductor, carry charges
Q1 and Q2. The plates have finite thickness. Show that the charge densities on the
two adjecent inside surfaces are equal and opposite while that on the two outside
surfaces are equal. (Hint : Field inside the plates due to four charged surfaces
must be zero.)
Poisson’s and Laplace’s Equations
Differential form of Gauss’s law,
∇˙·E˙ =
ρ
s0
5
Using E˙ = −∇V,
∇˙E˙ = −∇ ·(∇V ) = −∇2
V
6. so that
∇2
V = −
ρ
s0
This is Poisson equation. In cartesian form,
∂2V ∂2V ∂2V ρ
∂x2 +
∂y2 +
∂z2 = −
s0
For field free region, the equation becomes Laplace’s equation
∇2
V = 0
Equipotential surface
Equipotential surfaces are defined as surfaces over which the potential is constant
V(r˙)= constant
At each point on the surface, the electric field is perpendicular to the surface since
the electric field, being the gradient of potential, does not have component along
a surface of constant potential.
• We have seen that any charge on a conductor must reside on its surface.
These charges would move along the surface if there were a tangential com-
ponent of the electric field. The electric field must therefore be along the
normal to the surface of a conductor. The conductor surface is, therefore,
an equipotential surface.
• Electric field lines are perpendicular to equipotential surfaces (or curves)
and point in the direction from higher potential to lower potential.
• In the region where the electric field is strong, the equipotentials are closely
packed as the gradient is large.
6
7. −0.5 kV
3 kV
2 kV
1 kV
P x
−1 kV
0
−2 kV
The electric field strength at the point P may be found by finding the slope of the
potential at the point P. If ∆x is the distance between two equipotential curves
close to P,
E = −
∆V
∆x
where ∆V is the difference between the two equipotential curves near P.
Example 3:
Determine the equipotential surface for a point charge.
Solution : Let the point charge qbe located at the origin. The equation to the
equipotential surface is given by
V(x, y, z) =
1 q
4πs0
,
x 2 + y2 + z2
7
= V0 = constant
8. Equipotential surfaces (magenta)
and field lines (blue) for a positive
charge.
Thus the surfaces are concentric spheres with the origin (the location of the charge)
as the centre and radii given by
R =
q
4πs0φ0
The equipotential surfaces of an electric dipole is shown below.
Electric Field and Equipotential lines
for an electric dipole
8
9. Example 4 :
Determine the equipotential surface of an infinite line charge carrying a positive
charge density λ.
Solution :
Let the line charge be along the z- axis. The potential due to a line charge at a
point P is given by
λ
2πs0
V (r) = − lnr
where r is the distance of the point P from the line charge. Since the line charge
along the z-axis, r =
,
x 2 + y2 so that
V(r) = −
λ
4πs0
ln(x2
+ y2
)
The surface V = constant = V0 is givenby
ln(x2
+ y2
) = −
4πs0V0
λ
i.e.
x2
+ y2
= e−
4πs0V0
λ
which represent cylinders with axis along the z-axis with radii
r = e−
2πs0V0
λ
1
9
++++++++++++++++++++++++++++++
z−axis
2
As V0 increases, radius becomes
smaller. Thus the cylinders are
packed closer around the axis, show-
ing that the field is stronger near the
axis.
10. Example 5 :
Consider a charged sphere of radius R containing charge q, completely enclosed
by a spherical cavity of inner radius a and outer radius b.Calculate the charge
density on all surfaces and potential everywhere.
Solution :
R
ab
+q
−q
+q
As field inside the conductor is zero,
by taking a Gaussian surface com-
pletely in the region a < r < b, we
must have net charge enclosed by
such a surface to be zero. To com-
pensate for the charge q that exists
on the surface of the inner sphere,
the charge on the inside surface of
the shell must be −q. Since the shell
is charge neutral, a charge +q must,
therefore, appear on the outside sur- Gaussian surface
face of the shell.
For r > b, the field is
E˙ =
4πs0 r2
1 q
rˆ
The corresponding potential is
V(r) = −
¸ r
∞ 4πs0 r2
1 q
dr =
1 q
4πs0 r
At r = b,the potential is
V(b) =
1 q
4πs0 b
Since the field between a and b is zero, this is also the potential at all points from
r = bto r = a.
1 q
4πs0 b
V(a ≤ r ≤ b)=
For R < r < a, the potential is given by
V(r) =
1 q
4πs0 b
−
¸ R
a
1 q
4πs0 r2
=
1 . q q
+ −
4πs0 b R a
10
q.
11. If the outer surface is grounded, the potential on the shell becomes zero. There is
no charge on the outer surface. However the inner surface must have a charge −q
to keep the field in the shell zero,
R
V (0) = V (R) = −
¸
E˙·d˙l =
a
1
4πs0 R a
.
1 1
.
−
Exercise :
Determine the equipotential surface of an infinite plane with charge density σ.
Laplace’s Equation
Let us look at Laplace’s equation in one dimension. It becomes
d2V
dx2
= 0
which has the solution
V = mx + c
The solution shows two important characteristics of the solution of Laplace’s
equation, which are not immediately obvious in higher dimensions. The first
property is the potential at a point can be expressed asaverage of potentials at
neighbouring points. For instance,
1
V (x) = (V (x + x0) + V(x − x0))
2
This also illustrates the second property of the solutions, viz., the solution has no
local minimum or maximum. If it did, it would not be possible to express the
function as average of values at neigbouring points.
To see this consider a function
a
f (x, y) = (x2
+ y2
)
4
in two dimensions, which does not satisfy Laplace’s equation as
∂x2
2 2
∇2
V =
∂ f
+
∂ f
∂y2 = a
11
The function has a positive curvature everywhere and there exists a local minimum
at x = 0,y = 0.The function looks like thefollowing.
12. 0
2
4
6
8
10
12
14
16
18
-3 -2 -1 0 1 2
-2
3-3
-1
0 1
01..015523..235545..455567..675589..8955110111..0155112311..2355114511..4555116711..675518
x
2 3
y
V(x,y)
Consider, on the other hand, a function V(x, y) that satisfies Laplace’s equation
a
V (x, y) = (x2
− y2
)
12
4
The function has no minimum or maximum and looks like the following. It has a
saddle point at x = 0,y = 0.
13. -10
-8
-6
-4
-2
0
2
4
6
8
10
-3 -2 -1 0 1 2
-2
3-3
-1
0 1
--87--..9855--65--..7655--43--..5455--21--..3255-00-.015512..125534..345556..565578..78559
x
13
2 3
y
V(x,y)
An interesting consequence of Laplace equation is Earnshaw Theorem which
states that a charge cannot be held in stable equilibrium only by electrostatic
forces.
For instance, suppose we position a charge Q exactly at the centre of a cube which
has a positive charge q at each of its eight corners. We would expect the charge to
be in equilibrium as it is being pulled equally in all directions. However, this will
not be a stable equilibrium because at the centre, there being no charge density,
Laplace equation is obeyed. Thus there cannot be a minimum of the potential V
and hence of potential energy QV of the charge at the centre.
Consider again the case of cavity in a conductor. If the interior of the cavity does
not contain any charge, Laplace equation is obeyed. Thus the potential has no
minimum or maximum inside the cavity. Further, since the boundary of the cavity
is an equipotential, the potential inside the cavity is also constant.
Uniqueness Theorem :
This theorem states that the solution of Laplace’s equation is uniquely deter-
mined by the values of potential on the boundaries.
Suppose V1 and V2 are two potentials which satisfy Laplace’s equation in some
region with identical coundary conditions, i,e V1(boundary) = V2(boundary).
Consider a function V3 = V1 −V2. This satisfies Laplaces equation with thecondi-
14. tion V3(boundary) = 0. However, as V3 does not have a minimum or a maximum
in the region, its value has to be the same value as its value at the boundary, i.e.
V3 is constant. Hence V1 = V2.
Laplace’s Equations in 3-dimensions
We will consider the solutions of Laplace’s equations in problems with spherical
geometry having azimuthal symmetry. The equation to be solved is
∇2
V =
1
r2 ∂r
∂
.
r2
+
∂V
.
1
∂r r2 sin θ∂θ
∂
. 2∂V
.
1 ∂ V
sin θ + = 0
∂θ r2 sin θ∂φ2
where we have explicitly written down the Laplacian operator in spherical polar
coordinates.
For problems with azimuthal symmetry, ∂V/∂φ = 0so that we have
∂
.
r 2 ∂V
∂r ∂r
.
+
.
1 ∂
sinθ ∂θ ∂θ
.∂V
sin θ = 0
The equation above is conveniently solved by a technique called separation of
variables where we write the function V (r, θ) as a product of two functions, one
R(r) which is a function of radial variable r only and the other a function Θ(θ)
which is a function of the angle variable θ alone. Writing V (r, θ) = R(r)Θ(θ)
and dividing throghout by RΘ, weget
r21 ∂
.
∂R
.
1
R ∂r ∂r Θsin θ ∂θ
∂
.
+ sinθ
∂Θ
.
∂θ
= 0
Since the two terms on the left depend on two independent variables, this equation
can be satisfied only if each of the term equals to constants of opposite sign. We
write
r21 ∂
.
∂R
.
= l(l + 1)
1
Θsin θ ∂θ
R ∂r
∂
.
sinθ
∂r
∂Θ
.
∂θ
14
= −l(l + 1)
We will not attempt to solve these equations but merely quote the results. The
solution of the angular equation is in terms of what are known as Legendre Poly-
nomials. Pl(cos θ).These are polynomials of degree lin cosine of angle θ. The
first few polynomials are as follows :
15. P0(cos θ) = 1
P1(cos θ) = cos θ
1
P2(cos θ) =
P3(cos θ) =
2
1
2
(3 cos2
θ − 1)
(5 cos3
θ −3cos θ)
-0.5
0
0.5
1
-1
-1 -0.5
Pl(cos)
0 0.5 1
cos
P0
P1
P2
P3
P4
V(r, θ)=
The solution of radial equation is consists of a power series in r and 1/r. The
complete solution is
∞
.
i=0
.
l
l
A r +
Bl
rl+1
.
lP (cosθ) (A)
We will illustrate the use of these solution by an example.
Example 6 :
Consider an uncharged conducting sphere in a uniform electric field and deter-
mine the potential at all points in space.
Solution :
The sphere, being a conductor, is an
equipotential. Let the potential be
zero. Far from the sphere, the field
is uniform. Let the field strength be
E0 and be in z-direction,
The boundary conditions are :
V = 0at r = R
V = −E0z = −E0 cos θ for r R. −
−
− − − −−
−
+
+ + +
+
+
Using Eqn. (A) and substituting the first boundary condition, we get a relationship
between Al and Bl
Al Rl
+
Bl
Rl+1
15
= 0
16. Thus Bl = −AlR2l+1. Thus, wehave
∞ .
V (r, θ) =
.
Al rl
+
i=0
rl+1
R2l+1.
Pl(cosθ)
For r R, we may neglect the second term in bracket and get
∞.
Alrl
Pl(cos θ) = −E0r cosθ
i=0
On comparing both sides, we get l = 1 which gives A1 = −E0. Substituting these
we get
R3
r2
V (r, θ) = −E0(r − )cos θ
The induced charge density is
∂V
σ = −s0 |r +R = 3s0E0 cosθ
16
∂r
It can be seen that the charge density is positive in the upper hemisphere and
negative in the lower hemisphere.
Dielectrics
A conductor is characterized by existence of free electrons. These are electrons
in the outermost shells of atoms (the valence electrons) which get detatched from
the parent atoms during the formation of metallic bonds and move freely in the
entire medium in such way that the conductor becomes an equipotential volume.
In contrast, in dielectrics (insulators), the outer electrons remain bound to the
atoms or molecules to which they belong. Both conductors and dielectric, on the
whole, are charge neutral. However, in case of dielectrics, the charge neutrality is
satisfied over much smaller regions (e.g. at molecular level).
2.9.1 Polar and non-polar molecules :
A dielectric consists of molecules which remain locally charge neutral. The
molecules may be polar or non-polar. In non-polar molecules, the charge cen-
tres of positive and negative charges coincide so that the net dipole moment of
each molecule is zero. Carbon dioxide molecule is an example of a non-polar
molecule.
17. +6e+8e
Oxygen atom
+8e
Oxygen atomCarbon atom
+8e
17
+e
+e
Hydrogen atom
Hydrogen atom
In a polar molecules, the arrange-
ment of atoms is such that the
molecule has a permanent dipole
moment because of charge separa-
tion. Water molecule is an example
of a polar molecule.
Oxygen atom
When a non-polar molecule is put in an electric field, the electric forces cause a
small separation of the charges. The molecule thereby acquires an induced dipole
moment.
A polar molecule, which has a dipole moment in the absence of the electric field,
gets its dipole moment aligned in the direction of the field. In addition, the mag-
nitude of the dipole moment may also increase because of increased separation of
the charges.
18. A non−polar molecule
in an Electric Field
E=0
E
+
− +
E=0
E
A polar molecule in an
Electric Field
Dielectric in an Electric Field
A dielectric consists of molecules which may (polar) or may not (non-polar) have
permanent dipole moment. Even in the former case, the dipoles in a dielectric
are randomly oriented because dipole energies are at best comparable to thermal
energy.
+
+
+
+
+
+
+ +
+
+
18
+
+
+
+
+ +
+
+
+ + +
++
+ +
+
+
Randomly oriented dipole in a dielectric (E=0)
Polarised Dipoles in an electric field
When a dielectric is placed in an electric field the dipoles get partially aligned in
the direction of the field. The charge separation is opposed by a restoring force
19. due to attaraction between the charges until the forces are balanced. Since the
dipoles are partially aligned, there is a net dipole moment of the dielectric which
opposes the electric field. However, unlike in the case of the conductors, the net
field is not zero. The opposing dipolar field reduces the electric field inside the
dielectric.
Dielectric Polarization
Electric polarization is defined as the dipole moment per unit volume in a di-
electric medium. Since the distribution of dipole moment in the medium is not
uniform, the polarization P˙ is a function of position. If p˙(r˙) is the sum of the
dipole moment vectors in a volume element dτ located at the position ˙r,
p˙(r˙)= P˙(r˙)dτ
It can be checked that the dimension of P˙is same as that of electric field divided
by permittivity s0. Thus the source of polarization field is also electric charge,
except that the charges involved in producing polarization are bound charges.
Denoting the local bound charge density by ρb, one can write
∇˙·P˙ = −ρb
The equation above is obtained in a manner that is identical to the way we de-
rived the equation ∇˙ ·E˙ = ρ/s0. The absence of the factor so in the equation is
because of the dimensional difference between E˙ and P˙ while the minus sign
arises because the dipole moment vector (and hence the polarization) is defined to
be directed from negative to positive charge as against E˙ which is directed from
positive to negative charge. Clearly, if polarization is uniform, the volume den-
sity of bound charges is equal to zero. Even in this case, there are surface bound
charges given by the normal component of the polarization vector. Summarizing,
we have,
∇ ·P˙ = −ρb P˙ ·nˆ= σb
We will derive these relations shortly.
Free and Bound Charges
The charge density of a medium consists of free charges, which represent a surplus
or deficit of electrons in the medium, and bound charges. The term free charge is
19
20. used to denote any charge other than that due to polarization effect. For instance,
the valence charges in a metal or charges of ions embedded in a dielectric are
considered as free charges.
The total charge density of a medium is a sum of free and bound charges
ρ = ρf + ρb
Gauss’s Law takes the form
∇˙·E˙ =
ρ
=
ρf + ρb
s0 s0
Potential due to a dielectric
Consider the dielectric to be built up of volume elements dτ . The dipole moment
of the volume element is P˙dτ .
The potential at a point S, whose po-
sition vector with respect to the vol-
ume element is ˙ris
dV =
4πso
1 P˙ ·rˆ
r2
dτ
P d
r
S
The potential due to the whole volume is
V =
1
4πs0
¸
volume
P˙·rˆ
r2
dτ =
1
4πs0
¸
volume
1
r
P˙ ·∇( )dτ
where, we have used
1
r
∇( ) =
rˆ
r2
19
Use the vector identity
∇˙·(A˙f(r)) = A˙·∇f (r) + f (r)∇˙ ·A˙
Substituting A˙ = P˙ and f (r) = 1/r,
21. ∇˙·(
P˙
r
) = P˙ ·∇( ) +
1 1
r r
∇˙·P˙
we get
V =
1
4πso
¸
vol
P˙
r
∇˙ ·( )dτ −
1
4πso
¸
1
vol r
∇˙·P˙dτ
The first integral can be converted to a surface integral using the divergence theo-
rem giving,
V =
1
4πso
¸
P˙
surface r
·dS˙−
1
4πs0
¸
1
vol r
∇˙·P˙dτ
The first term is the potential that one would expect for a surface charge density
σb where
σb = P˙ ·nˆ
where nˆis the unit vector along outward normal to the surface. The second term
is the potential due to a volume charge density ρb given by
ρb = −∇˙ ·P˙
The potential due to the dielectric is, therefore, given by
V =
1
4πso
¸
surf ace
σbdS
r
+
1
4πs0
¸
vol
ρbdτ
r
and the electric field
E˙ = −∇V
1
=
4πs0
¸
dS +
1
4πs0
¸
ρbrˆσbrˆ
surface r2
vol r2
29
dτ
Electric Displacement Vector D˙
The electri displacement vector D˙ is defined by
D˙ = s0E˙ + P˙
which has the same dimension as that of P˙. The equation satisfied by D˙ is
22. ∇˙ ·D˙ = s0∇˙ ·E˙ + ∇˙ ·P˙ = ρ − ρb = ρf
which is the differential form of Gauss’s law for a dielectric medium.
Integrating over the dielectric volume,
¸
∇˙·D˙dτ =
¸
volume volume
ρfdτ = Qf
where Qf is the free charge enclosed in the volume. The volume integral can be
converted to a surface integral using the divergence theorem, which gives
¸
surface
D˙ ·dS˙ = Qf
Thus the flux over the vector D˙over a closed surface is equal to the free charged
enclosed by the surface.
Example 5:
An uncharged spherical dielectric has polarization vector given by P˙= k˙r. Find
the electric field both outside and inside the dielectric.
Solution :
The dielectric has both bound surface charge and volume charge. The surface
charge density is σb = P˙·nˆ= kR where R is the radius of the sphere. The
volume charge density is
ρb = −∇ ·P˙ = −k∇ ·˙r= −3k
. The field inside the dielectric is given by Gauss’s law,
4πr2
E =
Q
s0
encl 34πr ρb
=
3s0
which gives
rρb
3s0
E = = −
kr
s0
22
The field outside is zero.
Example 6 :
Consider a spherical dielectric shell of inner radius a and outer radius b.The
space in the region between r = a and r = b is filled with a dielectric hasving
r
polarization P˙ = k rˆ. Determine the field inside and outside the shell.
23. Solution :
The charge densities are,
σouter−surface
b
k= P˙ ·nˆ=
b
σinner−surface
b
k= −P˙ ·nˆ= −
a
ρb = −∇ ·P˙ = −
k
r2
For r < a, no charge is included, hence the field is zero. For a < r < b, the
charges enclosed by a Gaussian surface are the surface bound charges on the inner
surface and the volume charge within the region. Thus
Qencl
r
a
b= 4πb2
σin
+
¸
ρ 4πr2
dr
a
r
a
= −4πa2 k
+
¸
(−k/r2
)4πr2
dr
= −4πka + 4πk(a − r) = −4πkr
Thus E = −(k/s0r)rˆ. For a Gaussian surface outside, the total charge enclosed
can be similarly calculated to be zero, so that field is zero.
Example 7 :Electric Field Due to Uniformly polarized sphere :
Since the polarization is uniform,
the bound charge density is zero.
Only on the surface, there are bound
charges. We have
σb = P˙ ·nˆ= P cosθ
where θ is the angle between the
direction of the external field (z-
direction) and a point on the sphere.
P
23
n^
z^
This is, once again, a problem with azimuthal symmetry with no charges inside
or outside the sphere. Hence Laplace’s equation is satisfied both in the interior of
24. the sphere and outside.
V(r, θ)=
∞
.
l=0
.
l
l
A r +
Bl
rl+1
.
lP (cosθ)
For r < R, the second term must vanish since the potential cannot become infinity
at the origin. Similarly, for r > R, the first tem must vanish as the potential must
be well defined at large distances.
For r < R,
V(r, θ) = Alrl
Pl(cos θ)
and, for r > R
V(r, θ)=
Bl
rl+1
Pl(cosθ)
At r = R, the potential is continuous. Hence,
Bl = AlR2l+1
At r = R, while the tangential component of the field is continuous, the normal
component has a discontinuity,
E above
n
below σ
0
− En =
s
nˆ
Using E˙ = −∇V,
∂Vabove
∂r
−
∂Vbelow
∂r
= −
σ
s0
Thus,
−
∞
.
l=0
.
B l
rl+2 − Alr l− 1
.
Pl(cos θ) |r=R= −
P cosθ
s0
Comparing both sides, we see that only l = 1 term is non-zero. We get,
2B1 P
R3
+ A1 =
s0
Using B1 = A1R3, we get A1 = P/3s0 and B1 = PR3/3s0 Finally, we get
V(r, θ) = r cos θ for r < R
=
P
3s0
P R3
3s0 r3
24
cos θ for r > R
25. The electric field inside the sphere is uniform and is equal to −∇V= −P/3s0zˆ.
Outside the sphere, the potential has the same form as that of a giant dipole with
dipole moment equal to volume of the sphere times the polarization vector, located
at the centre, because,
V =
P R3
3s0 r2
cosθ
=
3s0
3p/4πR3 R3
r2
cosθ
=
4πs0
1 pcos θ
r2
=
4πs0
1 p˙·rˆ
r2
Constitutive Relation
Electric displacement vector D˙ helps us to calculate fields in the presence of a
dielectric. This is possible only if a relationship between E˙ and D˙ is known.
For a weak to moderate field strength, the electric polarization P˙ is found to be
directly proportional to the external electric field E˙. We define Electric Suscepti-
bility χ through
P˙ = s0χE˙
so that
D˙ = s0E˙ + P˙
= so(1 + χ)E˙ = s0sr E˙ = sE˙
where κ ≡ sr = 1 + χ is called the relative permittivity or the dielectric constant
and s is the permittivity of the medium. Using differential form of Gauss’s law for
D˙, we get
∇˙·E˙ =
1
∇˙ ·D˙ =
ρf
s s
25
Thus the electric field produced in the medium has the same form as that in free
space, except that the field strength is reduced by a factor equal to the dielectric
constant κ.
Capacitance filled with Dielectric
If a material of dielectric constant κ is inserted between the plates of a capacitor,
26. the field E˙is reduced by a factor κ. The potential between the plates also reduces
by the same factor κ.
φ −→
φ
κ
Thus the capacitance
C =
Q
φ
+Q
−Q
increases by a factor κ.
Example:
A parallel plate capacitor with plate separation 3.54mm and area 2m2 is initially
charged to a potential difference of 1000 volts. The charging batteries are then
disconnected. A dielectric sheet with the same thickness as that of the separation
between the plates and having a dielectric constant of 2 is then inserted between
the capacitor plates. Determine (a) the capacitance , (b) potential difference across
the capacitor plates, (c) surface charge density (d) the electric field and (e) dis-
placement vector , before and after the insertion of the dielectric .
Solution :
(a) The capacitance before insertion of the dielectric is
A
d
C = s0 = 8.85×10−12 2
3.54 ×10−3
= 5 ×10−9
F
After the insertion the capacitance doubles and becomes 10−8 F.
(b)Potential difference between the plates before insertion is given to be 1000 V.
On introducing the dielectric it becomes half, i.e. 500 V.
(c)The charge on each capacitor plate was Q = CV = 5 × 10−6 coulomb, giving
a surface charge density of 2.5 × 10−6 C/m2. The free charge density remains the
same on introduction of the dielectric.
(d) The electric field strength E is given by
σ
E = = 2.8× 105
volts/meter
26
s0
The electric field strength is reduced to 1.4 × 105 volt/meter oninsertion.
(e) The displacement vector remains the same in both cases as the free charge
density is not altered. It is given by D = σ = 2.5 × 10−6 C/m2.
Example :
The parallel plates of a capacitor of plate dimensions a × b and separation d are
27. Forcex
x−axis
charged to a potential difference φ and battery is disconnected. A dielectric slab
of relative permittivity κ is inserted between the plates of the capacitor such that
the left hand edge of the slab is at a distance x from the left most edge of the
capacitor. Calculate (a) the capacitance and (b) the force on the dielectric.
y−axis
a
b
d
z−axis
Solution :
Since the battery is disconnected, the potential difference between the plates will
change while the charge remains the same. Since the capacitance of the part of
the capacitor occupied by the dielectric is increased by a factor κ, the effective
capacitance is due to two capacitances in parallel ,
b
C = s0 [x + (a −x)κ]
d
The energy stored in the capacitor is
U = =
Q2 Q2
d 1
2C 2 bs0 x + (a −x)κ
Let F be the force we need to apply in the x-direction to keep the dielectric in
place. For an infinitisimal increment dx of x, we have to do an amount of work
Fdx , which will increase the energy strored in the field by dU , so that
F =
dU
dx
27
28. the differentiation is to be done, keeping the charge Q constant. Thus
F =
dQ2 κ− 1
2bs0 [x + (a − x)κ]2
Since κ > 1, F is positive. This means the electric field pulls the dielectric inward
so that an external agency has to apply an outward force to keep the dielectric in
position. Since the initial potential difference φ is given by Q/C, one can express
the force in terms of this potential
F =
s0b
2d
φ2
(κ −1)
This is the force that the external agency has to apply to keep the left edge of the
dielectric at x. The force with which the capacitor pulls the dielectric in has the
same magnitude.
Example 22 :
In the above example, what would be the force if the battery remained connected
?
Solution :
If the battery remained connected Q does not remain the same, the potential φ
does. The battery must do work to keep the potential constant. It may be realised
that the force exerted on the dielectric in a particular position depends on the
charge distribution (of both free and bound charges) existing in that position and
the force is independent of whether the battery stays connected or is disconnected.
However, in order to calculate the force with battery remaining connected, one
must, explicitly take into account the work done by the battery in computing the
total energy of the system. The total energy U now has two parts, one the work
done by the external agency Fdx and the other the work done by the battery, viz.,
φdQ where dQ is the extra charge supplied by the battery to keep the potential
constant. Thus
U= Fdx + φdQ
which gives
F =
dU
dx
−φ
dQ
dx
Since φ is constant, we have
1
2
28
cφ2
U =
Q = Cφ
29. Using these
F = φ21 dC dC
− φ2
= − φ21 dC
2 dx dx 2 dx
(Note that if the work done by the battery were negnected, the direction of F will
be wrong, though, because we have used linear dielectrics, the magnitude,
accidentally, turns out to be correct !)
In the previous example, we have seen that
bs0
d
C = [x + (a − x)κ]
giving
dC bs0
dx d
= (1 −κ)
which is negative. Thus F is positive, as before,
F =
bs0φ2
d
(κ −1)
Example :
The space between the plates of a parallel plate capacitor is filled with two differ-
ent dielectrics, as shown. Find the effective capacitance.
1
2
d1
d2
29
Solution :
30. Take a Gaussian pill-box as shown.
Wehave
¸
D˙ ·d˙S = ρfree
= 0
as there are no free charges inside
the dielectric. Contribution to the in-
tegral comes only from the faces of
the pill-box parallel to the plates and
d˙S1 = d˙S2.Hence,
D1 = D2 = σ
d1
d2
1
2
dS
dS2
1
dz
Let φ1
where σ is the surface density of free
charges.
be the potential difference between the upper plate and the interface between the
dielectric and φ2 that between the interface and the lower plate. We have
φ = φ1 + φ2
= E1d1 + E2d2
=
D1
κ1s0
d1 +
D2
κ2s0
d2
= +
σd1 σd2
κ1s0 κ2s0
Thus the effective capacitance is given by
C =
Q
φ
=
σA
σ
.
d1 d2
.
=
s0 κ1
+ κ2
As0
+d1 d2
κ κ
=
1 2
C1C2
C1 + C2
30
where C1 and C2 are the capacitances for parallel plate capacitors with one type
of dielectric with separations d1 and d2 between the plates respectively.
31. Example :
A capacitor consists of an inner conducting sphere of radius R and an outer con-
ducting shell of radius 2R. The space between the spheres is filled with two dif-
ferent linear dielectrics, one with a dielectric constant κ from r = R to r = 1.5R
and the other with dielectric constant 2κ from r = 1.5R to r = 2R. The outer
shell has a charge −Q while the inner conductor has a charge +Q. Determine the
electric field for r > 0and find the effective capacitance.
Solution :
The electric field is radially symmet-
ric and may be obtained by apply-
ing Gauss’s law for the displacement
vector
¸
D˙ ·d˙S = 4πr2
D = Qfree
where Qfree is the free charge en-
closed within a sphere of radius r.
For r < R, the field is zero as the
free charges are only on the surface
of the inner cylinder.
+
+
+
+ +
+
+ +
+ −
−
−
−
−
−
−
−
−
−
−
R
1.5R
2R
For R <
r < 1.5R, the electric fieldis
E = =
D Q
κs0 4πs0κr2
and for 1.5R < r < 2R,
E = =
D Q
2κs0 8πs0κr2
30
For r > 2R, the field is zero. The fields are radial with the ineer sphere at a
higher potential. The potential difference is calculated by taking the taking the
32. line integral of the electric field along any radial line.
¸
R
2R
∆φ = E˙ ·d˙l =
¸
E dr
=
¸ 1.5R
R
Q
4πκs0r2
dr +
¸ 2R
Q
1.5R 8πκs0r2
dr
=
5Q
48πκs0
The effective capacitance is
C = =
Q 48πκs0
∆φ 5
D = s0κE = s0
.
1 +
x
d
.
E
As the insertion of dielectric does
not affect free charges, the displace-
ment vector D˙ is remains the same
as it would in the absence of the di-
electric. Thus D˙ = ˆıσ.
Example :
Aparallel plate capacitor has charge densities ±σ on its plates which are separated
by a disance d. The space between the capacitor plates is filled with a linear but
inhomogeneous dielectric. The dielectric constant varies with distance from the
positive plate linearly from a value 1 to a value 2 at the negative plate. Determine
the effective capacitance.
2
As the dielectric is linear,
1
d
x
0
distance from positive plate
dielectricconstant
++++++
____
Thusthe
electric field E˙ is given by
E˙ = ˆı
σa
s0(x + d)
31
The field close to x = d is given by E = σ/2s0, which shows that adjacent to
the negative plate there is a positive charge density σ/2. To find the effective
33. capacitance, we find the potential difference between the plates by integrating the
electric field
0 s0
φ =
¸
Edx =
σd
¸d d
0
dx
d + x s0
σd
= ln2
so that
C = =
Q Aσ
φ φ
=
As0
dln 2
The polarization P is given by
P = D − s0E =
σx
x + d
The volume density of bound charges, given by ∇ ·P˙ = −ρb is found as follows:
ρb = −σ
dx
d
.
x
.
σd= −
x + d (x + d)2
The bound charge density on the surface, given by nˆ·P˙= P , has a value σ/2 on
the dielectric adjacent to the negative plate (x = d). As the dielectric is charge
neutral, this requires a net volume charge of −σ/2 in the dielectric. This can be
verified by integrating over the volume charge density ρb given above.
Exercise :
A parallel plate capacitor of plate area S and separation d, contains a dielectric of
thickness d/2 and of dielectric constant 2, resting on the negatve plate.
d/2
d−
33
+
A potential difference of φ is maintained between the plates. Calculate the electric
field in the region between the plates and the density of bound charges on the
surface of the dielectric. [Ans. field in empty region = 4φ/3d, within dielectric
= 2φ/3d, bound charge density = 2s0φ/3d]
34. Exercise :
The permittivity of a medium filling the space between the plates of a spherical
capacitor with raddi a and b (b > a) is given by
s =
.
2s0a ≤ r ≤ (a+ b)/2
4s0 (a + b)/2 ≤ r ≤ b
Find the capacitance of the capacitor, distribution of surface bound charges and
.
1
the total bound charges in the dielectric. [Ans. C = 8πs0 −
1
a a + b 2b
34
1
. − 1
− ,
bound charges on dielectric surface with radii a, (a + b)/2 and b are respectively
−σ/2, 3σa2/(a + b)2 and 3σa2/4b2]