This document summarizes key concepts about electric potential and potential energy from Chapter 3:
1) Electric potential is defined as the work required per unit charge to move a test charge between two points in an electric field, similar to how gravitational potential is defined. 2) In a uniform electric field, the electric potential decreases as one moves in the direction of the field lines, corresponding to a decrease in potential energy for a positive charge. 3) The change in electric potential between two points depends only on the endpoints and not the path taken, as electric fields are conservative.
This document describes electric potential and its relationship to electric field and potential energy. It begins by introducing electric potential and defining it as the work required per unit charge to move a test charge between two points against an electric field. The electric potential due to point charges and continuous charge distributions is then derived. Methods for calculating electric potential and field from each other are presented, along with examples such as charged rods, rings, and disks. The chapter concludes with problem-solving strategies and additional practice problems.
Movimiento De Una Partícula Cargada En Un Campo Eléctrico Y En Un Campo Magné...Gregory Zuñiga
The document discusses the mathematical modeling of the motion of a charged particle in electric and magnetic fields. It describes three cases: 1) motion in a uniform electric field, where the particle experiences a constant acceleration; 2) motion in a uniform magnetic field, where the particle travels in a circular path with radius determined by the magnetic field strength; and 3) motion in both uniform electric and magnetic fields perpendicular to each other, where the particle follows a helical path with the forces from the two fields balancing each other out under certain conditions. The modeling of the particle motion incorporates equations for forces, fields, energy, and kinematics.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Paramagnetism has been explained using the classical approach. Derivation of Magnetization and Susceptibility in case of paramagnetism using Langevin Theory of Paramagnetism.
This document section provides an introduction to magnetic fields, including:
- Defining a magnetic field B as the force per unit charge on a moving charged particle.
- Explaining that magnetic force BF is perpendicular to both the particle's velocity v and the magnetic field B.
- Deriving the magnetic force on a current-carrying wire segment and using this to calculate the total magnetic force on wires of different shapes, including closed loops.
- Working through an example problem to calculate the magnetic forces on different parts of a semi-circular current loop placed in a uniform magnetic field.
Analysis of Stress and Deflection of Cantilever Beam and its Validation Using...IJERA Editor
This study investigates the deflection and stress distribution in a long, slender cantilever beam of uniform
rectangular cross section made of linear elastic material properties that are homogeneous and isotropic. The
deflection of a cantilever beam is essentially a three dimensional problem. An elastic stretching in one direction
is accompanied by a compression in perpendicular directions. The beam is modeled under the action of three
different loading conditions: vertical concentrated
load applied at the free end, uniformly distributed load and uniformly varying load which runs over the whole
span. The weight of the beam is assumed to be negligible. It is also assumed that the beam is inextensible and so
the strains are also negligible. Considering this assumptions at first using the Bernoulli-Euler’s bendingmoment
curvature relationship, the approximate solutions of the cantilever beam was obtained from the general
set of equations. Then assuming a particular set of dimensions, the deflection and stress values of the beam are
calculated analytically. Finite element analysis of the beam was done considering various types of elements
under different loading conditions in ANSYS 14.5. The various numerical results were generated at different
nodal points by taking the origin of the Cartesian coordinate system at the fixed end of the beam. The nodal
solutions were analyzed and compared. On comparing the computational and analytical solutions it was found
that for stresses the 8 node brick element gives the most consistent results and the variation with the analytical
results is minimum.
1. Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
2. The electric field intensity is defined as the force per unit charge. The electric field intensity due to a single point charge is calculated using Coulomb's law. For multiple point charges, the total electric field is calculated using the principle of superposition.
3. Continuous charge distributions can also produce electric fields. These include line charge distributions with linear charge density, surface charge distributions with surface charge density, and volume charge distributions with volume charge density. The electric field due to a uniform line charge distribution can be calculated by treating
This document describes electric potential and its relationship to electric field and potential energy. It begins by introducing electric potential and defining it as the work required per unit charge to move a test charge between two points against an electric field. The electric potential due to point charges and continuous charge distributions is then derived. Methods for calculating electric potential and field from each other are presented, along with examples such as charged rods, rings, and disks. The chapter concludes with problem-solving strategies and additional practice problems.
Movimiento De Una Partícula Cargada En Un Campo Eléctrico Y En Un Campo Magné...Gregory Zuñiga
The document discusses the mathematical modeling of the motion of a charged particle in electric and magnetic fields. It describes three cases: 1) motion in a uniform electric field, where the particle experiences a constant acceleration; 2) motion in a uniform magnetic field, where the particle travels in a circular path with radius determined by the magnetic field strength; and 3) motion in both uniform electric and magnetic fields perpendicular to each other, where the particle follows a helical path with the forces from the two fields balancing each other out under certain conditions. The modeling of the particle motion incorporates equations for forces, fields, energy, and kinematics.
Ekeeda Provides Online Civil Engineering Degree Subjects Courses, Video Lectures for All Engineering Universities. Video Tutorials Covers Subjects of Mechanical Engineering Degree.
Paramagnetism has been explained using the classical approach. Derivation of Magnetization and Susceptibility in case of paramagnetism using Langevin Theory of Paramagnetism.
This document section provides an introduction to magnetic fields, including:
- Defining a magnetic field B as the force per unit charge on a moving charged particle.
- Explaining that magnetic force BF is perpendicular to both the particle's velocity v and the magnetic field B.
- Deriving the magnetic force on a current-carrying wire segment and using this to calculate the total magnetic force on wires of different shapes, including closed loops.
- Working through an example problem to calculate the magnetic forces on different parts of a semi-circular current loop placed in a uniform magnetic field.
Analysis of Stress and Deflection of Cantilever Beam and its Validation Using...IJERA Editor
This study investigates the deflection and stress distribution in a long, slender cantilever beam of uniform
rectangular cross section made of linear elastic material properties that are homogeneous and isotropic. The
deflection of a cantilever beam is essentially a three dimensional problem. An elastic stretching in one direction
is accompanied by a compression in perpendicular directions. The beam is modeled under the action of three
different loading conditions: vertical concentrated
load applied at the free end, uniformly distributed load and uniformly varying load which runs over the whole
span. The weight of the beam is assumed to be negligible. It is also assumed that the beam is inextensible and so
the strains are also negligible. Considering this assumptions at first using the Bernoulli-Euler’s bendingmoment
curvature relationship, the approximate solutions of the cantilever beam was obtained from the general
set of equations. Then assuming a particular set of dimensions, the deflection and stress values of the beam are
calculated analytically. Finite element analysis of the beam was done considering various types of elements
under different loading conditions in ANSYS 14.5. The various numerical results were generated at different
nodal points by taking the origin of the Cartesian coordinate system at the fixed end of the beam. The nodal
solutions were analyzed and compared. On comparing the computational and analytical solutions it was found
that for stresses the 8 node brick element gives the most consistent results and the variation with the analytical
results is minimum.
1. Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
2. The electric field intensity is defined as the force per unit charge. The electric field intensity due to a single point charge is calculated using Coulomb's law. For multiple point charges, the total electric field is calculated using the principle of superposition.
3. Continuous charge distributions can also produce electric fields. These include line charge distributions with linear charge density, surface charge distributions with surface charge density, and volume charge distributions with volume charge density. The electric field due to a uniform line charge distribution can be calculated by treating
11. kinetics of particles work energy methodEkeeda
The document provides information about work, kinetic energy, work energy principle, and conservation of energy. It defines key terms like work, kinetic energy, spring force, weight force, friction force, power, and efficiency. It explains:
- Work is the product of force and displacement in the direction of force. Work by various forces can be used to solve kinetics problems.
- Kinetic energy is the energy of motion and is defined as one-half mass times velocity squared.
- The work energy principle states that the total work done by forces on an object equals its change in kinetic energy.
- For conservative forces acting on a particle, the mechanical energy (sum of kinetic and potential energy) is
Control of vibration through an innovative isolation technique of a multistor...eSAT Journals
Abstract
The paper presents a simple and an innovative idea regarding vibration isolation technique of a multistory building using magnetic
field and is proposed to application in the field of civil engineering structures. This idea is fully based on hypothetical concept, but the
objective is to explore this idea or concept in to the entire community which may be a potential sector for the effective vibration
isolation system. The technique is described based on the magnetic properties of materials and a magnetic field characteristic which
depends on the amount of current flow, number of turns in coil, distance between electromagnet and the magnetic materials.
Therefore using magnetic field, the attraction induced between magnet and magnetic materials, a force could finally be developed on
the floor of building. This force would become effective until and unless the power is cutoff. Therefore this induced force is actually
acts as a live load on the structure which indirectly increases the weight and this has been described in this paper.
Keywords: - Isolation, vibration, frequency, magnetic field, performance based design, damping
Strain energy is a type of potential energy that is stored in a structural member as a result of elastic deformation. The external work done on such a member when it is deformed from its unstressed state is transformed into (and considered equal to the strain energy stored in it.
Principale of super position and maxwell reciprocal therom KEVINMISTRY8
1. The principle of superposition states that the displacements or deflections caused by individual loads or forces acting on a structure can be added to determine the total displacement caused by all loads acting together.
2. Maxwell's reciprocal theorem states that the deflection at one point on a structure due to a unit load at another point is equal to the deflection at the second point due to a unit load at the first point.
3. Betti's law states that the virtual work done by one system of forces during displacements caused by a second system of forces is equal to the virtual work done by the second system of forces during displacements caused by the first system of forces.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
1) A bar magnet falling through a copper pipe reaches a terminal velocity due to eddy currents induced in the pipe walls from the changing magnetic field.
2) By equating the loss of potential energy to energy dissipated through Joule heating, an expression for terminal velocity as a function of magnet and pipe properties is derived.
3) Only considering the magnetic dipole contribution results in an estimated copper conductivity around 40% lower than published values, showing higher order effects are needed.
The document discusses the principle of minimum potential energy in solid mechanics. It begins by defining the total potential energy of a system as the sum of the internal strain energy and external work done by applied loads. The principle then states that the equilibrium configuration is the one that minimizes this total potential energy. Numerical techniques like the Rayleigh-Ritz method use a "trial function" to approximate the displacement field and find the minimum of the total potential energy to obtain approximate solutions. Several examples are provided to illustrate applying this principle to problems involving axial members, beams, and statically indeterminate structures.
Complimentary Energy Method in structural analysisMahdi Damghani
This document discusses the energy method for structural analysis. It begins with an introduction to strain energy and complementary energy. For linear elastic materials, the strain energy is equal to the complementary energy. The principle of stationary complementary energy is then presented, stating that the true internal forces and reactions are those that make the total complementary energy stationary. Application examples are provided for determining deflections in truss and beam structures using the complementary energy approach. Indeterminate structures are also discussed.
This document discusses Castigliano's theorem for calculating deflections in elastic structures. Castigliano's theorem states that the deflection of a point on a structure due to a load is equal to the partial derivative of the total strain energy with respect to that load. The document provides examples of using Castigliano's theorem to calculate deflections for structures undergoing bending, shear, tension, and compression. It also discusses using a "dummy load" and setting the partial derivative of energy to the load equal to zero to solve for an unknown force.
1. The document discusses the derivation of the permittivity constant (ε0) and permeability constant (μ0) from first principles, which has never been done before.
2. It shows that ε0 and μ0 can be derived from dimensional analysis and have dimensions of meters per second, corresponding to the speed of light.
3. The product of ε0 and μ0 is inversely proportional to the square of the speed of light, allowing the speed of light to be calculated from Maxwell's equations.
This document is a project report submitted by Shubham Patel for the partial fulfillment of an M.Sc. in Physics. The report introduces Galilean electromagnetism and constrained Hamiltonian systems. In part one, the report discusses various Galilean limits of Maxwell's equations including the electric limit, magnetic limit, and Carrollian limit. It also discusses formulations of these limits that are invariant under different systems of units. In part two, the report discusses Maxwell's field theory from a Hamiltonian perspective and constraints that arise in the formulation. It also discusses a higher order field tensor Lagrangian and its Hamiltonian formulation.
This document discusses energy methods in elasticity theory. It defines external work done by forces and internal strain energy density due to stresses in elastic materials. Equations are presented relating external work to changes in internal strain energy for various loads, including normal stresses, shear stresses, bending moments, and torsional loads. Castigliano's theorem and the principle of virtual work are also introduced as methods to determine displacements and slopes in elastic structures.
This document discusses multi-degree-of-freedom (MDOF) systems and their analysis. It introduces concepts such as flexibility and stiffness matrices, natural frequencies and mode shapes, orthogonality of modes, and equations of motion. Methods for analyzing free and forced vibration of MDOF systems in the time domain are presented, including modal superposition and direct integration. An example 3DOF system is analyzed to illustrate the concepts.
The use of Calculus is very important in every aspects of engineering.
The use of Differential equation is very much applied in the concept of Elastic beams.
1. The document presents an overview of fracture mechanics, including an atomic view of fracture, stress concentration effects of flaws, the Griffith energy balance approach, and the energy release rate.
2. It discusses how cracks propagate when sufficient stress is applied at the atomic level to break atomic bonds, and how flaws concentrate stress which can cause fracture at lower overall stresses than theoretical strength estimates.
3. The Griffith energy balance approach is summarized, showing how the energy required to create new crack surfaces must be balanced by the strain energy released from the material as the crack extends.
12. kinetics of particles impulse momentum methodEkeeda
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes.
https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document provides solutions to 15 problems related to magnetic materials and magnetism. It begins by deriving expressions for how the angular frequency and radius of a classical electron orbit change with the application of a magnetic field using the Lorentz force. It then calculates magnetic moments and susceptibilities for various systems. Other problems cover topics like Lenz's law, diamagnetism, paramagnetism, Bohr orbits, susceptibility measurements, and predicting magnetic behaviors in different materials. Detailed step-by-step workings are shown for each problem.
Analysis of Simple Maglev System using SimulinkArslan Guzel
The document is a PowerPoint presentation analyzing a simple maglev system using Simulink. It includes:
- An introduction to maglev systems and their applications.
- Circuit diagrams and explanations of the position sensor, electromagnet actuator, and other system components.
- Derivations of the system's mathematical model and equations in state space form for both nonlinear and linear controller approaches.
- Descriptions of the Simulink models created to simulate the system, including blocks for the generalized system, nonlinear controller, and determining voltage and current.
- Analysis of simulation results for two air gap scenarios, comparing graphs of position, velocity, acceleration, and other signals between the scenarios.
N. Schlager - Study Materials for MIT Course [8.02T] - Electricity and Magnet...cfisicaster
This document provides a summary of topics covered in Class 1 of the physics course 8.02, which included an introduction to TEAL (Technology Enhanced Active Learning), fields, a review of gravity, and the electric field. Key points include:
1) The course focuses on electricity and magnetism, specifically how charges interact through fields. Gravity and electric fields are introduced as the first examples of fields.
2) Scalar and vector fields are defined and examples of representing each type of field visually are given.
3) Gravity is reviewed as an example of a physical vector field, with masses creating gravitational fields and other masses feeling forces due to those fields.
4) Electric charges are described
The document discusses electric field, potential, and energy. It defines electric potential as the work done to move a unit positive charge from infinity to a point in an electric field. Electric potential is a scalar quantity measured in volts. Equipotentials are regions in space where the electric potential has a constant value, forming equipotential surfaces or lines. Analogies are drawn between electric and gravitational fields, such as both following inverse square laws and having field lines and equipotentials.
1) Newton proposed that all objects with mass attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
2) An object in free fall experiences a sensation of weightlessness because both it and the elevator/spacecraft it is in are accelerating downward at the same rate due to gravity, so there is no relative acceleration between them.
3) A satellite in orbit around a planet is weightless because the centripetal acceleration needed to maintain its orbit exactly counteracts the acceleration due to gravity, resulting in no net acceleration felt by objects in the spacecraft.
11. kinetics of particles work energy methodEkeeda
The document provides information about work, kinetic energy, work energy principle, and conservation of energy. It defines key terms like work, kinetic energy, spring force, weight force, friction force, power, and efficiency. It explains:
- Work is the product of force and displacement in the direction of force. Work by various forces can be used to solve kinetics problems.
- Kinetic energy is the energy of motion and is defined as one-half mass times velocity squared.
- The work energy principle states that the total work done by forces on an object equals its change in kinetic energy.
- For conservative forces acting on a particle, the mechanical energy (sum of kinetic and potential energy) is
Control of vibration through an innovative isolation technique of a multistor...eSAT Journals
Abstract
The paper presents a simple and an innovative idea regarding vibration isolation technique of a multistory building using magnetic
field and is proposed to application in the field of civil engineering structures. This idea is fully based on hypothetical concept, but the
objective is to explore this idea or concept in to the entire community which may be a potential sector for the effective vibration
isolation system. The technique is described based on the magnetic properties of materials and a magnetic field characteristic which
depends on the amount of current flow, number of turns in coil, distance between electromagnet and the magnetic materials.
Therefore using magnetic field, the attraction induced between magnet and magnetic materials, a force could finally be developed on
the floor of building. This force would become effective until and unless the power is cutoff. Therefore this induced force is actually
acts as a live load on the structure which indirectly increases the weight and this has been described in this paper.
Keywords: - Isolation, vibration, frequency, magnetic field, performance based design, damping
Strain energy is a type of potential energy that is stored in a structural member as a result of elastic deformation. The external work done on such a member when it is deformed from its unstressed state is transformed into (and considered equal to the strain energy stored in it.
Principale of super position and maxwell reciprocal therom KEVINMISTRY8
1. The principle of superposition states that the displacements or deflections caused by individual loads or forces acting on a structure can be added to determine the total displacement caused by all loads acting together.
2. Maxwell's reciprocal theorem states that the deflection at one point on a structure due to a unit load at another point is equal to the deflection at the second point due to a unit load at the first point.
3. Betti's law states that the virtual work done by one system of forces during displacements caused by a second system of forces is equal to the virtual work done by the second system of forces during displacements caused by the first system of forces.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
1) A bar magnet falling through a copper pipe reaches a terminal velocity due to eddy currents induced in the pipe walls from the changing magnetic field.
2) By equating the loss of potential energy to energy dissipated through Joule heating, an expression for terminal velocity as a function of magnet and pipe properties is derived.
3) Only considering the magnetic dipole contribution results in an estimated copper conductivity around 40% lower than published values, showing higher order effects are needed.
The document discusses the principle of minimum potential energy in solid mechanics. It begins by defining the total potential energy of a system as the sum of the internal strain energy and external work done by applied loads. The principle then states that the equilibrium configuration is the one that minimizes this total potential energy. Numerical techniques like the Rayleigh-Ritz method use a "trial function" to approximate the displacement field and find the minimum of the total potential energy to obtain approximate solutions. Several examples are provided to illustrate applying this principle to problems involving axial members, beams, and statically indeterminate structures.
Complimentary Energy Method in structural analysisMahdi Damghani
This document discusses the energy method for structural analysis. It begins with an introduction to strain energy and complementary energy. For linear elastic materials, the strain energy is equal to the complementary energy. The principle of stationary complementary energy is then presented, stating that the true internal forces and reactions are those that make the total complementary energy stationary. Application examples are provided for determining deflections in truss and beam structures using the complementary energy approach. Indeterminate structures are also discussed.
This document discusses Castigliano's theorem for calculating deflections in elastic structures. Castigliano's theorem states that the deflection of a point on a structure due to a load is equal to the partial derivative of the total strain energy with respect to that load. The document provides examples of using Castigliano's theorem to calculate deflections for structures undergoing bending, shear, tension, and compression. It also discusses using a "dummy load" and setting the partial derivative of energy to the load equal to zero to solve for an unknown force.
1. The document discusses the derivation of the permittivity constant (ε0) and permeability constant (μ0) from first principles, which has never been done before.
2. It shows that ε0 and μ0 can be derived from dimensional analysis and have dimensions of meters per second, corresponding to the speed of light.
3. The product of ε0 and μ0 is inversely proportional to the square of the speed of light, allowing the speed of light to be calculated from Maxwell's equations.
This document is a project report submitted by Shubham Patel for the partial fulfillment of an M.Sc. in Physics. The report introduces Galilean electromagnetism and constrained Hamiltonian systems. In part one, the report discusses various Galilean limits of Maxwell's equations including the electric limit, magnetic limit, and Carrollian limit. It also discusses formulations of these limits that are invariant under different systems of units. In part two, the report discusses Maxwell's field theory from a Hamiltonian perspective and constraints that arise in the formulation. It also discusses a higher order field tensor Lagrangian and its Hamiltonian formulation.
This document discusses energy methods in elasticity theory. It defines external work done by forces and internal strain energy density due to stresses in elastic materials. Equations are presented relating external work to changes in internal strain energy for various loads, including normal stresses, shear stresses, bending moments, and torsional loads. Castigliano's theorem and the principle of virtual work are also introduced as methods to determine displacements and slopes in elastic structures.
This document discusses multi-degree-of-freedom (MDOF) systems and their analysis. It introduces concepts such as flexibility and stiffness matrices, natural frequencies and mode shapes, orthogonality of modes, and equations of motion. Methods for analyzing free and forced vibration of MDOF systems in the time domain are presented, including modal superposition and direct integration. An example 3DOF system is analyzed to illustrate the concepts.
The use of Calculus is very important in every aspects of engineering.
The use of Differential equation is very much applied in the concept of Elastic beams.
1. The document presents an overview of fracture mechanics, including an atomic view of fracture, stress concentration effects of flaws, the Griffith energy balance approach, and the energy release rate.
2. It discusses how cracks propagate when sufficient stress is applied at the atomic level to break atomic bonds, and how flaws concentrate stress which can cause fracture at lower overall stresses than theoretical strength estimates.
3. The Griffith energy balance approach is summarized, showing how the energy required to create new crack surfaces must be balanced by the strain energy released from the material as the crack extends.
12. kinetics of particles impulse momentum methodEkeeda
Learn Online Courses of Subject Engineering Mechanics of First Year Engineering. Clear the Concepts of Engineering Mechanics Through Video Lectures and PDF Notes.
https://ekeeda.com/streamdetails/subject/Engineering-Mechanics
This document provides solutions to 15 problems related to magnetic materials and magnetism. It begins by deriving expressions for how the angular frequency and radius of a classical electron orbit change with the application of a magnetic field using the Lorentz force. It then calculates magnetic moments and susceptibilities for various systems. Other problems cover topics like Lenz's law, diamagnetism, paramagnetism, Bohr orbits, susceptibility measurements, and predicting magnetic behaviors in different materials. Detailed step-by-step workings are shown for each problem.
Analysis of Simple Maglev System using SimulinkArslan Guzel
The document is a PowerPoint presentation analyzing a simple maglev system using Simulink. It includes:
- An introduction to maglev systems and their applications.
- Circuit diagrams and explanations of the position sensor, electromagnet actuator, and other system components.
- Derivations of the system's mathematical model and equations in state space form for both nonlinear and linear controller approaches.
- Descriptions of the Simulink models created to simulate the system, including blocks for the generalized system, nonlinear controller, and determining voltage and current.
- Analysis of simulation results for two air gap scenarios, comparing graphs of position, velocity, acceleration, and other signals between the scenarios.
N. Schlager - Study Materials for MIT Course [8.02T] - Electricity and Magnet...cfisicaster
This document provides a summary of topics covered in Class 1 of the physics course 8.02, which included an introduction to TEAL (Technology Enhanced Active Learning), fields, a review of gravity, and the electric field. Key points include:
1) The course focuses on electricity and magnetism, specifically how charges interact through fields. Gravity and electric fields are introduced as the first examples of fields.
2) Scalar and vector fields are defined and examples of representing each type of field visually are given.
3) Gravity is reviewed as an example of a physical vector field, with masses creating gravitational fields and other masses feeling forces due to those fields.
4) Electric charges are described
The document discusses electric field, potential, and energy. It defines electric potential as the work done to move a unit positive charge from infinity to a point in an electric field. Electric potential is a scalar quantity measured in volts. Equipotentials are regions in space where the electric potential has a constant value, forming equipotential surfaces or lines. Analogies are drawn between electric and gravitational fields, such as both following inverse square laws and having field lines and equipotentials.
1) Newton proposed that all objects with mass attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
2) An object in free fall experiences a sensation of weightlessness because both it and the elevator/spacecraft it is in are accelerating downward at the same rate due to gravity, so there is no relative acceleration between them.
3) A satellite in orbit around a planet is weightless because the centripetal acceleration needed to maintain its orbit exactly counteracts the acceleration due to gravity, resulting in no net acceleration felt by objects in the spacecraft.
1) Gravitational and electric fields can be described by their field strength, which is defined as the force exerted per unit mass or charge.
2) Coulomb's law and Newton's law of gravitation describe the relationship between field strength and distance from the source of the field. Field strength decreases with the inverse square of the distance.
3) Electric and gravitational potential are scalar quantities that represent the potential energy per unit mass or charge. Potential increases as distance from the source decreases. Equipotential lines represent regions of constant potential.
1) Gravitational and electric fields can be described by their field strength, which is defined as the force exerted per unit mass or charge.
2) Gravitational field strength is calculated using Newton's law of universal gravitation, while electric field strength uses Coulomb's law.
3) The electric potential at a point is defined as the work required to move a unit charge from infinity to that point, and equipotentials are surfaces or lines of constant potential.
1. The document provides 11 physics problems related to work, power, and energy. The problems include calculations of velocity, force, acceleration, energy, and power for objects moving under the influence of forces or friction.
2. Sample questions calculate the velocity of blocks sliding down inclined planes, the heat generated when dragging an object, and the speed of a mass attached to a string passing over a pulley.
3. The problems involve concepts such as kinetic and static friction, gravitational potential energy, spring force constants, and conservation of energy. Solutions require applying physics equations like work, power, kinetic and potential energy to analyze motion under different force conditions.
The document discusses electric fields and electrostatics. It explains that when objects are rubbed together, electrons are transferred causing objects to become charged. It then discusses Coulomb's law which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It provides equations for calculating electric field strength, potential, and force experienced by charges in fields.
This document discusses the motion of charged particles in electric and magnetic fields. It begins by introducing the topic and importance of understanding single particle motion. It then examines the motion of a particle in (1) a constant, uniform electric field, finding the maximum velocity it can attain, and (2) a constant, uniform magnetic field, deriving that it will travel in a circular orbit with Larmor radius. The document further derives the equations of motion and orbit for a particle in a magnetic field. It concludes by discussing drift motions such as the E×B drift caused by combined electric and magnetic fields.
1) In an elastic collision between two bodies in one dimension, both linear momentum and kinetic energy are conserved.
2) By applying the laws of conservation of momentum and kinetic energy, equations relating the velocities of the bodies before and after collision can be derived.
3) These equations allow calculating the unknown velocities if the masses of the bodies and their velocities before collision are known.
This document discusses energy methods for structural analysis. It introduces key energy theorems like Betti's theorem, Maxwell's theorem, and Castigliano's theorem. It also discusses potential energy and stationary potential energy. Various energy terms are defined, such as external work, strain energy, and their relationships through the principle of conservation of energy. Examples are given of calculating energy for axial forces, shear forces, torsional moments, and different structural elements. Castigliano's first theorem is introduced for determining deflections and slopes using partial derivatives of internal strain energy.
La iluminación requiere menor espacio, y además es conveniente colocarla por
encima de la altura del gálibo. La señalización vertical se suele colocar sobre las
aceras o por encima del gálibo en el caso de paneles luminosos. Por otra parte, las
canalizaciones para cables y otras instalaciones se suelen colocar bajo la acera o
adheridas al hastial en bandejas porta-cables (ver en el capítulo 13).
Okay, let's break this down step-by-step:
1) The electric field E is uniform, so E has a constant magnitude and direction.
2) The path icf consists of two segments: ic and cf.
3) For ic, the displacement is at a 90° angle to E so no work is done along that segment.
4) For cf, the displacement forms a 45° angle with E.
So the key is to integrate E×ds along just the cf segment.
This document describes electromagnetic waves and Maxwell's equations. It discusses the displacement current, Gauss's law for magnetism, and Maxwell's four equations. Plane electromagnetic waves are examined, with the electric field in the y-direction and magnetic field in the z-direction. Maxwell's equations predict that changing electric and magnetic fields produce each other, leading to electromagnetic waves that travel at the speed of light.
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.
The document describes concepts from a lecture on electric fields, including:
1. The electric field E at a point is defined as the electric force F on a test charge divided by the charge q. Field lines depict the direction and strength of the field.
2. The electric field due to a point charge is directed away from a positive charge and toward a negative charge.
3. The field of a dipole is non-uniform, exerting a torque and giving the dipole a potential energy depending on its orientation in the field.
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Chapter3
1. Chapter 3
Electric Potential
3.1 Potential and Potential Energy................................................................................. 1
3.2 Electric Potential in a Uniform Field....................................................................... 4
3.3 Electric Potential due to Point Charges ................................................................... 5
3.3.1 Potential Energy in a System of Charges.......................................................... 7
3.4 Continuous Charge Distribution .............................................................................. 8
3.5 Deriving Electric Field from the Electric Potential ................................................. 9
3.5.1 Gradient and Equipotentials............................................................................ 10
Example 3.1: Uniformly Charged Rod .................................................................... 12
Example 3.2: Uniformly Charged Ring ................................................................... 14
Example 3.3: Uniformly Charged Disk ................................................................... 15
Example 3.4: Calculating Electric Field from Electric Potential............................. 17
3.6 Summary................................................................................................................ 17
3.7 Problem-Solving Strategy: Calculating Electric Potential..................................... 19
3.8 Solved Problems .................................................................................................... 21
3.8.1
3.8.2
3.8.3
3.8.4
Electric Potential Due to a System of Two Charges....................................... 21
Electric Dipole Potential ................................................................................. 22
Electric Potential of an Annulus ..................................................................... 23
Charge Moving Near a Charged Wire ............................................................ 24
3.9 Conceptual Questions ............................................................................................ 25
3.10 Additional Problems ............................................................................................ 26
3.10.1 Cube .............................................................................................................. 26
3.10.2 Three Charges ............................................................................................... 26
3.10.3 Work Done on Charges................................................................................. 26
3.10.4 Calculating E from V .................................................................................... 27
3.10.5 Electric Potential of a Rod ............................................................................ 27
3.10.6 Electric Potential........................................................................................... 28
3.10.7 Calculating Electric Field from the Electric Potential .................................. 28
3.10.8 Electric Potential and Electric Potential Energy........................................... 29
3.10.9. Electric Field, Potential and Energy ............................................................. 29
0
2. Electric Potential
3.1 Potential and Potential Energy
In the introductory mechanics course, we have seen that gravitational force from the
Earth on a particle of mass m located at a distance r from Earth’s center has an inversesquare form:
Fg = −G
Mm
ˆ
r
r2
(3.1.1)
ˆ
where G = 6.67 ×10− 11 N ⋅ m 2 /kg 2 is the gravitational constant and r is a unit vector
pointing radially outward. The Earth is assumed to be a uniform sphere of mass M. The
corresponding gravitational field g , defined as the gravitational force per unit mass, is
given by
g=
Fg
m
=−
GM
ˆ
r
r2
(3.1.2)
Notice that g only depends on M, the mass which creates the field, and r, the distance
from M.
Figure 3.1.1
Consider moving a particle of mass m under the influence of gravity (Figure 3.1.1). The
work done by gravity in moving m from A to B is
⎡ GMm
GMm
Wg = ∫ Fg ⋅ d s = ∫ ⎛ − 2 ⎞ dr =
⎜
⎟
rA ⎝
r
⎠
⎢
⎣
rB
r
⎤
⎥
⎦
rB
rA
⎛ 1 1 ⎞
= GMm ⎜ − ⎟
⎝ rB rA ⎠
(3.1.3)
The result shows that Wg is independent of the path taken; it depends only on the
endpoints A and B. It is important to draw distinction between Wg , the work done by the
1
3. field and Wext , the work done by an external agent such as you. They simply differ by a
negative sign: Wg = −Wext .
Near Earth’s surface, the gravitational field g is approximately constant, with a
magnitude g = GM / rE 2 ≈ 9.8 m/s 2 , where rE is the radius of Earth. The work done by
gravity in moving an object from height y A to yB (Figure 3.1.2) is
B
B
yB
A
A
yA
Wg = ∫ Fg ⋅ d s = ∫ mg cos θ ds = − ∫ mg cos φ ds = − ∫ mg dy = − mg ( yB − y A )
(3.1.4)
Figure 3.1.2 Moving a mass m from A to B.
The result again is independent of the path, and is only a function of the change in
vertical height yB − y A .
In the examples above, if the path forms a closed loop, so that the object moves around
and then returns to where it starts off, the net work done by the gravitational field would
be zero, and we say that the gravitational force is conservative. More generally, a force F
is said to be conservative if its line integral around a closed loop vanishes:
∫ F⋅d s = 0
(3.1.5)
When dealing with a conservative force, it is often convenient to introduce the concept of
potential energy U. The change in potential energy associated with a conservative force
F acting on an object as it moves from A to B is defined as:
B
∆U = U B − U A = − ∫ F ⋅ d s = −W
A
(3.1.6)
where W is the work done by the force on the object. In the case of gravity, W = Wg and
from Eq. (3.1.3), the potential energy can be written as
Ug = −
GMm
+ U0
r
(3.1.7)
2
4. where U 0 is an arbitrary constant which depends on a reference point. It is often
convenient to choose a reference point where U 0 is equal to zero. In the gravitational
case, we choose infinity to be the reference point, with U 0 (r = ∞) = 0 . Since U g depends
on the reference point chosen, it is only the potential energy difference ∆U g that has
physical importance. Near Earth’s surface where the gravitational field g is
approximately constant, as an object moves from the ground to a height h, the change in
potential energy is ∆U g = + mgh , and the work done by gravity is Wg = − mgh .
A concept which is closely related to potential energy is “potential.” From ∆U , the
gravitational potential can be obtained as
∆Vg =
∆U g
m
B
B
A
A
= − ∫ (Fg / m) ⋅ d s = − ∫ g ⋅ d s
(3.1.8)
Physically ∆Vg represents the negative of the work done per unit mass by gravity to
move a particle from A to B .
Our treatment of electrostatics is remarkably similar to gravitation. The electrostatic force
F e given by Coulomb’s law also has an inverse-square form. In addition, it is also
conservative. In the presence of an electric field E , in analogy to the gravitational field
g , we define the electric potential difference between two points A and B as
B
B
A
A
∆V = − ∫ (Fe / q0 ) ⋅ d s = − ∫ E ⋅ d s
(3.1.9)
where q0 is a test charge. The potential difference ∆V represents the amount of work
done per unit charge to move a test charge q0 from point A to B, without changing its
kinetic energy. Again, electric potential should not be confused with electric potential
energy. The two quantities are related by
∆ U = q0 ∆ V
(3.1.10)
The SI unit of electric potential is volt (V):
1volt = 1 joule/coulomb (1 V= 1 J/C)
(3.1.11)
When dealing with systems at the atomic or molecular scale, a joule (J) often turns out to
be too large as an energy unit. A more useful scale is electron volt (eV), which is defined
as the energy an electron acquires (or loses) when moving through a potential difference
of one volt:
3
5. 1eV = (1.6 × 10−19 C)(1V) = 1.6 ×10−19 J
(3.1.12)
3.2 Electric Potential in a Uniform Field
j
Consider a charge + q moving in the direction of a uniform electric field E = E0 (−ˆ) , as
shown in Figure 3.2.1(a).
(b)
(a)
Figure 3.2.1 (a) A charge q which moves in the direction of a constant electric field E .
(b) A mass m that moves in the direction of a constant gravitational field g .
Since the path taken is parallel to E , the potential difference between points A and B is
given by
B
B
A
A
∆V = VB − VA = − ∫ E ⋅ d s = − E0 ∫ ds = − E0 d < 0
(3.2.1)
implying that point B is at a lower potential compared to A. In fact, electric field lines
always point from higher potential to lower. The change in potential energy is
∆ U = U B − U A = − qE0 d . Since q > 0, we have ∆U < 0 , which implies that the potential
energy of a positive charge decreases as it moves along the direction of the electric field.
The corresponding gravitational analogy, depicted in Figure 3.2.1(b), is that a mass m
loses potential energy ( ∆ U = − mgd ) as it moves in the direction of the gravitational
field g .
Figure 3.2.2 Potential difference due to a uniform electric field
What happens if the path from A to B is not parallel to E , but instead at an angle θ, as
shown in Figure 3.2.2? In that case, the potential difference becomes
4
6. B
∆V = VB − VA = − ∫ E ⋅ d s = −E ⋅ s = − E0 s cos θ = − E0 y
A
(3.2.2)
Note that y increase downward in Figure 3.2.2. Here we see once more that moving along
the direction of the electric field E leads to a lower electric potential. What would the
change in potential be if the path were A → C → B ? In this case, the potential difference
consists of two contributions, one for each segment of the path:
∆V = ∆VCA + ∆VBC
(3.2.3)
When moving from A to C, the change in potential is ∆VCA = − E0 y . On the other hand,
when going from C to B, ∆VBC = 0 since the path is perpendicular to the direction of E .
Thus, the same result is obtained irrespective of the path taken, consistent with the fact
that E is conservative.
Notice that for the path A → C → B , work is done by the field only along the segment
AC which is parallel to the field lines. Points B and C are at the same electric potential,
i.e., VB = VC . Since ∆U = q∆V , this means that no work is required in moving a charge
from B to C. In fact, all points along the straight line connecting B and C are on the same
“equipotential line.” A more complete discussion of equipotential will be given in
Section 3.5.
3.3 Electric Potential due to Point Charges
Next, let’s compute the potential difference between two points A and B due to a charge
ˆ
ˆ
+Q. The electric field produced by Q is E = (Q / 4πε 0 r 2 )r , where r is a unit vector
pointing toward the field point.
Figure 3.3.1 Potential difference between two points due to a point charge Q.
ˆ
From Figure 3.3.1, we see that r ⋅ d s = ds cos θ = dr , which gives
∆ V = VB − VA = − ∫
B
A
Q
4πε 0 r
2
ˆ
r ⋅ d s = −∫
B
A
Q
4πε 0 r
2
dr =
Q ⎛ 1 1 ⎞
⎜ − ⎟
4πε 0 ⎝ rB rA ⎠
(3.3.1)
5
7. Once again, the potential difference ∆V depends only on the endpoints, independent of
the choice of path taken.
As in the case of gravity, only the difference in electrical potential is physically
meaningful, and one may choose a reference point and set the potential there to be zero.
In practice, it is often convenient to choose the reference point to be at infinity, so that the
electric potential at a point P becomes
P
VP = − ∫ E ⋅ d s
(3.3.2)
∞
With this reference, the electric potential at a distance r away from a point charge Q
becomes
1
Q
4πε 0 r
V (r ) =
(3.3.3)
When more than one point charge is present, by applying the superposition principle, the
total electric potential is simply the sum of potentials due to individual charges:
V (r ) =
1
4πε 0
qi
∑r
i
i
= ke ∑
i
qi
ri
(3.3.4)
A summary of comparison between gravitation and electrostatics is tabulated below:
Gravitation
Electrostatics
Mass m
Charge q
Gravitational force Fg = −G
Mm
ˆ
r
r2
Gravitational field g = Fg / m
B
Potential energy change ∆ U = − ∫ Fg ⋅ d s
A
B
Gravitational potential Vg = − ∫ g ⋅ d s
A
For a source M: Vg = −
GM
r
| ∆ U g | = mg d (constant g )
Coulomb force Fe = ke
Qq
ˆ
r
r2
Electric field E = Fe / q
B
Potential energy change ∆ U = − ∫ Fe ⋅ d s
A
B
Electric Potential V = − ∫ E ⋅ d s
A
For a source Q: V = ke
Q
r
| ∆ U | = qEd (constant E )
6
8. 3.3.1 Potential Energy in a System of Charges
If a system of charges is assembled by an external agent, then ∆ U = −W = +Wext . That is,
the change in potential energy of the system is the work that must be put in by an external
agent to assemble the configuration. A simple example is lifting a mass m through a
height h. The work done by an external agent you, is + mgh (The gravitational field
does work − mgh ). The charges are brought in from infinity without acceleration i.e. they
are at rest at the end of the process. Let’s start with just two charges q1 and q2 . Let the
potential due to q1 at a point P be V1 (Figure 3.3.2).
Figure 3.3.2 Two point charges separated by a distance r12 .
The work W2 done by an agent in bringing the second charge q2 from infinity to P is
then W2 = q2V1 . (No work is required to set up the first charge and W1 = 0 ). Since
V1 = q1 / 4πε 0 r12 , where r12 is the distance measured from q1 to P, we have
U12 = W2 =
1
q1q2
4πε 0 r12
(3.3.5)
If q1 and q2 have the same sign, positive work must be done to overcome the electrostatic
repulsion and the potential energy of the system is positive, U12 > 0 . On the other hand,
if the signs are opposite, then U12 < 0 due to the attractive force between the charges.
Figure 3.3.3 A system of three point charges.
To add a third charge q3 to the system (Figure 3.3.3), the work required is
7
9. W3 = q3 ( V1 + V2 ) =
q3 ⎛ q1 q2 ⎞
⎜ +
⎟
4πε 0 ⎝ r13 r23 ⎠
(3.3.6)
The potential energy of this configuration is then
U = W2 + W3 =
1 ⎛ q1q2 q1q3 q2 q3
+
+
⎜
4πε 0 ⎝ r12
r13
r23
⎞
⎟ = U12 + U13 + U 23
⎠
(3.3.7)
The equation shows that the total potential energy is simply the sum of the contributions
from distinct pairs. Generalizing to a system of N charges, we have
U=
1
4πε 0
N
N
∑∑
i =1 j =1
j >i
qi q j
rij
(3.3.8)
where the constraint j > i is placed to avoid double counting each pair. Alternatively,
one may count each pair twice and divide the result by 2. This leads to
U=
1
8πε 0
N
N
∑∑
i =1 j =1
j ≠i
qi q j
rij
⎛
1 N ⎜ 1
= ∑ qi
2 i =1 ⎜ 4πε 0
⎜
⎝
⎞
qj ⎟ 1 N
∑ r ⎟ = 2 ∑ qiV (ri )
j =1 ij ⎟
i =1
j ≠i
⎠
N
(3.3.9)
where V (ri ) , the quantity in the parenthesis, is the potential at ri (location of qi) due to all
the other charges.
3.4 Continuous Charge Distribution
If the charge distribution is continuous, the potential at a point P can be found by
summing over the contributions from individual differential elements of charge dq .
Figure 3.4.1 Continuous charge distribution
8
10. Consider the charge distribution shown in Figure 3.4.1. Taking infinity as our reference
point with zero potential, the electric potential at P due to dq is
dV =
1
dq
4πε 0 r
(3.4.1)
Summing over contributions from all differential elements, we have
V=
1
4πε 0
∫
dq
r
(3.4.2)
3.5 Deriving Electric Field from the Electric Potential
In Eq. (3.1.9) we established the relation between E and V. If we consider two points
which are separated by a small distance ds , the following differential form is obtained:
dV = −E ⋅ d s
(3.5.1)
ˆ
ˆ
i
j
In Cartesian coordinates, E = Ex ˆ + E y ˆ + Ez k and d s = dx ˆ + dyˆ + dz k , we have
i
j
(
)(
)
ˆ
ˆ
dV = Ex ˆ + E y ˆ + Ez k ⋅ dx ˆ + dyˆ + dz k = Ex dx + E y dy + Ez dz
i
j
i
j
(3.5.2)
which implies
Ex = −
∂V
∂V
∂V
, Ey = −
, Ez = −
∂x
∂y
∂z
(3.5.3)
By introducing a differential quantity called the “del (gradient) operator”
∇≡
∂ ˆ ∂ ˆ ∂ ˆ
j+ k
i+
∂x
∂y
∂z
(3.5.4)
the electric field can be written as
⎛ ∂V ˆ ∂V ˆ ∂V ˆ ⎞
⎛ ∂ ˆ ∂ ˆ ∂ ˆ⎞
ˆ
E = Ex ˆ + E y ˆ + Ez k = − ⎜
i
j
i+
j+
k ⎟ = −⎜
i+
j + k ⎟ V = −∇ V
∂y
∂z ⎠
∂y
∂z ⎠
⎝ ∂x
⎝ ∂x
E = −∇ V
(3.5.5)
Notice that ∇ operates on a scalar quantity (electric potential) and results in a vector
quantity (electric field). Mathematically, we can think of E as the negative of the
gradient of the electric potential V . Physically, the negative sign implies that if
9
11. V increases as a positive charge moves along some direction, say x, with ∂V / ∂x > 0 ,
then there is a non-vanishing component of E in the opposite direction (− Ex ≠ 0) . In the
case of gravity, if the gravitational potential increases when a mass is lifted a distance h,
the gravitational force must be downward.
If the charge distribution possesses spherical symmetry, then the resulting electric field is
ˆ
a function of the radial distance r, i.e., E = Er r . In this case, dV = − Er dr. If V ( r ) is
known, then E may be obtained as
⎛ dV ⎞ ˆ
ˆ
E = Er r = − ⎜
⎟r
⎝ dr ⎠
(3.5.6)
For example, the electric potential due to a point charge q is V (r ) = q / 4πε 0 r . Using the
ˆ
above formula, the electric field is simply E = (q / 4πε 0 r 2 )r .
3.5.1 Gradient and Equipotentials
Suppose a system in two dimensions has an electric potential V ( x, y ) . The curves
characterized by constant V ( x, y ) are called equipotential curves. Examples of
equipotential curves are depicted in Figure 3.5.1 below.
Figure 3.5.1 Equipotential curves
In three dimensions we have equipotential surfaces and they are described by
V ( x, y , z ) =constant. Since E = −∇ V , we can show that the direction of E is always
perpendicular to the equipotential through the point. Below we give a proof in two
dimensions. Generalization to three dimensions is straightforward.
Proof:
Referring to Figure 3.5.2, let the potential at a point P ( x, y ) be V ( x, y ) . How much is
V changed at a neighboring point P ( x + dx, y + dy ) ? Let the difference be written as
10
12. dV = V ( x + dx, y + dy ) − V ( x, y )
⎡
∂V
∂V
= ⎢ V ( x, y ) +
dx +
dy +
∂x
∂y
⎣
⎤
∂V
∂V
⎥ − V ( x, y ) ≈ ∂ x dx + ∂ y dy
⎦
(3.5.7)
Figure 3.5.2 Change in V when moving from one equipotential curve to another
With the displacement vector given by d s = dx ˆ + dyˆ , we can rewrite dV as
i
j
⎛ ∂V ˆ ∂V
dV = ⎜
i+
∂y
⎝ ∂x
ˆ ⎞ ⋅ dx ˆ + dyˆ = (∇ V ) ⋅ d s = −E ⋅ d s
j⎟
i
j
⎠
(
)
(3.5.8)
If the displacement ds is along the tangent to the equipotential curve through P(x,y),
then dV = 0 because V is constant everywhere on the curve. This implies that E ⊥ d s
along the equipotential curve. That is, E is perpendicular to the equipotential. In Figure
3.5.3 we illustrate some examples of equipotential curves. In three dimensions they
become equipotential surfaces. From Eq. (3.5.8), we also see that the change in potential
dV attains a maximum when the gradient ∇ V is parallel to d s :
⎛ dV ⎞
max ⎜
⎟ = ∇V
⎝ ds ⎠
(3.5.9)
Physically, this means that ∇ V always points in the direction of maximum rate of change
of V with respect to the displacement s.
Figure 3.5.3 Equipotential curves and electric field lines for (a) a constant E field, (b) a
point charge, and (c) an electric dipole.
11
13. The properties of equipotential surfaces can be summarized as follows:
(i)
The electric field lines are perpendicular to the equipotentials and point from
higher to lower potentials.
(ii)
By symmetry, the equipotential surfaces produced by a point charge form a family
of concentric spheres, and for constant electric field, a family of planes
perpendicular to the field lines.
(iii) The tangential component of the electric field along the equipotential surface is
zero, otherwise non-vanishing work would be done to move a charge from one
point on the surface to the other.
(iv) No work is required to move a particle along an equipotential surface.
A useful analogy for equipotential curves is a topographic map (Figure 3.5.4). Each
contour line on the map represents a fixed elevation above sea level. Mathematically it is
expressed as z = f ( x, y ) = constant . Since the gravitational potential near the surface of
Earth is Vg = gz , these curves correspond to gravitational equipotentials.
Figure 3.5.4 A topographic map
Example 3.1: Uniformly Charged Rod
Consider a non-conducting rod of length having a uniform charge density λ . Find the
electric potential at P , a perpendicular distance y above the midpoint of the rod.
Figure 3.5.5 A non-conducting rod of length
and uniform charge density λ .
12
14. Solution:
Consider a differential element of length dx′ which carries a charge dq = λ dx′ , as shown
in Figure 3.5.5. The source element is located at ( x′, 0) , while the field point P is located
on the y-axis at (0 , y ) . The distance from dx′ to P is r = ( x′2 + y 2 )1/ 2 . Its contribution to
the potential is given by
dq
1
λ dx′
=
4πε 0 r
4πε 0 ( x′2 + y 2 )1/ 2
1
dV =
Taking V to be zero at infinity, the total potential due to the entire rod is
V=
λ
4πε 0
∫
/2
− /2
dx′
x ′2 + y 2
=
λ
ln ⎡ x′ + x′2 + y 2 ⎤
⎦
4πε 0 ⎣
⎡ ( / 2) + ( / 2) 2 + y 2
λ
=
ln ⎢
4πε 0 ⎢ −( / 2) + ( / 2) 2 + y 2
⎣
⎤
⎥
⎥
⎦
/2
− /2
(3.5.10)
where we have used the integration formula
∫
dx′
x′ + y
2
2
(
= ln x′ + x′2 + y 2
A plot of V ( y ) / V0 , where V0 = λ / 4πε 0 , as a function of y /
)
is shown in Figure 3.5.6
Figure 3.5.6 Electric potential along the axis that passes through the midpoint of a nonconducting rod.
In the limit
y , the potential becomes
13
15. ⎡ ( / 2) + / 2 1 + (2 y / ) 2 ⎤
⎡ 1 + 1 + (2 y / ) 2
λ
λ
V=
ln ⎢
ln ⎢
⎥=
4πε 0 ⎢ −( / 2) + / 2 1 + (2 y / ) 2 ⎥ 4πε 0 ⎢ − 1 + 1 + (2 y / ) 2
⎣
⎦
⎣
≈
⎛
2
λ
ln ⎜ 2
4πε 0 ⎝ 2 y /
=
⎛ ⎞
λ
ln ⎜ ⎟
2πε 0 ⎝ y ⎠
2
⎛ 2 ⎞
⎞
λ
ln ⎜ 2 ⎟
=
⎟
⎠ 4πε 0 ⎝ y ⎠
⎤
⎥
⎥
⎦
(3.5.11)
The corresponding electric field can be obtained as
Ey = −
∂V
λ
=
∂ y 2πε 0 y
/2
( / 2) 2 + y 2
in complete agreement with the result obtained in Eq. (2.10.9).
Example 3.2: Uniformly Charged Ring
Consider a uniformly charged ring of radius R and charge density λ (Figure 3.5.7). What
is the electric potential at a distance z from the central axis?
Figure 3.5.7 A non-conducting ring of radius R with uniform charge density λ .
Solution:
Consider a small differential element d = R dφ ′ on the ring. The element carries a
charge dq = λ d = λ R dφ ′ , and its contribution to the electric potential at P is
dV =
dq
1
λ R dφ ′
=
4πε 0 r
4πε 0 R 2 + z 2
1
The electric potential at P due to the entire ring is
14
16. V = ∫ dV =
λR
1
4πε 0
2πλ R
1
R2 + z2
∫ dφ ′ = 4πε
0
R2 + z2
=
1
4πε 0
Q
R2 + z2
(3.5.12)
where we have substituted Q = 2π Rλ for the total charge on the ring. In the limit z
the potential approaches its “point-charge” limit:
V≈
R,
1 Q
4πε 0 z
From Eq. (3.5.12), the z-component of the electric field may be obtained as
Ez = −
∂V
∂ ⎛ 1
=− ⎜
∂z
∂ z ⎝ 4πε 0
⎞
1
Qz
⎟=
2
2 3/ 2
2
2
R + z ⎠ 4πε 0 ( R + z )
Q
(3.5.13)
in agreement with Eq. (2.10.14).
Example 3.3: Uniformly Charged Disk
Consider a uniformly charged disk of radius R and charge density σ lying in the xyplane. What is the electric potential at a distance z from the central axis?
Figure 3.4.3 A non-conducting disk of radius R and uniform charge density σ.
Solution:
Consider a circular ring of radius r ′ and width dr ′ . The charge on the ring is
dq′ = σ dA′ = σ (2π r ′dr ′). The field point P is located along the z -axis a distance z
from the plane of the disk. From the figure, we also see that the distance from a point on
the ring to P is r = (r ′2 + z 2 )1/ 2 . Therefore, the contribution to the electric potential at P
is
dV =
dq
1 σ (2π r ′dr ′)
=
4πε 0 r
4πε 0 r ′2 + z 2
1
15
17. By summing over all the rings that make up the disk, we have
V=
In the limit | z |
σ
4πε 0
∫
R
2π r ′dr ′
r ′2 + z 2
0
=
σ ⎡ 2 2⎤
r′ + z
⎦
2ε 0 ⎣
R
=
0
σ ⎡ 2 2
R + z − | z |⎤
⎦
2ε 0 ⎣
(3.5.14)
R,
1/ 2
⎛
R2 ⎞
R + z =| z | ⎜ 1 + 2 ⎟
z ⎠
⎝
2
2
⎛
R2
=| z | ⎜ 1 + 2 +
2z
⎝
⎞
⎟,
⎠
and the potential simplifies to the point-charge limit:
V≈
σ R2
1 σ (π R 2 )
1 Q
⋅
=
=
2ε 0 2 | z | 4πε 0 | z |
4πε 0 | z |
As expected, at large distance, the potential due to a non-conducting charged disk is the
same as that of a point charge Q. A comparison of the electric potentials of the disk and a
point charge is shown in Figure 3.4.4.
Figure 3.4.4 Comparison of the electric potentials of a non-conducting disk and a point
charge. The electric potential is measured in terms of V0 = Q / 4πε 0 R .
Note that the electric potential at the center of the disk ( z = 0 ) is finite, and its value is
Vc =
σR
Q
R
1 2Q
=
⋅
=
= 2V0
2
2ε 0 π R 2ε 0 4πε 0 R
(3.5.15)
This is the amount of work that needs to be done to bring a unit charge from infinity and
place it at the center of the disk.
The corresponding electric field at P can be obtained as:
Ez = −
σ
∂V
=
∂ z 2ε 0
⎡ z
⎤
z
−
⎢
⎥
R2 + z 2 ⎦
⎣| z|
(3.5.16)
16
18. which agrees with Eq. (2.10.18). In the limit R z , the above equation becomes
Ez = σ / 2ε 0 , which is the electric field for an infinitely large non-conducting sheet.
Example 3.4: Calculating Electric Field from Electric Potential
Suppose the electric potential due to a certain charge distribution can be written in
Cartesian Coordinates as
V ( x, y, z ) = Ax 2 y 2 + Bxyz
where A , B and C are constants. What is the associated electric field?
Solution:
The electric field can be found by using Eq. (3.5.3):
∂V
= −2 Axy 2 − Byz
∂x
∂V
Ey = −
= −2 Ax 2 y − Bxz
∂y
∂V
Ez = −
= − Bxy
∂z
Ex = −
ˆ
Therefore, the electric field is E = ( −2 Axy 2 − Byz ) ˆ − (2 Ax 2 y + Bxz ) ˆ − Bxy k .
i
j
3.6 Summary
•
A force F is conservative if the line integral of the force around a closed loop
vanishes:
∫ F⋅d s = 0
•
The change in potential energy associated with a conservative force F acting on an
object as it moves from A to B is
B
∆U = U B − U A = − ∫ F ⋅ d s
A
17
19. •
The electric potential difference ∆V between points A and B in an electric field
E is given by
∆V = VB − VA =
B
∆U
= −∫ E ⋅ ds
A
q0
The quantity represents the amount of work done per unit charge to move a test
charge q0 from point A to B, without changing its kinetic energy.
•
The electric potential due to a point charge Q at a distance r away from the charge is
V=
1 Q
4πε 0 r
For a collection of charges, using the superposition principle, the electric potential is
V=
•
1
4πε 0
Qi
∑r
i
i
The potential energy associated with two point charges q1 and q2 separated by a
distance r12 is
U=
•
1 q1q2
4πε 0 r12
From the electric potential V , the electric field may be obtained by taking the
gradient of V :
E = −∇V
In Cartesian coordinates, the components may be written as
Ex = −
•
∂V
∂V
∂V
, Ey = −
, Ez = −
∂x
∂y
∂z
The electric potential due to a continuous charge distribution is
V=
1
4πε 0
∫
dq
r
18
20. 3.7 Problem-Solving Strategy: Calculating Electric Potential
In this chapter, we showed how electric potential can be calculated for both the discrete
and continuous charge distributions. Unlike electric field, electric potential is a scalar
quantity. For the discrete distribution, we apply the superposition principle and sum over
individual contributions:
V = ke ∑
i
qi
ri
For the continuous distribution, we must evaluate the integral
V = ke ∫
dq
r
In analogy to the case of computing the electric field, we use the following steps to
complete the integration:
(1) Start with dV = ke
dq
.
r
(2) Rewrite the charge element dq as
⎧λ dl
⎪
dq = ⎨σ dA
⎪ ρ dV
⎩
(length)
(area)
(volume)
depending on whether the charge is distributed over a length, an area, or a volume.
(3) Substitute dq into the expression for dV .
(4) Specify an appropriate coordinate system and express the differential element (dl, dA
or dV ) and r in terms of the coordinates (see Table 2.1.)
(5) Rewrite dV in terms of the integration variable.
(6) Complete the integration to obtain V.
Using the result obtained for V , one may calculate the electric field by E = −∇V .
Furthermore, the accuracy of the result can be readily checked by choosing a point P
which lies sufficiently far away from the charge distribution. In this limit, if the charge
distribution is of finite extent, the field should behave as if the distribution were a point
charge, and falls off as 1/ r 2 .
19
21. Below we illustrate how the above methodologies can be employed to compute the
electric potential for a line of charge, a ring of charge and a uniformly charged disk.
Charged Rod
Charged Ring
Charged disk
dq = λ dx′
dq = λ dl
dq = σ dA
Figure
(2) Express dq in
terms of charge
density
(3) Substitute dq
into expression for
dV
(4) Rewrite r and the
differential element
in terms of the
appropriate
coordinates
Derive E from V
Point-charge
for E
limit
r
λ dl
dV = ke
r
σ dA
r
dl = R dφ ′
dA = 2π r ′ dr ′
r = x′2 + y 2
V=
(6) Integrate to get V
dV = ke
dx′
r = R2 + z2
r = r ′2 + z 2
dV = ke
(5) Rewrite dV
λ dx′
dV = ke
λ
4πε0 ∫−
/2
/2
λ dx′
2 1/ 2
V = ke
dx′
x′2 + y2
⎡ ( / 2) + ( / 2)2 + y2
λ
ln ⎢
=
4πε0 ⎢ −( / 2) + ( / 2)2 + y2
⎣
Ey = −
∂V
∂y
λ
/2
=
2πε 0 y ( / 2) 2 + y 2
Ey ≈
dV = ke
( x′ + y )
2
ke Q
y2
y
⎤
⎥
⎥
⎦
= ke
= ke
Ez = −
Ez ≈
λ R dφ ′
(R + z )
2
2 1/ 2
Rλ
( R + z 2 )1/ 2
(2π Rλ )
2
∫ dφ ′
R +z
keQz
∂V
=
∂z ( R 2 + z 2 )3/ 2
ke Q
z2
z
R
R
0
=
2
2πσ r ′ dr ′
(r ′2 + z 2 )1/ 2
V = ke 2πσ ∫
= 2keπσ
R2 + z 2
Q
2
dV = ke
Ez = −
(
r′ dr′
(r′ + z2 )1/2
2
)
z2 + R2 − | z |
)
(
2keQ 2 2
z + R −| z |
R2
⎞
∂V 2keQ ⎛ z
z
= 2 ⎜
−
⎟
R ⎝| z |
∂z
z 2 + R2 ⎠
Ez ≈
ke Q
z2
z
R
20
22. 3.8 Solved Problems
3.8.1 Electric Potential Due to a System of Two Charges
Consider a system of two charges shown in Figure 3.8.1.
Figure 3.8.1 Electric dipole
Find the electric potential at an arbitrary point on the x axis and make a plot.
Solution:
The electric potential can be found by the superposition principle. At a point on the x
axis, we have
V ( x) =
1 (−q )
1 ⎤
q
q ⎡ 1
+
=
⎢| x − a | − | x + a | ⎥
4πε 0 | x − a | 4πε 0 | x + a | 4πε 0 ⎣
⎦
1
The above expression may be rewritten as
V ( x)
1
1
=
−
V0
| x / a − 1| | x / a + 1|
where V0 = q / 4πε 0 a . The plot of the dimensionless electric potential as a function of x/a.
is depicted in Figure 3.8.2.
Figure 3.8.2
21
23. As can be seen from the graph, V ( x ) diverges at x / a = ±1 , where the charges are
located.
3.8.2 Electric Dipole Potential
Consider an electric dipole along the y-axis, as shown in the Figure 3.8.3. Find the
electric potential V at a point P in the x-y plane, and use V to derive the corresponding
electric field.
Figure 3.8.3
By superposition principle, the potential at P is given by
V = ∑ Vi =
i
1 ⎛ q q ⎞
⎜ − ⎟
4πε 0 ⎝ r+ r− ⎠
where r± 2 = r 2 + a 2 ∓ 2ra cos θ . If we take the limit where r
a, then
−1/ 2
1 1
1⎡ 1
= ⎡ 1 + (a / r ) 2 ∓ 2(a / r ) cos θ ⎤
= ⎢ 1 − (a / r ) 2 ± (a / r ) cos θ +
⎣
⎦
r± r
r⎣ 2
⎤
⎥
⎦
and the dipole potential can be approximated as
1
⎡ 1
2
2
⎢ 1 − 2 (a / r ) + (a / r ) cos θ − 1 + 2 (a / r ) + (a / r ) cos θ +
4πε 0 r ⎣
ˆ
q
p ⋅r
2a cos θ p cos θ
≈
⋅
=
=
2
r
4πε 0 r
4πε 0 r
4πε 0 r 2
V=
q
⎤
⎥
⎦
where p = 2aq ˆ is the electric dipole moment. In spherical polar coordinates, the gradient
j
operator is
∇=
∂
1 ∂ ˆ
1
∂
ˆ
ˆ
r+
θ+
φ
∂r
r ∂θ
r sin θ ∂ φ
22
24. Since the potential is now a function of both r and θ , the electric field will have
ˆ
ˆ
components along the r and θ directions. Using E = −∇V , we have
Er = −
∂ V p cos θ
1 ∂ V p sin θ
=
, Eθ = −
=
, Eφ = 0
3
∂ r 2πε 0 r
r ∂ θ 4πε 0 r 3
3.8.3 Electric Potential of an Annulus
Consider an annulus of uniform charge density σ , as shown in Figure 3.8.4. Find the
electric potential at a point P along the symmetric axis.
Figure 3.8.4 An annulus of uniform charge density.
Solution:
Consider a small differential element dA at a distance r away from point P. The amount
of charge contained in dA is given by
dq = σ dA = σ ( r ' dθ ) dr '
Its contribution to the electric potential at P is
dV =
dq
1 σ r ' dr ' dθ
=
4πε 0 r
4πε 0 r '2 + z 2
1
Integrating over the entire annulus, we obtain
V=
σ
4πε 0
b
2π
a
0
∫∫
r ' dr ' dθ
r '2 + z 2
=
2πσ
4πε 0
∫
b
a
r ' ds
r '2 + z 2
=
σ ⎡ 2 2
b + z − a2 + z2 ⎤
⎣
⎦
2ε 0
where we have made used of the integral
23
25. ds s
∫
s +z
2
2
= s2 + z 2
Notice that in the limit a → 0 and b → R , the potential becomes
V=
σ ⎡ 2 2
R + z − | z |⎤
⎦
2ε 0 ⎣
which coincides with the result of a non-conducting disk of radius R shown in Eq.
(3.5.14).
3.8.4 Charge Moving Near a Charged Wire
A thin rod extends along the z-axis from z = −d to z = d . The rod carries a positive
charge Q uniformly distributed along its length 2d with charge density λ = Q / 2d .
(a) Calculate the electric potential at a point z > d along the z-axis.
(b) What is the change in potential energy if an electron moves from z = 4d to z = 3d ?
(c) If the electron started out at rest at the point z = 4d , what is its velocity at z = 3d ?
Solutions:
(a) For simplicity, let’s set the potential to be zero at infinity, V (∞) = 0 . Consider an
infinitesimal charge element dq = λ dz ′ located at a distance z ' along the z-axis. Its
contribution to the electric potential at a point z > d is
dV =
λ dz '
4πε 0 z − z '
Integrating over the entire length of the rod, we obtain
V ( z) =
λ z −d dz'
λ
⎛ z+d ⎞
⎟
∫ z+d z − z' = 4πε 0 ln ⎜ z − d ⎠
4πε 0
⎝
(b) Using the result derived in (a), the electrical potential at z = 4d is
V ( z = 4d ) =
λ
λ
⎛ 4d + d ⎞
⎛5⎞
ln ⎜
ln ⎜ ⎟
⎟=
4πε 0 ⎝ 4d − d ⎠ 4πε 0 ⎝ 3 ⎠
Similarly, the electrical potential at z = 3d is
24
26. V ( z = 3d ) =
λ
λ
⎛ 3d + d ⎞
ln ⎜
ln 2
⎟=
4πε 0 ⎝ 3d − d ⎠ 4πε 0
The electric potential difference between the two points is
∆V = V ( z = 3d ) − V ( z = 4d ) =
λ
⎛6⎞
ln ⎜ ⎟ > 0
4πε 0 ⎝ 5 ⎠
Using the fact that the electric potential difference ∆V is equal to the change in potential
energy per unit charge, we have
∆U = q∆V = −
|e|λ ⎛6⎞
ln ⎜ ⎟ < 0
4πε 0 ⎝ 5 ⎠
where q = − | e | is the charge of the electron.
(c) If the electron starts out at rest at z = 4d then the change in kinetic energy is
1
∆K = mv f 2
2
By conservation of energy, the change in kinetic energy is
∆K = −∆U =
|e|λ ⎛6⎞
ln ⎜ ⎟ > 0
4πε 0 ⎝ 5 ⎠
Thus, the magnitude of the velocity at z = 3d is
vf =
2|e| λ ⎛ 6⎞
ln ⎜ ⎟
4πε 0 m ⎝ 5 ⎠
3.9 Conceptual Questions
1. What is the difference between electric potential and electric potential energy?
2. A uniform electric field is parallel to the x-axis. In what direction can a charge be
displaced in this field without any external work being done on the charge?
3. Is it safe to stay in an automobile with a metal body during severe thunderstorm?
Explain.
25
27. 4. Why are equipotential surfaces always perpendicular to electric field lines?
5. The electric field inside a hollow, uniformly charged sphere is zero. Does this imply
that the potential is zero inside the sphere?
3.10 Additional Problems
3.10.1 Cube
How much work is done to assemble eight identical point charges, each of magnitude q,
at the corners of a cube of side a?
3.10.2 Three Charges
Three charges with q = 3.00 × 10−18 C and q1 = 6 × 10 −6 C are placed on the x-axis, as
shown in the figure 3.10.1. The distance between q and q1 is a = 0.600 m.
Figure 3.10.1
(a) What is the net force exerted on q by the other two charges q1?
(b) What is the electric field at the origin due to the two charges q1?
(c) What is the electric potential at the origin due to the two charges q1?
3.10.3 Work Done on Charges
Two charges q1 = 3.0 µ C and q2 = −4.0 µ C initially are separated by a distance
r0 = 2.0 cm . An external agent moves the charges until they are rf = 5.0 cm apart.
(a) How much work is done by the electric field in moving the charges from r0 to rf ? Is
the work positive or negative?
(b) How much work is done by the external agent in moving the charges from r0 to rf ?
Is the work positive or negative?
26
28. (c) What is the potential energy of the initial state where the charges are r0 = 2.0 cm
apart?
(d) What is the potential energy of the final state where the charges are rf = 5.0 cm apart?
(e) What is the change in potential energy from the initial state to the final state?
3.10.4 Calculating E from V
Suppose in some region of space the electric potential is given by
V ( x, y, z ) = V0 − E0 z +
E0 a 3 z
( x 2 + y 2 + z 2 )3/ 2
where a is a constant with dimensions of length. Find the x, y, and the z-components of
the associated electric field.
3.10.5 Electric Potential of a Rod
A rod of length L lies along the x-axis with its left end at the origin and has a nonuniform charge density λ = α x ,where α is a positive constant.
Figure 3.10.2
(a) What are the dimensions of α ?
(b) Calculate the electric potential at A.
(c) Calculate the electric potential at point B that lies along the perpendicular bisector of
the rod a distance b above the x-axis.
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29. 3.10.6 Electric Potential
Suppose that the electric potential in some region of space is given by
V ( x, y, z ) = V0 exp(− k | z |) cos kx .
Find the electric field everywhere. Sketch the electric field lines in the x − z plane.
3.10.7 Calculating Electric Field from the Electric Potential
Suppose that the electric potential varies along the x-axis as shown in Figure 3.10.3
below.
Figure 3.10.3
The potential does not vary in the y- or z -direction. Of the intervals shown (ignore the
behavior at the end points of the intervals), determine the intervals in which Ex has
(a) its greatest absolute value. [Ans: 25 V/m in interval ab.]
(b) its least. [Ans: (b) 0 V/m in interval cd.]
(c) Plot Ex as a function of x.
(d) What sort of charge distributions would produce these kinds of changes in the
potential? Where are they located? [Ans: sheets of charge extending in the yz direction
located at points b, c, d, etc. along the x-axis. Note again that a sheet of charge with
charge per unit area σ will always produce a jump in the normal component of the
electric field of magnitude σ / ε 0 ].
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30. 3.10.8 Electric Potential and Electric Potential Energy
A right isosceles triangle of side a has charges q, +2q and −q arranged on its vertices, as
shown in Figure 3.10.4.
Figure 3.10.4
(a) What is the electric potential at point P, midway between the line connecting the +q
and − q charges, assuming that V = 0 at infinity? [Ans: q/ 2 πεoa.]
(b) What is the potential energy U of this configuration of three charges? What is the
significance of the sign of your answer? [Ans: −q2/4 2 πεoa, the negative sign means
that work was done on the agent who assembled these charges in moving them in from
infinity.]
(c) A fourth charge with charge +3q is slowly moved in from infinity to point P. How
much work must be done in this process? What is the significance of the sign of your
answer? [Ans: +3q2/ 2 πεoa, the positive sign means that work was done by the agent
who moved this charge in from infinity.]
3.10.9. Electric Field, Potential and Energy
Three charges, +5Q, −5Q, and +3Q are located on the y-axis at y = +4a, y = 0, and
y = −4a , respectively. The point P is on the x-axis at x = 3a.
(a) How much energy did it take to assemble these charges?
(b) What are the x, y, and z components of the electric field E at P?
(c) What is the electric potential V at point P, taking V = 0 at infinity?
(d) A fourth charge of +Q is brought to P from infinity. What are the x, y, and z
components of the force F that is exerted on it by the other three charges?
(e) How much work was done (by the external agent) in moving the fourth charge +Q
from infinity to P?
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