Gauss's Law. Gauss's Law in 3, 2, and 1 Dimension
|
|
- Deirdre George
- 7 years ago
- Views:
Transcription
1 [ Assignment View ] [ Eðlisfræði 2, vor Gauss' Law Assignment is due at 2:00am on Wednesday, January 31, 2007 Credit for problems submitted late will decrease to 0% after the deadline has passed. The wrong answer penalty is 2% per part. Multiple choice questions are penalized as described in the online help. The unopened hint bonus is 2% per part. You are allowed 4 attempts per answer. Gauss' Law Gauss's Law Learning Goal: To understand the meaning of the variables in Gauss's law, and the conditions under which the law is applicable. Gauss's law is usually written where is the permittivity of vacuum. How should the integral in Gauss's law be evaluated? Answer not displayed Gauss's Law in 3, 2, and 1 Dimension Gauss's law relates the electric flux through a closed surface to the total charge enclosed by the surface:. You can use Gauss's law to determine the charge enclosed inside a closed surface on which the electric field is known. However, Gauss's law is most frequently used to determine the electric field from a symmetric charge distribution. The simplest case in which Gauss's law can be used to determine the electric field is that in which the charge is localized at a point, a line, or a plane. When the charge is localized at a point, so that the electric field radiates in three-dimensional space, the Gaussian surface is a sphere, and computations can be done in spherical coordinates. Now consider extending all elements of the problem (charge, Gaussian surface, boundary conditions) infinitely along some direction, say along the z axis. In this case, the point has been extended to a line, namely, the z axis, and the resulting electric field has cylindrical symmetry. Consequently, the problem reduces to two dimensions, since the field varies only with x and y, or with and in cylindrical coordinates. A one-dimensional problem may be achieved by extending the problem uniformly in two directions. In this case, the point is extended to a plane, and consequently, it has planar symmetry. Three dimensions Consider a point charge in three-dimensional space. Symmetry requires the electric field to point directly away from the charge in all directions. To find, the magnitude of the field at distance from the charge, the logical 1 of 13 17/4/07 14:59
2 Gaussian surface is a sphere centered at the charge. The electric field is normal to this surface, so the dot product of the electric field and an infinitesimal surface element involves. The flux integral is therefore reduced to surface., where is the magnitude of the electric field on the Gaussian surface, and is the area of the Determine the magnitude by applying Gauss's law..1 Find the area of the surface Express in terms of some or all of the variables/constants,, and. Two dimensions Now consider the case that the charge has been extended along the z axis. This is generally called a line charge. The usual variable for a line charge density (charge per unit length) is, and it has units (in the SI system) of coulombs per meter. By symmetry, the electric field must point radially outward from the wire at each point; that is, the field lines lie in planes perpendicular to the wire. In solving for the magnitude of the radial electric field produced by a line charge with charge density, one should use a cylindrical Gaussian surface whose axis is the line charge. The length of the cylindrical surface should cancel out of the expression for. Apply Gauss's law to this situation to find an expression for..1 Find the surface area of a Gaussian cylinder.2 Find the enclosed charge Express in terms of some or all of the variables,, and any needed constants. 2 of 13 17/4/07 14:59
3 One dimension Now consider the case with one effective direction. In order to make a problem effectively one-dimensional, it is necessary to extend a charge to infinity along two orthogonal axes, conventionally taken to be x and y. When the charge is extended to infinity in the xy plane (so that by symmetry, the electric field will be directed in the z direction and depend only on z), the charge distribution is sometimes called a sheet charge. The symbol usually used for two-dimensional charge density is either, or. In this problem we will use. has units of coulombs per meter squared. In solving for the magnitude of the electric field produced by a sheet charge with charge density, use the planar symmetry since the charge distribution doesn't change if you slide it in any direction of xy plane parallel to the sheet. Therefore at each point, the electric field is perpendicular to the sheet and must have the same magnitude at any given distance on either side of the sheet. To take advantage of these symmetry properties, use a Gaussian surface in the shape of a cylinder with its axis perpendicular to the sheet of charge, with ends of area which will cancel out of the expression for in the end. The result of applying Gauss's law to this situation then gives an expression for for both and..1 Find the total electric flux out of the cylinder.2 Find the charge within the Gaussian surface Express for in terms of some or all of the variables/constants,, and. In this problem, the electric field from a distribution of charge in 3, 2, and 1 dimension has been found using Gauss's law. The most noteworthy feature of the three solutions is that in each case, there is a different relation of the field strength to the distance from the source of charge. In each case, the field strength varies inversely as an integral power of the distance from the charge. In the case of a point charge (spherical symmetry, field in three dimensions), the field strength varies as. In the case of a line charge (cylindrical symmetry, field in two dimensions), the field strength varies as in one dimension), the field varies as the sheet!. Finally, in the case of a sheet charge (planar symmetry, field ; that is, the strength of the field is independent of the distance from If you visualize the electric field using field lines, this result shows that as the number of directions in which the electric field can point is reduced, the field lines have one dimension fewer in which to to spread out, and the field therefore falls off less rapidly with distance. In a one-dimensional problem (sheet charge), the extension of the charge in the xy plane means that all field lines are parallel to the z axis, and so the field strength does not change with distance. Such a situation, of course, is impossible in the real world: In reality, 3 of 13 17/4/07 14:59
4 the planar charge is not infinite, so the field will in fact fall off over long distances. The Electric Field and Surface Charge at a Conductor Learning Goal: To understand the behavior of the electric field at the surface of a conductor, and its relationship to surface charge on the conductor. A conductor is placed in an external electrostatic field. The external field is uniform before the conductor is placed within it. The conductor is completely isolated from any source of current or charge. Which of the following describes the electric field inside this conductor? It is in the same direction as the original external field. It is in the opposite direction from that of the original external field. It has a direction determined entirely by the charge on its surface. It is always zero. The net electric field inside a conductor is always zero. If the net electric field were not zero, a current would flow inside the conductor. This would build up charge on the exterior of the conductor. This charge would oppose the field, ultimately (in a few nanoseconds for a metal) canceling the field to zero. The charge density inside the conductor is: 0 non-zero; but uniform non-zero; non-uniform infinite You already know that there is a zero net electric field inside a conductor; therefore, if you surround any internal point with a Gaussian surface, there will be no flux at any point on this surface, and hence the surface will enclose zero net charge. This surface can be imagined around any point inside the conductor with the same result, so the charge density must be zero everywhere inside the conductor. This argument breaks down at the surface of the conductor, because in that case, part of the Gaussian surface must lie outside the conducting object, where there is an electric field. Assume that at some point just outside the surface of the conductor, the electric field has magnitude and is directed toward the surface of the conductor. What is the charge density on the surface of the conductor at that point?.1 How to approach the problem Which of the following is the best way to solve this problem? Answer not displayed.2 Calculate the flux through the top of the cylinder.3 Calculate the flux through the bottom of the box.4 What is the charge inside the Gaussian surface? 4 of 13 17/4/07 14:59
5 Hint C.5 Apply Gauss's law Express your answer in terms of and. The Electric Field inside and outside a Charged Insulator A slab of insulating material of uniform thickness, lying between along the x axis, extends infinitely in the y and z directions, as shown in the figure. The slab has a uniform charge density. The electric field is zero in the middle of the slab, at. to Which of the following statements is true of the electric field at the surface of one side of the slab? Answer not displayed Part D Concept and Exercises on Electric Flux Calculating Electric Flux through a Disk Suppose a disk with area is placed in a uniform electric field of magnitude. The disk is oriented so that the vector normal to its surface,, makes an angle with the electric field, as shown in the figure. 5 of 13 17/4/07 14:59
6 Calculating Flux for Hemispheres of Different Radii Learning Goal: To understand the definition of electric flux, and how to calculate it. Flux is the amount of a vector field that "flows" through a surface. We now discuss the electric flux through a surface (a quantity needed in Gauss's law):, where is the flux through a surface with differential area element, and is the electric field in which the surface lies. There are several important points to consider in this expression: 1. It is an integral over a surface, involving the electric field at the surface. 2. is a vector with magnitude equal to the area of an infinitesmal surface element and pointing in a direction normal (and usually outward) to the infinitesmal surface element. 3. The scalar (dot) product implies that only the component of normal to the surface contributes to the integral. That is,, where is the angle between and. When you compute flux, try to pick a surface that is either parallel or perpendicular to, so that the dot product is easy to compute. Two hemispherical surfaces, 1 and 2, of respective radii and, are centered at a point charge and are facing each other so that their edges define an annular ring (surface 3), as shown. The field at position due to the point charge is: where is a constant proportional to the charge,, and is the unit vector in the radial direction. What is the electric flux through the annular ring, surface 3? Hint A.1 Apply the definition of electric flux Express your answer in terms of,,, and any constants. What is the electric flux through surface 1? Hint B.1 Apply the definition of electric flux.2 Find the area of surface 1 Express in terms of,,, and any needed constants. 6 of 13 17/4/07 14:59
7 What is the electric flux passing outward through surface 2? Hint C.1 Apply the definition of electric flux.2 Find the area of surface 2 Express in terms of,,, and any constants or other known quantities. Flux through a Cube A cube has one corner at the origin and the opposite corner at the point. The sides of the cube are parallel to the coordinate planes. The electric field in and around the cube is given by. Find the total electric flux through the surface of the cube. Hint A.1 Definition of flux.2 Flux through the face.3 Flux through the face.4 Flux through the face.5 Flux through the face Hint A.6 Putting it together Express your answer in terms of,,, and. 7 of 13 17/4/07 14:59
8 What is the net charge inside the cube? Hint C.1 Gauss's law Express your answer in terms of,,,, and. The Charge Inside a Conductor A spherical cavity is hollowed out of the interior of a neutral conducting sphere. At the center of the cavity is a point charge, of positive charge. What is the total surface charge on the interior surface of the conductor (i.e., on the wall of the cavity)? Hint A.1 Gauss's law and properties of conductors What is the total surface charge on the exterior surface of the conductor? Hint B.1 Properties of the conductor What is the magnitude charge? Let, as usual, denote. Hint C.1 How to approach the problem of the electric field inside the cavity as a function of the distance from the point The net electric field inside the conductor has three contributions: 1. from the charge ; 2. from the charge on the cavity's walls ; 3. from the charge on the outer surface of the spherical conductor. However, the net electric field inside the conductor must be zero. How must and be distributed for this to happen? Here's a clue: the first two contributions above cancel each other out, outside the cavity. Then the electric field produced by inside the spherical conductor must separately be zero also. How must be distributed for this to happen? 8 of 13 17/4/07 14:59
9 After you have figured out how and are distributed, it will be easy to find the field in the cavity, either by adding field contributions from all charges, or using Gauss's Law..2 Charge distributions and finding the electric field and are both uniformly distributed. Unfortunately there is no easy way to determine this, that is why a clue was given in the last hint. You might hit upon it by assuming the simplest possible distribution (i.e., uniform) or by trial and error, and check that it works (gives no net electric field inside the conductor). If is distributed uniformly over the surface of the conducting sphere, it will not produce a net electric field inside the sphere. What are the characteristics of the field produces inside the cavity? zero the same as the field produced by a point charge located at the center of the sphere the same as the field produced by a point charge located at the position of the charge in the cavity 0 Part D What is the electric field outside the conductor? Hint D.1 How to approach the problem The net electric field inside the conductor has three contributions: 1. from the charge ; 2. from the charge on the cavity's walls ; 3. from the charge on the outer surface of the spherical conductor. However, the net electric field inside the conductor must be zero. How must and be distributed for this to happen? Here's a helpful clue: the first two contributions above cancel each other out, outside the cavity. Then the electric field produced by inside the spherical conductor must be separately be zero also. How must be distributed for this to happen? What sort of field would such a distribution produce outside the conductor? Hint D.2 The distribution of If is distributed uniformly over the surface of the conducting sphere, it will not produce a net electric field inside the sphere. What are the characteristics of the field it produces outside the sphere? zero the same as the field produced by a point charge located at the center of the sphere the same as the field produced by a point charge located at the position of the charge in the cavity Now a second charge,, is brought near the outside of the conductor. Which of the following quantities would change? Part E The total surface charge on the wall of the cavity, : Hint E.1 Canceling the field due to the charge The net electric field inside a conductor is always zero. The charges on the inner conductor cavity will always arrange themselves so that the field lines due to charge do not penetrate into the conductor. 9 of 13 17/4/07 14:59
10 would change would not change Part F The total surface charge on the exterior of the conductor, : Hint F.1 Canceling the field due to the charge The net electric field inside a conductor is always zero. The charges on the outer surface of the conductor will rearrange themselves to shield the external field completely. Does this require the net charge on the outer surface to change? would change would not change Part G The electric field within the cavity, : would change would not change Part H The electric field outside the conductor, : would change would not change Finding E-Fields Using Gauss' law The Electric Field of a Ball of Uniform Charge Density A solid ball of radius has a uniform charge density. What is the magnitude of the electric field at a distance from the center of the ball? Hint A.1 Gauss's law.2 Find Express your answer in terms of,,, and. What is the magnitude of the electric field at a distance from the center of the ball?.1 How does this situation compare to that of the field outside the ball? Express your answer in terms of,,, and. 10 of 13 17/4/07 14:59
11 Let represent the electric field due to the charged ball throughout all of space. Which of the following statements about the electric field are true? The maximum electric field occurs when. 5. The maximum electric field occurs when. 6. The maximum electric field occurs as Hint C.1 Plot the electric field Enter t (true) or f (false) for each statement. Separate your answers with commas. Answer not displayed A Conducting Shell around a Conducting Rod An infinitely long conducting cylindrical rod with a positive charge per unit length is surrounded by a conducting cylindrical shell (which is also infinitely long) with a charge per unit length of and radius, as shown in the figure. What is, the radial component of the electric field between the rod and cylindrical shell as a function of the distance from the axis of the cylindrical rod? Hint A.1 The implications of symmetry Because the cylinder and rod are cylindrically symmetric, the magnitude of the electric field cannot vary as a function of angle around the rod, nor as a function of longitudinal position along the rod (typically represented by the spatial variables and ). By symmetry, the magnitude of the electric field can only depend on the distance from the axis of the rod (the spatial variable ). Hint A.2 Apply Gauss' law Gauss's law states that, where is the electric flux through a Gaussian surface, and is the total charge enclosed by the surface. Construct a cylindrical Gaussian surface with radius and length with. coaxial with the rod, 11 of 13 17/4/07 14:59
12 What is, the surface charge density (charge per unit area) on the inner surface of the conducting shell?.1 Apply Gauss's law The magnitude of the net force on charges within a conductor is always zero. This implies that the magnitude of the electric field within the conductor is zero. Think about a cylindrical Gaussian surface of length whose radius lies at the middle of the outer cylindrical shell. Since the electric field inside a conductor is zero and the Gaussian surface lies within the conductor, the electric flux across the Gaussian surface must be zero. What, then, must, the total charge inside this Gaussian surface, be? 0.2 Find the charge contribution from the surface What is, the total charge on the inner surface of the cylindrical shell that is contained within the Gaussian surface? Express your answer in terms of and. To obtain the charge density per unit area, divide that is contained within the Gaussian surface. by the area of the inner surface of the conducting shell What is, the surface charge density on the outside of the conducting shell? (Recall from the problem statement that the conducting shell has a total charge per unit length given by.).1 What is the charge on the cylindrical shell? What is, the total surface charge (the sum of charges on the inner and outer surfaces) of a portion of the shell of length? Since the charge on the inner surface of the cylinder is and the total charge on the cylinder is, it is now easy to obtain the charge on the outer surface of the cylinder. Then divide this result by the surface area of the portion of the cylinder that you took to obtain your result. Part D What is the radial component of the electric field,, outside the shell? Hint D.1 How to approach the problem Part D.2 Find the charge within the Gaussian surface Part D.3 Find the flux in terms of the electric field 12 of 13 17/4/07 14:59
13 A Charged Sphere with a Cavity An insulating sphere of radius, centered at the origin, has a uniform volume charge density. Find the electric field inside the sphere (for < ) in terms of the position vector. Hint A.1 How to approach the problem.2 Determine the enclosed charge.3 Calculate the integral over the Gaussian surface Express your answer in terms of,, and. Summary 4 of 11 problems complete (35.13% avg. score) of 20 points 13 of 13 17/4/07 14:59
Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings
1 of 11 9/7/2012 1:06 PM Logged in as Julie Alexander, Instructor Help Log Out Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings Course Home Assignments Roster Gradebook Item Library
More informationChapter 22: Electric Flux and Gauss s Law
22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we
More informationCHAPTER 24 GAUSS S LAW
CHAPTER 4 GAUSS S LAW 4. The net charge shown in Fig. 4-40 is Q. Identify each of the charges A, B, C shown. A B C FIGURE 4-40 4. From the direction of the lines of force (away from positive and toward
More informationChapter 18. Electric Forces and Electric Fields
My lecture slides may be found on my website at http://www.physics.ohio-state.edu/~humanic/ ------------------------------------------------------------------- Chapter 18 Electric Forces and Electric Fields
More informationExam 1 Practice Problems Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8 Spring 13 Exam 1 Practice Problems Solutions Part I: Short Questions and Concept Questions Problem 1: Spark Plug Pictured at right is a typical
More informationHW6 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case.
HW6 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 22.P.053 The figure below shows a portion of an infinitely
More informationChapter 4. Electrostatic Fields in Matter
Chapter 4. Electrostatic Fields in Matter 4.1. Polarization A neutral atom, placed in an external electric field, will experience no net force. However, even though the atom as a whole is neutral, the
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline
More informationChapter 22: The Electric Field. Read Chapter 22 Do Ch. 22 Questions 3, 5, 7, 9 Do Ch. 22 Problems 5, 19, 24
Chapter : The Electric Field Read Chapter Do Ch. Questions 3, 5, 7, 9 Do Ch. Problems 5, 19, 4 The Electric Field Replaces action-at-a-distance Instead of Q 1 exerting a force directly on Q at a distance,
More informationThe Electric Field. Electric Charge, Electric Field and a Goofy Analogy
. The Electric Field Concepts and Principles Electric Charge, Electric Field and a Goofy Analogy We all know that electrons and protons have electric charge. But what is electric charge and what does it
More informationPhysics 202, Lecture 3. The Electric Field
Physics 202, Lecture 3 Today s Topics Electric Field Quick Review Motion of Charged Particles in an Electric Field Gauss s Law (Ch. 24, Serway) Conductors in Electrostatic Equilibrium (Ch. 24) Homework
More informationForce on Moving Charges in a Magnetic Field
[ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after
More informationLecture 5. Electric Flux and Flux Density, Gauss Law in Integral Form
Lecture 5 Electric Flux and Flux ensity, Gauss Law in Integral Form ections: 3.1, 3., 3.3 Homework: ee homework file LECTURE 5 slide 1 Faraday s Experiment (1837), Flux charge transfer from inner to outer
More informationExam 2 Practice Problems Part 1 Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Exam Practice Problems Part 1 Solutions Problem 1 Electric Field and Charge Distributions from Electric Potential An electric potential V ( z
More informationElectromagnetism Laws and Equations
Electromagnetism Laws and Equations Andrew McHutchon Michaelmas 203 Contents Electrostatics. Electric E- and D-fields............................................. Electrostatic Force............................................2
More informationElectromagnetism - Lecture 2. Electric Fields
Electromagnetism - Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More informationCHAPTER 26 ELECTROSTATIC ENERGY AND CAPACITORS
CHAPTER 6 ELECTROSTATIC ENERGY AND CAPACITORS. Three point charges, each of +q, are moved from infinity to the vertices of an equilateral triangle of side l. How much work is required? The sentence preceding
More informationProblem 1 (25 points)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2012 Exam Three Solutions Problem 1 (25 points) Question 1 (5 points) Consider two circular rings of radius R, each perpendicular
More informationEdmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).
INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes
More informationChapter 7: Polarization
Chapter 7: Polarization Joaquín Bernal Méndez Group 4 1 Index Introduction Polarization Vector The Electric Displacement Vector Constitutive Laws: Linear Dielectrics Energy in Dielectric Systems Forces
More informationELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES
ELECTRIC FIELD LINES AND EQUIPOTENTIAL SURFACES The purpose of this lab session is to experimentally investigate the relation between electric field lines of force and equipotential surfaces in two dimensions.
More information1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 80 µ T at the loop center. What is the loop radius?
CHAPTER 3 SOURCES O THE MAGNETC ELD 1. A wire carries 15 A. You form the wire into a single-turn circular loop with magnetic field 8 µ T at the loop center. What is the loop radius? Equation 3-3, with
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. 8.02 Spring 2013 Conflict Exam Two Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 802 Spring 2013 Conflict Exam Two Solutions Problem 1 (25 points): answers without work shown will not be given any credit A uniformly charged
More informationShape Dictionary YR to Y6
Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use
More informationSolution. Problem. Solution. Problem. Solution
4. A 2-g ping-pong ball rubbed against a wool jacket acquires a net positive charge of 1 µc. Estimate the fraction of the ball s electrons that have been removed. If half the ball s mass is protons, their
More informationVector surface area Differentials in an OCS
Calculus and Coordinate systems EE 311 - Lecture 17 1. Calculus and coordinate systems 2. Cartesian system 3. Cylindrical system 4. Spherical system In electromagnetics, we will often need to perform integrals
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationEðlisfræði 2, vor 2007
[ Assignment View ] [ Print ] Eðlisfræði 2, vor 2007 30. Inductance Assignment is due at 2:00am on Wednesday, March 14, 2007 Credit for problems submitted late will decrease to 0% after the deadline has
More informationExercises on Voltage, Capacitance and Circuits. A d = (8.85 10 12 ) π(0.05)2 = 6.95 10 11 F
Exercises on Voltage, Capacitance and Circuits Exercise 1.1 Instead of buying a capacitor, you decide to make one. Your capacitor consists of two circular metal plates, each with a radius of 5 cm. The
More informationSURFACE TENSION. Definition
SURFACE TENSION Definition In the fall a fisherman s boat is often surrounded by fallen leaves that are lying on the water. The boat floats, because it is partially immersed in the water and the resulting
More informationElectric Fields in Dielectrics
Electric Fields in Dielectrics Any kind of matter is full of positive and negative electric charges. In a dielectric, these charges cannot move separately from each other through any macroscopic distance,
More informationA METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS
A METHOD OF CALIBRATING HELMHOLTZ COILS FOR THE MEASUREMENT OF PERMANENT MAGNETS Joseph J. Stupak Jr, Oersted Technology Tualatin, Oregon (reprinted from IMCSD 24th Annual Proceedings 1995) ABSTRACT The
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationReview B: Coordinate Systems
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B-2 B.1.1 Infinitesimal Line Element... B-4 B.1.2 Infinitesimal Area Element...
More informationGauss Formulation of the gravitational forces
Chapter 1 Gauss Formulation of the gravitational forces 1.1 ome theoretical background We have seen in class the Newton s formulation of the gravitational law. Often it is interesting to describe a conservative
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationDOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS RAYLEIGH-SOMMERFELD DIFFRACTION INTEGRAL OF THE FIRST KIND
DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS RAYLEIGH-SOMMERFELD DIFFRACTION INTEGRAL OF THE FIRST KIND THE THREE-DIMENSIONAL DISTRIBUTION OF THE RADIANT FLUX DENSITY AT THE FOCUS OF A CONVERGENCE BEAM
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More informationReview Questions PHYS 2426 Exam 2
Review Questions PHYS 2426 Exam 2 1. If 4.7 x 10 16 electrons pass a particular point in a wire every second, what is the current in the wire? A) 4.7 ma B) 7.5 A C) 2.9 A D) 7.5 ma E) 0.29 A Ans: D 2.
More informationExperiment 7: Forces and Torques on Magnetic Dipoles
MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics 8. Spring 5 OBJECTIVES Experiment 7: Forces and Torques on Magnetic Dipoles 1. To measure the magnetic fields due to a pair of current-carrying
More informationAs customary, choice (a) is the correct answer in all the following problems.
PHY2049 Summer 2012 Instructor: Francisco Rojas Exam 1 As customary, choice (a) is the correct answer in all the following problems. Problem 1 A uniformly charge (thin) non-conucting ro is locate on the
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationMagnetic Field of a Circular Coil Lab 12
HB 11-26-07 Magnetic Field of a Circular Coil Lab 12 1 Magnetic Field of a Circular Coil Lab 12 Equipment- coil apparatus, BK Precision 2120B oscilloscope, Fluke multimeter, Wavetek FG3C function generator,
More informationThe purposes of this experiment are to test Faraday's Law qualitatively and to test Lenz's Law.
260 17-1 I. THEORY EXPERIMENT 17 QUALITATIVE STUDY OF INDUCED EMF Along the extended central axis of a bar magnet, the magnetic field vector B r, on the side nearer the North pole, points away from this
More informationLecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More informationElectric Field Mapping Lab 3. Precautions
HB 09-25-07 Electric Field Mapping Lab 3 1 Electric Field Mapping Lab 3 Equipment mapping board, U-probe, resistive boards, templates, dc voltmeter (431B), 4 long leads, 16 V dc for wall strip Reading
More informationChapter 27 Magnetic Field and Magnetic Forces
Chapter 27 Magnetic Field and Magnetic Forces - Magnetism - Magnetic Field - Magnetic Field Lines and Magnetic Flux - Motion of Charged Particles in a Magnetic Field - Applications of Motion of Charged
More information6 J - vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J - vector electric current density (A/m2 ) M - vector magnetic current density (V/m 2 ) Some problems
More informationVOLUME AND SURFACE AREAS OF SOLIDS
VOLUME AND SURFACE AREAS OF SOLIDS Q.1. Find the total surface area and volume of a rectangular solid (cuboid) measuring 1 m by 50 cm by 0.5 m. 50 1 Ans. Length of cuboid l = 1 m, Breadth of cuboid, b
More informationChapter 19. Mensuration of Sphere
8 Chapter 19 19.1 Sphere: A sphere is a solid bounded by a closed surface every point of which is equidistant from a fixed point called the centre. Most familiar examples of a sphere are baseball, tennis
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of
More informationElectrostatic Fields: Coulomb s Law & the Electric Field Intensity
Electrostatic Fields: Coulomb s Law & the Electric Field Intensity EE 141 Lecture Notes Topic 1 Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University
More informationExperiments on the Basics of Electrostatics (Coulomb s law; Capacitor)
Experiments on the Basics of Electrostatics (Coulomb s law; Capacitor) ZDENĚK ŠABATKA Department of Physics Education, Faculty of Mathematics and Physics, Charles University in Prague The physics textbooks
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationThe potential (or voltage) will be introduced through the concept of a gradient. The gradient is another sort of 3-dimensional derivative involving
The potential (or voltage) will be introduced through the concept of a gradient. The gradient is another sort of 3-dimensional derivative involving the vector del except we don t take the dot product as
More informationStack Contents. Pressure Vessels: 1. A Vertical Cut Plane. Pressure Filled Cylinder
Pressure Vessels: 1 Stack Contents Longitudinal Stress in Cylinders Hoop Stress in Cylinders Hoop Stress in Spheres Vanishingly Small Element Radial Stress End Conditions 1 2 Pressure Filled Cylinder A
More informationMagnetism. d. gives the direction of the force on a charge moving in a magnetic field. b. results in negative charges moving. clockwise.
Magnetism 1. An electron which moves with a speed of 3.0 10 4 m/s parallel to a uniform magnetic field of 0.40 T experiences a force of what magnitude? (e = 1.6 10 19 C) a. 4.8 10 14 N c. 2.2 10 24 N b.
More informationBALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle
Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and
More informationChapter 6. Current and Resistance
6 6 6-0 Chapter 6 Current and Resistance 6.1 Electric Current... 6-2 6.1.1 Current Density... 6-2 6.2 Ohm s Law... 6-5 6.3 Summary... 6-8 6.4 Solved Problems... 6-9 6.4.1 Resistivity of a Cable... 6-9
More information6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationTeaching Electromagnetic Field Theory Using Differential Forms
IEEE TRANSACTIONS ON EDUCATION, VOL. 40, NO. 1, FEBRUARY 1997 53 Teaching Electromagnetic Field Theory Using Differential Forms Karl F. Warnick, Richard H. Selfridge, Member, IEEE, and David V. Arnold
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationForce on a square loop of current in a uniform B-field.
Force on a square loop of current in a uniform B-field. F top = 0 θ = 0; sinθ = 0; so F B = 0 F bottom = 0 F left = I a B (out of page) F right = I a B (into page) Assume loop is on a frictionless axis
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics
More informationMECHANICS OF SOLIDS - BEAMS TUTORIAL 1 STRESSES IN BEAMS DUE TO BENDING. On completion of this tutorial you should be able to do the following.
MECHANICS OF SOLIDS - BEAMS TUTOIAL 1 STESSES IN BEAMS DUE TO BENDING This is the first tutorial on bending of beams designed for anyone wishing to study it at a fairly advanced level. You should judge
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationarxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014
Theory of Electromagnetic Fields Andrzej Wolski University of Liverpool, and the Cockcroft Institute, UK arxiv:1111.4354v2 [physics.acc-ph] 27 Oct 2014 Abstract We discuss the theory of electromagnetic
More information1. Units of a magnetic field might be: A. C m/s B. C s/m C. C/kg D. kg/c s E. N/C m ans: D
Chapter 28: MAGNETIC FIELDS 1 Units of a magnetic field might be: A C m/s B C s/m C C/kg D kg/c s E N/C m 2 In the formula F = q v B: A F must be perpendicular to v but not necessarily to B B F must be
More informationPre-lab Quiz/PHYS 224 Magnetic Force and Current Balance. Your name Lab section
Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance Your name Lab section 1. What do you investigate in this lab? 2. Two straight wires are in parallel and carry electric currents in opposite directions
More informationIntroduction to CATIA V5
Introduction to CATIA V5 Release 16 (A Hands-On Tutorial Approach) Kirstie Plantenberg University of Detroit Mercy SDC PUBLICATIONS Schroff Development Corporation www.schroff.com www.schroff-europe.com
More informationChapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.
Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems
More informationMagnetic fields of charged particles in motion
C H A P T E R 8 Magnetic fields of charged particles in motion CONCEPTS 8.1 Source of the magnetic field 8. Current loops and spin magnetism 8.3 Magnetic moment and torque 8.4 Ampèrian paths QUANTTATVE
More informationThe small increase in x is. and the corresponding increase in y is. Therefore
Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that
More informationUnderstanding astigmatism Spring 2003
MAS450/854 Understanding astigmatism Spring 2003 March 9th 2003 Introduction Spherical lens with no astigmatism Crossed cylindrical lenses with astigmatism Horizontal focus Vertical focus Plane of sharpest
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationChapter 6 Circular Motion
Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example
More informationDiscovering Math: Exploring Geometry Teacher s Guide
Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 4 AREAS AND VOLUMES This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More informationSection V.3: Dot Product
Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,
More informationA Survival Guide to Vector Calculus
A Survival Guide to Vector Calculus Aylmer Johnson When I first tried to learn about Vector Calculus, I found it a nightmare. Eventually things became clearer and I discovered that, once I had really understood
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationVersion 001 Electrostatics I tubman (12125) 1
Version 001 Electrostatics I tubman (115) 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. AP EM 1993 MC 55
More informationGeometric Optics Converging Lenses and Mirrors Physics Lab IV
Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The
More informationConceptual: 1, 3, 5, 6, 8, 16, 18, 19. Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65. Conceptual Questions
Conceptual: 1, 3, 5, 6, 8, 16, 18, 19 Problems: 4, 6, 8, 11, 16, 20, 23, 27, 34, 41, 45, 56, 60, 65 Conceptual Questions 1. The magnetic field cannot be described as the magnetic force per unit charge
More informationActivity Set 4. Trainer Guide
Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES
More informationpotential in the centre of the sphere with respect to infinity.
Umeå Universitet, Fysik 1 Vitaly Bychkov Prov i fysik, Electricity and Waves, 2006-09-27, kl 16.00-22.00 Hjälpmedel: Students can use any book. Define the notations you are using properly. Present your
More informationAngles that are between parallel lines, but on opposite sides of a transversal.
GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,
More informationPhysics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam
Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry
More informationCalculation of gravitational forces of a sphere and a plane
Sphere and plane 1 Calculation of gravitational forces of a sphere and a plane A paper by: Dipl. Ing. Matthias Krause, (CID) Cosmological Independent Department, Germany, 2007 Objective The purpose of
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationF B = ilbsin(f), L x B because we take current i to be a positive quantity. The force FB. L and. B as shown in the Figure below.
PHYSICS 176 UNIVERSITY PHYSICS LAB II Experiment 9 Magnetic Force on a Current Carrying Wire Equipment: Supplies: Unit. Electronic balance, Power supply, Ammeter, Lab stand Current Loop PC Boards, Magnet
More informationLesson 26: Reflection & Mirror Diagrams
Lesson 26: Reflection & Mirror Diagrams The Law of Reflection There is nothing really mysterious about reflection, but some people try to make it more difficult than it really is. All EMR will reflect
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationD Alembert s principle and applications
Chapter 1 D Alembert s principle and applications 1.1 D Alembert s principle The principle of virtual work states that the sum of the incremental virtual works done by all external forces F i acting in
More informationProblem set on Cross Product
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j - 3 k ) 2 Calculate the vector product of i - j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
More informationChapter 16. Mensuration of Cylinder
335 Chapter 16 16.1 Cylinder: A solid surface generated by a line moving parallel to a fixed line, while its end describes a closed figure in a plane is called a cylinder. A cylinder is the limiting case
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More information