Physics
Gauss Theorem
Gauss's theorem, also known as Gauss's law, states that the total electric flux through a closed surface is equal to the total charge enclosed by the surface divided by the permittivity of the medium. In simpler terms, it relates the electric field to the distribution of electric charge. This theorem is a fundamental concept in the study of electromagnetism.
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9 Key excerpts on "Gauss Theorem"
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
6 | GAUSS'S LAW Figure 6.1 This chapter introduces the concept of flux, which relates a physical quantity and the area through which it is flowing. Although we introduce this concept with the electric field, the concept may be used for many other quantities, such as fluid flow. (credit: modification of work by “Alessandro”/Flickr) Chapter Outline 6.1 Electric Flux 6.2 Explaining Gauss’s Law 6.3 Applying Gauss’s Law 6.4 Conductors in Electrostatic Equilibrium Introduction Flux is a general and broadly applicable concept in physics. However, in this chapter, we concentrate on the flux of the electric field. This allows us to introduce Gauss’s law, which is particularly useful for finding the electric fields of charge distributions exhibiting spatial symmetry. The main topics discussed here are 1. Electric flux. We define electric flux for both open and closed surfaces. 2. Gauss’s law. We derive Gauss’s law for an arbitrary charge distribution and examine the role of electric flux in Gauss’s law. 3. Calculating electric fields with Gauss’s law. The main focus of this chapter is to explain how to use Gauss’s law to find the electric fields of spatially symmetrical charge distributions. We discuss the importance of choosing a Gaussian surface and provide examples involving the applications of Gauss’s law. 4. Electric fields in conductors. Gauss’s law provides useful insight into the absence of electric fields in conducting materials. So far, we have found that the electrostatic field begins and ends at point charges and that the field of a point charge varies inversely with the square of the distance from that charge. These characteristics of the electrostatic field lead to an important mathematical relationship known as Gauss’s law. This law is named in honor of the extraordinary German mathematician and scientist Karl Friedrich Gauss (Figure 6.2).
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
LEARNING OBJECTIVES Gauss’ Law What Is Physics? In the preceding chapter we found the electric field at points near extended charged objects, such as rods. Our technique was labor-intensive: We split the charge distribution up into charge elements dq, found the field dE → due to an ele- ment, and resolved the vector into components. Then we determined whether the components from all the elements would end up canceling or adding. Finally we summed the adding components by integrating over all the elements, with several changes in notation along the way. One of the primary goals of physics is to find simple ways of solving such labor-intensive problems. One of the main tools in reaching this goal is the use of symmetry. In this chapter we discuss a beautiful relationship between charge and electric field that allows us, in certain symmetric situations, to find the electric field of an extended charged object with a few lines of algebra. The relationship is called Gauss’ law, which was developed by German mathematician and physicist Carl Friedrich Gauss (1777–1855). Let’s first take a quick look at some simple examples that give the spirit of Gauss’ law. Figure 23.1.1 shows a particle with charge +Q that is surrounded by an 660 CHAPTER 23 Gauss’ Law 23.1.5 Calculate the flux Φ through a surface by integrating the dot product of the electric field vector E → and the area vector dA → (for patch elements) over the surface, in magnitude-angle notation and unit- vector notation. 23.1.6 For a closed surface, explain the algebraic signs associated with inward flux and outward flux. 23.1.7 Calculate the net flux Φ through a closed surface, algebraic sign included, by integrating the dot product of the electric field vector E → and the area vector dA → (for patch elements) over the full surface. 23.1.8 Determine whether a closed surface can be broken up into parts (such as the sides of a cube) to simplify the integration that yields the net flux through the surface.
eBook - PDF- David Halliday, Robert Resnick, Kenneth S. Krane(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
It is fair to say that whereas Coulomb’s law provides the workhorse of electrostatics, Gauss’ law provides the insight. E B E B CHAPTER 27 CHAPTER 27 equations of electromagnetism (Maxwell’s equations, which we discuss in Chapter 38). Before we introduce Gauss’ law, we first need to define and discuss a new quantity, the flux of the electric field. The flux is a mathematical property of any field represented by vectors that is determined by the surface integral of the field vector over a particular area. There is also a geometri- cal interpretation of the flux that is based on the number of field lines that pass through the area. 27-2 THE FLUX OF A VECTOR FIELD The word “flux” comes from a Latin word meaning “to flow,” and you can consider the flux of a vector field to be a measure of the flow or penetration of the field vectors through an imaginary fixed element of surface in the field. Later we consider the flux of the electric field, but for now we consider a more familiar example, the velocity field of a flowing fluid. Imagine a stream of fluid in steady flow, in which we represent the flow by specifying the velocity vector at each point. Figure 27-1 shows a uniform flow; the velocity vec- tors are parallel throughout the fluid. Suppose we place into the stream a wire bent into a square loop of area A. In Fig. 27-1a, the loop is placed so that its plane is perpendicular to the direction of flow. We define the flux of the velocity field so that its magnitude is given by (27-1) where v is the magnitude of the velocity at the location of the loop. The flux has units of m 3 /s and might be consid- ered to represent the rate at which fluid passes through the loop; in terms of the field concept (and for the purpose of introducing Gauss’ law), however, it is convenient to con- sider the flux as a measure of the number of field lines pass- ing through the loop.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Keep in mind that we want to determine the net flux through a surface because that is what Gauss’ law relates to the charge enclosed by the surface. (The law is coming up in the next module.) Note that flux is a scalar (yes, we talk about field vectors but flux is the amount of piercing field, not a vector itself). 23.2 Gauss’ law LEARNING OBJECTIVES After reading this module, you should be able to: 23.2.1 apply Gauss’ law to relate the net fux Φ through a closed surface to the net enclosed charge q enc 23.2.2 identify how the algebraic sign of the net enclosed charge corresponds to the direction (inward or outward) of the net fux through a Gaussian surface 23.2.3 identify that charge outside a Gaussian surface makes no contribution to the net fux through the closed surface 23.2.4 derive the expression for the magnitude of the electric feld of a charged particle by using Gauss’ law 23.2.5 identify that for a charged particle or uniformly charged sphere, Gauss’ law is applied with a Gaussian surface that is a concentric sphere. KEY IDEAS • Gauss’ law relates the net fux Φ penetrating a closed surface to the net charge q enc enclosed by the surface: 0 Φ = q enc . • Gauss’ law can also be written in terms of the electric feld piercing the enclosing Gaussian surface: 0 ∮ E ⋅ d A = q enc . Gauss’ law relates the net flux Φ of an electric field through a closed surface (a Gaussian surface) to the net charge q enc that is enclosed by that surface. It tells us that 0 Φ = q enc . (23.7) By substituting the definition of flux, Φ = ∮ E ⋅ d A, we can also write Gauss’ law as 0 ∮ E ⋅ d A = q enc . (23.8) The two equations for Gauss’ law hold only when the net charge is located in a vacuum or (what is the same for most practical purposes) in air. In chapter 25, we modify Gauss’ law to include situations in which a material such as mica, oil, or glass is present.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
569 569 C H A P T E R 2 3 Gauss’ Law 23-1 ELECTRIC FLUX Learning Objectives After reading this module, you should be able to . . . 23.01 Identify that Gauss’ law relates the electric field at points on a closed surface (real or imaginary, said to be a Gaussian surface) to the net charge enclosed by that surface. 23.02 Identify that the amount of electric field piercing a surface (not skimming along the surface) is the electric flux Φ through the surface. 23.03 Identify that an area vector for a flat surface is a vector that is perpendicular to the surface and that has a magnitude equal to the area of the surface. 23.04 Identify that any surface can be divided into area elements (patch elements) that are each small enough and flat enough for an area vector dA → to be assigned to it, with the vector perpendicular to the element and having a magnitude equal to the area of the element. 23.05 Calculate the flux Φ through a surface by integrating the dot product of the electric field vector E → and the area vector dA → (for patch elements) over the surface, in magnitude-angle notation and unit-vector notation. 23.06 For a closed surface, explain the algebraic signs associated with inward flux and outward flux. 23.07 Calculate the net flux Φ through a closed surface, algebraic sign included, by integrating the dot product of the electric field vector E → and the area vector dA → (for patch elements) over the full surface. 23.08 Determine whether a closed surface can be broken up into parts (such as the sides of a cube) to simplify the integration that yields the net flux through the surface. ● The electric flux Φ through a surface is the amount of electric field that pierces the surface. ● The area vector dA → for an area element (patch element) on a surface is a vector that is perpendicular to the element and has a magnitude equal to the area dA of the element.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Key Ideas ● The electric flux Φ through a surface is the amount of electric field that pierces the surface. ● The area vector dA → for an area element (patch element) on a surface is a vector that is perpendicular to the element and has a magnitude equal to the area dA of the element. ● The electric flux dΦ through a patch element with area vector dA → is given by a dot product: dΦ = E → ⋅ dA → . ● The total flux through a surface is given by Φ = E → ⋅ dA → (total flux), where the integration is carried out over the surface. ● The net flux through a closed surface (which is used in Gauss’ law) is given by Φ = ∮ E → ⋅ dA → (net flux), where the integration is carried out over the entire surface. What Is Physics? In the preceding chapter we found the electric field at points near extended charged objects, such as rods. Our technique was labor-intensive: We split the charge distribution up into charge elements dq, found the field dE → due to an ele- ment, and resolved the vector into components. Then we determined whether the components from all the elements would end up canceling or adding. Finally we summed the adding components by integrating over all the elements, with several changes in notation along the way. C H A P T E R 2 3 697 23.1 ELECTRIC FLUX One of the primary goals of physics is to find simple ways of solving such labor-intensive problems. One of the main tools in reaching this goal is the use of symmetry. In this chapter we discuss a beautiful relationship between charge and electric field that allows us, in certain symmetric situations, to find the electric field of an extended charged object with a few lines of algebra. The relationship is called Gauss’ law, which was developed by German mathematician and physicist Carl Friedrich Gauss (1777–1855). Let’s first take a quick look at some simple examples that give the spirit of Gauss’ law. Figure 23.1.1 shows a particle with charge +Q that is surrounded by an imaginary concentric sphere.
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2023(Publication Date)
- Società Editrice Esculapio(Publisher)
Gauss’s Law 3.1 Introduction Coulomb’s law has been obtained in the case of static fields assuming a point-like charge approximation and it has allowed the evaluation of electric fields generated by static charge distributions, but it is not valid anymore for time dependent electric fields. There- fore, it must not be regarded as a fundamental law of electromagnetism. Hence, a new law, which includes Coulomb’s law as its limit for point-like charges, but which is valid in gen- eral, must be determined. The deduction of such a law requires the introduction of new mathematical concepts. 3.2 Flux The flux concept is imported from hydrodynamics: it corresponds to the amount of mat- ter, i.e. its volume at constant density, passing through a surface per unit time. Consider an infinitesimal transverse section dA of a stream-tube, crossed by a fluid moving with velocity v. All the fluid contained in a volume dV = dAd = dAvdt passes through that surface in time dt, thus the infinitesimal flux through surface dA is dΦ = dV dt = vdA . If the section is not transverse, but it is inclined by an angle θ with respect to the velocity direction, the useful surface, or effective sur- face, is still the transverse section which keeps reducing as the inclination increases until it becomes null at θ = π/2. Then the effective cross sectional area is dA eff = dA cos θ , so, in general, an infinitesimal flux through an infinitesimal surface dA is given by dΦ = v dA cos θ . Consider the unit vector u n normal to surface dA, fixing its direction with respect to the surface arbitrarily. The angle between u n and v is, by construction, θ (or π–θ if the oppo- site direction was chosen), so that, apart from the sign, vi u n = v cos θ ⇒ dΦ = vdA cos θ = vidA u n . The vector defined as dA = dA u n is called oriented surface and the operation per- formed in order to choose the direction of unit vector u n is called surface orientation.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 6 Gauss's Law for Magnetism In physics, Gauss's law for magnetism is one of Maxwell's equations, the four equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist. Rather than magnetic charges, the basic entity for magnetism is the magnetic dipole. (Of course, if monopoles were ever found, the law would have to be modified, as elaborated below.) Gauss's law for magnetism can be written in two forms, a differential form and an integral form . These forms are equivalent due to the divergence theorem. Note that the terminology Gauss's law for magnetism is not universally used. The law is also called Absence of free magnetic poles. (or some variant); one reference even explicitly says the law has no name. It is also referred to as the transversality requirement because for plane waves it requires that the polarization be transverse to the direction of propagation. Differential form The differential form for Gauss's law for magnetism is the following: where denotes divergence, B is the magnetic field. Integral form The integral form of Gauss's law for magnetism states: ________________________ WORLD TECHNOLOGIES ________________________ where S is any closed surface (a closed surface is the boundary of some three-dimensional volume; the surface of a sphere or cube is a closed surface, but a disk is not), d A is a vector, whose magnitude is the area of an infinitesimal piece of the surface S , and whose direction is the outward-pointing surface normal. The left-hand side of this equation is called the net flux of the magnetic field out of the surface, and Gauss's law for magnetism states that it is always zero.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 8 Gauss's Law for Magnetism In physics, Gauss's law for magnetism is one of Maxwell's equations, the four equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero, in other words, that it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist. Rather than magnetic charges, the basic entity for magnetism is the magnetic dipole. (Of course, if monopoles were ever found, the law would have to be modified, as elaborated below.) Gauss's law for magnetism can be written in two forms, a differentia l form and an integral form . These forms are equivalent due to the divergence theorem. Note that the terminology Gauss's law for magnetism is not universally used. The law is also called Absence of free magnetic poles. (or some variant); one reference even explicitly says the law has no name. It is also referred to as the transversality requirement because for plane waves it requires that the polarization be transverse to the direction of propagation. Differential form The differential form for Gauss's law for magnetism is the following: where denotes divergence, B is the magnetic field. Integral form The integral form of Gauss's law for magnetism states: ________________________ WORLD TECHNOLOGIES ________________________ where S is any closed surface (a closed surface is the boundary of some three-dimensional volume; the surface of a sphere or cube is a closed surface, but a disk is not), d A is a vector, whose magnitude is the area of an infinitesimal piece of the surface S , and whose direction is the outward-pointing surface normal. The left-hand side of this equation is called the net flux of the magnetic field out of the surface, and Gauss's law for magnetism states that it is always zero.
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