What is Gauss's law: Definition and 380 Discussions

In physics and electromagnetism , Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating to the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.
The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1813, both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which forms the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law, and vice versa.

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  1. N

    Differential Gauss's Law & Coaxial Cable

    Homework Statement A long coaxial cable consists of an inner solid cylinder, radius a, and an outer thin coaxial cylindrical shell, radius b. The outer shell carries a uniform surface charge density σ. Find the uniform volume charge density ρ that the inner cylinder must have in order that...
  2. O

    Gauss's law for a thick plane

    Alright so I already checked the answer to this problem in my solution manual, but I can't figure out why our results are differing say we have an infinite plane slab of thickness 2d, with uniform charge density ρ. Find the function of the electric field with regards to y where y=0 is at the...
  3. S

    How correct is deriving Coulomb's Law from Gauss's Law

    How correct is "deriving" Coulomb's Law from Gauss's Law Homework Statement Here is a question that appeared in my school question paper: "Derive Coulomb's Law from Gauss's Law." 2. The attempt at a solution I tried the following: Consider a Gaussian Sphere or radius $r$ By Gass's...
  4. K

    Gauss's Law - point charge and charged sphere.

    Homework Statement A point charge q1 = -9.7 μC is located at the center of a thick conducting shell of inner radius a = 2.3 cm and outer radius b = 4.5 cm, The conducting shell has a net charge of q2 = 1.4 μC. 1) What is Ex(P), the value of the x-component of the electric field at...
  5. B

    Gauss's law in electrodynamics

    most(or all) proofs i have seen of gauss's law is based on coulumb's law. however coulumb's law is based off of electrostatics which certainly does not hold in electrodynamics. however gauss's law is used extensively in electrodynamics. is gauss's law derived otherwise or is it just a law like...
  6. B

    Electric Field Near a Plane of Charge: Why is the Total Field Not Zero?

    Homework Statement Find the electric field near a large nonconducting plane of charge density sigma. Homework Equations The Attempt at a Solution I am working through chapter 24 of Giancoli and am having trouble understanding the concepts. This is example 24-4, and the magnitude...
  7. K

    Can Gauss's Law Explain the Field of an Electron in a Closed Finite Universe?

    How does Gauss's divergence law work in a closed finite universe? Let's say the universe were a 4-sphere, with a single electron. How can I work out the field of the electron? If I draw a 3-sphere around the electron, then I split space into two regions. One region contains an electron, so...
  8. S

    Which of the following statements about Gauss's law are correct?

    Homework Statement There may be more than one correct choice a. Only charge enclosed within a Gaussian surface can produce an electric field at points on that surface. b. If a Gaussian surface is completely inside an electrostatic conductor, the electric field must always be zero at all points...
  9. B

    Gauss's law and nonuniform electric field

    Homework Statement A Gaussian surface is in the shape of a cube with edge length 1.40m. The electric field is E=[-4i+(6+3y)j]N/C. I got an answer, but the solution manual stated that we treat the electric field as E=3yj+E0, where E0=-4i+6j, which does not contribute to the flux. Why is...
  10. O

    Gauss's law and Moving Charges

    I am confused as to how Gauss's law can hold for moving charges. When discussing how to use Gauss's law to calculate the charge of a moving object, my textbook says: "At the instant the moving charge passes the center of the sphere, the force on each test charge is measured, and the average...
  11. V

    Who derived Gauss's law for gravity

    As the title states, my question is fairly simple. It is fairly well-known that one can derive a law similar to Gauss's law (for electric fields) for gravity, which is essentially equivalent to Newton's law of universal gravitation. But what I was wondering is who actually came up with it? I...
  12. L

    Gauss's law intuitive explanation?

    I'm not sure what Gauss's law really means. "The electric flux through any closed surface is proportional to the enclosed electric charge." How does this apply to finding the electric field? apcentral.collegeboard. com/apc/public/repository/ap11_frq_physics_cem.pdf Look at parts 1 a and b. Part...
  13. S

    Point-Source derivative form Gauss's law valid?

    I reviewed Maxwell's equations today and I was upset to find that the volume integral of the derivative form of Gauss's equation seems to be 0 for point sources when it should come out to be Q/(electro constant). This comes from the fact that the gradient is equal to 0 and anything else you do...
  14. O

    Gauss's Law for Magnetism Question

    Hey everyone, I'm new to these forums. Being an electrical engineering major, most of my teachers aren't very concerned with the "physics" side of things. I'm hoping I can gain some insight on Maxwell's equations. When first stating Gauss's Law for Magnetism, the only reason my electromagnetics...
  15. S

    Calculating Charge Density and Total Charge Using Gauss's Law

    Homework Statement Suppose the electric field in some region is found to be \vec{E} = kr^{3} \hat{r}, in spherical coordinates (k is some constant). (a) Find the charge density ρ. (b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different...
  16. H

    Is Gauss's Law Applicable in Atomic Scales?

    Hi all, Can we use Gauss's law to find the electric field near an electron, or is Gauss's law invalid in atomic scales? I ask this basic question because I know nothing about Quantum physics.
  17. B

    Gauss's law and surface electrons on a charged conductor, contradiction?

    My textbook (Halliday & Walker) explains that a charged conductor (a solid, of an arbitrary shape) in electrostatic equilibrium will have the electric field inside be 0 and all electrons will be on its surface. It proves this by saying that if the electric field inside was not 0, the free...
  18. F

    Gauss's Law - Insulating Material

    Gauss's Law -- Insulating Material Homework Statement A slab of insulating material of uniform thickness d, lying between -{d}/{2} to +{d}/{2} along the x axis, extends infinitely in the y and z directions. The slab has a uniform charge density rho. The electric field is zero in the middle of...
  19. G

    Solve Gauss's Law: Cube w/E Field & Calculate Flux

    Homework Statement Consider a cube with one corner at the origin and with sides of length 10 cm positioned along the xyz axes. There is an electric field E= (18, 235y, 0) through the region that has a constant x component and a y component that increases linearly with y. Start by...
  20. A

    Gauss's Law and Magnetic Reconnection

    Hi, I have a question regarding Gauss's law and magnetic reconnection. From what I have heard Gauss's law for magnetism (Div B = 0) clearly states that magnetic reconnection cannot occur. Furthermore, according to the THEMIS project, magnetic reconnection triggers an acceleration that creates...
  21. B

    Equivalence of Integral and Differential Forms of Gauss's Law?

    A sphere has charge density \rho=k\cdot r. Using the integral form of Gauss's Law, one easily finds that the electric field is E=\frac{k\cdot r^2}{4\epsilon} anywhere inside the sphere. However, \nabla\cdot E=\frac{k\cdot r}{2\epsilon}, which is half of what should be expected from the...
  22. W

    When should I use Gauss's Law for calculating electric fields?

    I'm trying to get a better understanding of when to use Gauss's Law and I would appreciate any help. I know so far that it can be easily used in cases of high symmetry and infinitely long charged objects. Does it matter if the surface is conducting or insulating? If I have, for example, a...
  23. D

    Understanding Gauss's Law: When to Apply and When it Fails

    This is not a homework question. I'm just having trouble understanding when it's appropriate to apply Gauss's law or not, and under what conditions does it fail. For example, suppose I have two point charges forming a dipole. If I draw a cylinder around them, then the total charge enclosed...
  24. A

    Gauss's Law in Matter: What is It?

    Hi, The Poisson equation (or Gauss Law) in a vacuum is given by \nabla^2\phi=-\frac{\rho}{\epsilon_0} where \rho \mbox{ and } \epsilon_0 are the charge density and vacuum permittivity or (electric constant of space). My question is what is the Gauss's Law in a dielectric material? Do...
  25. R

    Gauss's Law - Irregular Surfaces

    Gauss's Law -- Irregular Surfaces I don't fully understand why Gauss's Law holds for any Gaussian surface. My textbook clearly derives Gauss's Law from Coulomb's Law using a spherical surface, but it then extends the result to any Gaussian surface without sufficient explanation. Why does...
  26. E

    Gauss's Law and Charges on Conductors

    A hollow conducting sphere has no net charge. There is a positive point charge at the center of the spherical cavity within the sphere. You connect a conducting wire from the outside of the sphere to ground. Will you measure an electric field outside the sphere? I know that since the electric...
  27. F

    Concentric Metal Sphere - Gauss's Law

    1. Please see the attached problem, essentially a metal ball with charge Q1 is surrounded by a grounded shell q which in turn is surrounded by a further shell with charge Q2 Q1. The first question is what is the charge induced on q by Q1 and Q2 Q2. The second question is how is the charge...
  28. G

    Electric Field for a cylinder without using Gauss's law?

    I recently learned how to calculate the field from a cylinder (inside and outside the cylinder) using Gauss's law. I was wondering how I would be able to derive the same formula without using Gauss's law (just for practice). My idea is that you would need to integrate the electric field from...
  29. W

    How Does Gauss's Law Explain Zero Charge in Hollow Surfaces?

    Hi. My question is straightforward: how do I know when the charge inside a surface is zero? I've read my textbook chapter twice, but it doesn't offer any explanation on this. In one example problem with a hollow shell, it simply makes the automatic assumption that "there's no charge inside...
  30. T

    Gauss's Law: Charge Distribution in Cylinder Volume

    Homework Statement Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius R. a)Show that at a distance r from the cylinder (r<R): E= (ρ*r)/2ε° where ρ is the volume charge density and (b) write and for E when r>R Homework Equations The...
  31. E

    Application of Gauss's Law to Charged Insulators

    Homework Statement A cylindrical shell of radius 7.00 cm and length 240 cm has its charge uniformly distributed on its curved surface. The magnitude of the electric field at a point 19.0 cm ra- dially outward from its axis (measured from the midpoint of the shell) is 36.0 kN/C. Use approximate...
  32. V

    Gauss's Law and charged conductors

    A conductor carries a net charge +Q. There is a hollow cavity inside the conductor that contains a point charge -Q. In electrostatic equilibrium,what is the charge on the outer surface of the conductor? A. -2Q B. -Q C. 0 D. Q E. 2Q I know that the answer is 0, however, I cannot...
  33. W

    Deriving Electric Field Inside and Outside an Infinitely Long Charged Cylinder

    Not sure if this is advanced. Highly doubt it but oh well 1. Homework Statement Consider an infinitely long charged cylinder of radius R, carrying a charge whose density varies with radius as ρ(r) = ρo r. Derive expressions for the electric field (a) inside the cylinder (i.e. r<R), and (b)...
  34. W

    Electromagnetism: Gauss's Law

    Homework Statement Consider an infinitely long charged cylinder of radius R, carrying a charge whose density varies with radius as ρ(r) = ρ_{o} r. Derive expressions for the electric field (a) inside the cylinder (i.e. r<R), and (b) outside the cylinder (i.e. r>R). Homework Equations...
  35. H

    Electromagnetism problem using Gauss's Law

    Homework Statement Using Gauss's law calculate the electric field everywhere for the infinitely long insulating cylinder of radius R and charge density of rho = 3r^2(nc/m^3). SHOW ALL YOUR WORK INCLUDING DIAGRAMS. Homework Equations I am lost. The Attempt at a Solution I don't...
  36. jegues

    Gauss's Law, Sphere (Electric Field

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution I'm extremely confused. For r<1, The total charge enclosed should always be, \rho \frac{4}{3\epsilon_{0} \epsilon_{r}} \pi r^{3} = \oint _{S} \vec{E} \cdot \vec{dA} = I The electric...
  37. perplexabot

    Electricfield of a Disk using Gauss's law

    Electricfield of a Disk using Gauss's law! Hi all. Trying to find electric field of a disk with a charge density of D and a radius R (let us assume it has a thickness h). I know you can do so by breaking up the disk into concentric rings and integrating coulombs law. However, i would like to go...
  38. L

    Gauss's Law on a Charged Sphere

    Hi I'm sorry if this is posted in the wrong section or it's laid out wrong but I have a question that I need a bit of help with. Homework Statement I'm given: A spherically symmetrical charge distribution is contained within a sphere of radius a with no charge outside. At a distance r (r...
  39. F

    Gauss's Law with non-uniform E-field

    Homework Statement A long insulating cylinder has radius R, length l, and a non-uniform charge density per volume \rho = e^{ar} where r is the distance from the axis of the cylinder. Find the electric field from the center of the axis for i) r < R ii) r > R The Attempt at a Solution...
  40. S

    Gauss's law derivation using solid angle

    So, I was trying to find a rigorous mathematical derivation of gauss's law(please I don't want to hear again any field lines nonsense) and I stumbled upon jackson's proof which uses the solid angle concept and seems a solid enough proof(stupid joke:smile:).The problem is that it's the first time...
  41. D

    Using divergence theorem to prove Gauss's law

    i need to prove that div(R/r^3) = 4πδ where R is a vector and r is the magnitude of the vector R. also δ is the dirac delta function. so div(R/r^3) is 0 everywhere except for the origin. i need to show that the volume integral of div(R/r^3) = 4π as well. using the divergence theorem we...
  42. P

    Virtual particles and Gauss's Law

    Does Gauss's Law apply to virtual particles? For example, when computing the field around a real proton, is the net charge in Gauss's Law the proton charge plus the contribution of all the virtual charges within the closed surface? (I'm thinking about the screening of charges by virtual...
  43. F

    Rings of Electric Field and Gauss's Law

    Homework Statement Trying to find the E-field inside a conductor using rings even though my book tells me it is 0. I haven't learned how to do surface integrals yet but I think I only need Calc II to do this. The Attempt at a Solution...
  44. A

    Gauss's Law and electric field magnitude

    Homework Statement You have a solid, non-conducting cylinder of radius a, length L, and a total charge of Q. Concentric with this is an uncharged conducting cylindrical shell of inner radius b and outer radius c. Find the magnitude of the electric field for values of r where r < a...
  45. A

    Theory of Images, or Gauss's Law?

    Homework Statement Two conducting spheres are concentric, the radius of the inner being a and the outer being b. One sphere is grounded & the other is at a potential V0. Find the charge on each when the grounded sphere is a) the outer, b) the inner. 2. The attempt at a solution Is this...
  46. F

    Why is the E-field inside a conductor zero?

    Homework Statement Oh dear, veterens of PF who have tried to convince me that E = 0 inside conductors, I am having this problem again. OKay, let's say that "charges do arrange themselves such that the E-field inside a conductor is always zero", then how can wires in circuits work? If the...
  47. Z

    Gauss's Law and Electric Dipole

    Can we use Gauss's Law to calculate the field distribution around an electric dipole?
  48. B

    Conducting spheres Gauss's law

    Homework Statement There are two conducting spheres, one hollow outer sphere which originally had +60nC of charge on it, and a smaller inner sphere enclosed by the hollow outer sphere which contains +20nC of charge. Homework Equations INT(E.DA) = Qenclosed/epsilon0 The Attempt at a...
  49. F

    What Is the Flux Through a Disk-Shaped Area in a Solenoid?

    Homework Statement Calculate the flux through the surface of a disk-shaped area of radius R = 5.00 cm that is positioned perpendicular to and centered on the axis of the solenoid as shown below The Attempt at a Solution \oint \vec{B} \cdot \vec{dA} = \Phi_{B} \vec{B} =...
  50. S

    Gauss's Law Concept Question

    Homework Statement A spherical cavity of radius 4.50 cm is at the center of a metal sphere of radius 18.0 cm. A point charge Q = 6.40 µC rests at the very center of the cavity, whereas the metal conductor carries no net charge. Determine the electric field at the following points. (a) 2.0...
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